Solution and Characteristic Analysis of Fractional-Order Chaotic Systems 9789811932724, 9789811932731

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Solution and Characteristic Analysis of Fractional-Order Chaotic Systems

Kehui Sun · Shaobo He · Huihai Wang

Solution and Characteristic Analysis of Fractional-Order Chaotic Systems

Kehui Sun School of Physics and Electronics Central South University Changsha, Hunan, China

Shaobo He School of Physics and Electronics Central South University Changsha, Hunan, China

Huihai Wang School of Physics and Electronics Central South University Changsha, Hunan, China

ISBN 978-981-19-3272-4 ISBN 978-981-19-3273-1 (eBook) https://doi.org/10.1007/978-981-19-3273-1 Jointly published with Science Press The print edition is not for sale in China (Mainland). Customers from China (Mainland) please order the print book from: Science Press. Translation from the Chinese Simplified language edition: “Solution and Characteristic Analysis of Fractional-Order Chaotic Systems” by Kehui Sun et al., © Science Press 2020. Published by Science Press. All Rights Reserved. © Science Press 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

The study of fractional calculus theory has almost a long history as the study of integral calculus theory, which has existed for more than 300 years. However, the development of fractional calculus theory is slow due to the lack of practical application background for a long time. Compared with an integer-order differential equation, a fractional-order differential equation can describe natural phenomena more accurately in many applied scientific fields. Since Mandelbrot pointed out the existence of a large number of fractal dimensions in nature and many scientific and technological fields for the first time in 1983, fractional calculus has gained new development and has become one hot spot of nonlinear disciplines. At present, a fractional-order nonlinear chaotic system is an important aspect of fractional calculus research. In order to realize the application of fractional-order chaotic system in information security, we not only need to reveal the chaotic characteristics of the fractional nonlinear system, but also need to solve the key technical problems in the actual application of the fractional-order chaotic system, such as the precise numerical solution problem of fractional differential equation, the analysis of the fractional-order chaotic system and digital circuit implementation issues, etc. Therefore, it is necessary to study the chaotic dynamics theory, characteristic analysis, and experimental techniques of fractional-order chaotic systems. This book will focus on the numerical solution algorithm, characteristic analysis method, and circuit realization technology of fractional-order chaotic system, which lays a theoretical and experimental foundation for the application of fractional-order chaotic system in information security. The book consists of ten chapters, including the solution, characteristic analysis, and circuit realization of fractional-order chaotic system. In Chap. 1, the research progress of fractional-order chaotic systems is reviewed, and the analysis methods and application fields of fractional-order chaotic systems are introduced. The Chap. 2 describes the time-frequency domain algorithm of fractional-order chaotic system. In Chap. 3, the predictor–corrector algorithm of fractional-order chaotic systems is presented. In Chap. 4, the Adomian decomposition algorithm for solving fractionalorder chaotic systems is discussed. In Chap. 5, the performance of three different

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algorithms is compared. In Chap. 6, the characteristic analysis algorithm of fractionalorder chaotic system is discussed. The complexities of fractional-order chaotic systems are analyzed in Chap. 7. In Chap. 8, the circuit realization of fractional-order chaotic system is studied, including analog circuit and digital circuit. In Chap. 9, the application of fractional-order chaotic system in secure communication is studied. In Chap. 10, the solution and characteristic analysis of fractional-order discrete chaotic map are studied. In order to facilitate readers to grasp the research methods of fractional-order chaotic systems quickly, the authors have sorted out the relevant analysis programs accumulated by the research group as Appendix A of this book. This book is compiled by summarizing our recent research and teaching work on fractional-order chaos, and referring to a large number of domestic and foreign relevant literature. We would like to thank the National Natural Science Foundation of China (62071496, 61901530, 62061008) for supporting this research. Thanks to Prof. J. C. Sprott from the University of Wisconsin, Prof. Chunhua Wang from Hunan University, Prof. Simin Yu from Guangdong University of Technology, and Prof. Guangyi Wang from Hangzhou Electronic Science and Technology University for their guidance and helpful suggestions on the contents of this book. Thanks to the previous Master’s students Ren Jian, Wang Xia, Yang Jingli, Liu Xuan, Wang Yan, Ruan Jingya, Zhang Limin, Peng Dong, Wang Lingyu and Ph.D. student Peng Yuexi for their research and collation work. Due to the author’s limited knowledge, there are inadequacies in the book. We look forward to your comments. Changsha, China October 2021

Kehui Sun Shaobo He Huihai Wang

Introduction

Since Mandelbrot pointed out the existence of a large number of fractal dimensions in nature, many scientific and technological fields for the first time in 1983, the fractional calculus has gained new development and become a research hot spot of nonlinear disciplines. Fractional-order nonlinear chaotic system is an important aspect of fractional-order calculus. This book focuses on the numerical solution algorithm, characteristic analysis method, and circuit realization technology of fractionalorder chaotic system, which lay a theoretical and experimental foundation for the application of fractional-order chaotic system. The book consists of ten chapters, which are divided into three parts: numerical solution algorithm, dynamical characteristics analysis, and circuit implementation. In order to facilitate readers to quickly grasp the research method of fractional-order chaotic system, the author sorted out the relevant analysis procedures accumulated by the research group as Appendix A of this book. This book can be used as a textbook or reference book for undergraduates and graduate students, as well as for teachers and students of science and engineering universities, and for researchers in the field of natural science and engineering technology.

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Contents

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2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Overview of Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 The Origin and Development of Chaos . . . . . . . . . . . . . . . 1.1.2 Definition of Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Characteristics of Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Overview of Fractional-Order Chaotic Systems . . . . . . . . . . . . . . . 1.2.1 Definitions and Properties of Fractional Calculus . . . . . . 1.2.2 Overview of Solving Algorithms for Fractional-Order Chaotic Systems . . . . . . . . . . . . . . . . . . . 1.3 Review on the Analysis Methods of Fractional-Order Chaotic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Study on the Dynamical Behavior of Fractional-Order Chaotic Systems . . . . . . . . . . . . . . . . . . . 1.3.2 Complexity of Fractional-Order Chaotic System and Its Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Complexity Analysis of Fractional-Order Chaotic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Overview of the Application of Fractional Chaotic Systems . . . . 1.4.1 Circuit Implementation of Fractional-Order Chaotic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Synchronization Control of Fractional-Order Chaotic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Application of Fractional-Order Chaotic Systems in Secure Communication . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency-Domain Approximation Method . . . . . . . . . . . . . . . . . . . . . 2.1 Algorithm Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Dynamic Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Algorithm Instance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 3 4 5 6 11 13 13 14 17 18 19 21 23 24 27 27 30 31 34

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Predictor–Corrector Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Algorithm Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Algorithm Example: Dynamical Analysis of the Fractional-Order Simplified Lorenz Hyperchaotic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Dynamics with the Variation of the System Order . . . . . 3.3.2 Dynamics with the Variation of the Frequency ω . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 37 38

Adomian Decomposition Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Algorithm Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Algorithm Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Fractional-Order Lorenz Chaotic System . . . . . . . . . . . . . 4.2.2 Fractional-Order Simplified Lorenz Chaos System . . . . . 4.3 Adomian Decomposition Method for Fractional-Order Time-Delay Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Performance Comparison of Solution Algorithms . . . . . . . . . . . . . . . . 5.1 Uniqueness of Solutions of Fractional-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Accuracy and Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Influence of Different Factors on the Minimum Order of Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Minimum Order of Chaos with Different Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Simulation Step Size on the Minimum Order of Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Lowest Order of Chaos with Different Derivative Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 System Parameters on the Lowest Order of Chaos . . . . . 5.3.5 The Optimal Minimum Order of Chaos . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Dynamics of Fractional-Order Chaotic Systems . . . . . . . . . . . . . . . . . . 6.1 Lyapunov Exponent Spectrum Calculation Algorithm . . . . . . . . . 6.2 Dynamical Characterization of a Family of Fractional-Order Lorenz Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Fractional-Order Lorenz Chaotic System . . . . . . . . . . . . . 6.2.2 Fractional-Order Simplified Lorenz Chaotic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Fractional-Order Lorenz Hyperchaotic System . . . . . . . . 6.2.4 Fractional-Order Rössler Chaotic System . . . . . . . . . . . . . 6.2.5 Fractional-Order Lorenz–Stenflo Chaotic System . . . . . .

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Fractional-Order Simplified Lorenz Hyperchaotic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 7

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Complexity Analysis of Fractional-Order Chaotic System . . . . . . . . . 7.1 Behavioral Complexity Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Research on Multivariate Permutation Entropy Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Complexity Analysis of Fractional-Order Chaotic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Research on Improved Multi-Scale Permutation Entropy Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 Complexity Analysis of Fractional-Order Chaotic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Structural Complexity Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 SE and C0 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Structural Complexity Analysis of Fractional-Order Chaotic System . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Circuit Design and Realization of Fractional-Order Chaotic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Research on Analog Circuit of Fractional Chaos System . . . . . . . 8.2 Design and Implementation of Fractional-Order Chaotic System on DSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Algorithm for Solving Fractional-Order Simplified Lorenz System . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Hardware Design of Fractional-Order Chaotic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Software Design of Fractional Chaos System . . . . . . . . . 8.3 FPGA Design and Implementation of Fractional-Order Chaotic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Design and Optimization of Circuit Structure . . . . . . . . . 8.3.2 Design of Fractional Simplified Lorenz System Chaotic Signal Generator . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Design of Floating-Point Arithmetic Units . . . . . . . . . . . . 8.3.4 Experimental Results and Analysis . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117 118 118 126 129 132 134 134 137 140 143 143 148 148 151 153 156 156 159 160 163 166

Applications of Fractional-Order Chaotic Systems in Secure Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 9.1 Synchronous Control of the Fractional-Order Chaotic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 9.1.1 Coupled Synchronization of Fractional-Order Chaotic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

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9.1.2

Generalized Function Projection Synchronization for Fractional-Order Chaotic Systems . . . . . . . . . . . . . . . . 9.1.3 Network Synchronization of Fractional-Order Chaotic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Design of the Fractional-Order Chaotic Pseudo-Random Sequence Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Design of Chaotic Pseudo-Random Sequence Generator Based on the Fractional-Order Simplified Lorenz System . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Performance Tests for Chaotic Pseudo-Random Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Design of the Fractional-Order Lorenz Hyperchaotic Pseudo-Random Sequence Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Study of Fractional-Order Chaotic Image Encryption Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Image Encryption Algorithm Based on Fractional-Order Hyperchaotic System . . . . . . . . . . . . 9.3.2 Security Analysis of Image Encryption Algorithm . . . . . 9.4 Fractional-Order Chaotic Spread Spectrum Communication Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Spread Spectrum Codes in Spread Spectrum Communication Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Design of a Single-User Fractional-Order Chaotic Spread Spectrum Communication System Based on Correlated Reception . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3 Performance Analysis of the Spread Spectrum Communication Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.4 Design of a Multi-user Chaotic Spread Spectrum Communication System Based on Correlated Reception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.5 Design of a Multi-User Chaotic Spread Spectrum Communication System Based on the Rake Reception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Solution and Characteristic Analysis of Fractional-Order Discrete Chaotic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Research Progress of Fractional-Order Discrete Chaotic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Definition of Discrete Fractional-Order Difference . . . . . . . . . . . . 10.3 Characteristic Analysis of Fractional-Order Discrete Chaotic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 The Fractional-Order Logistic Map . . . . . . . . . . . . . . . . . . 10.3.2 The Fractional-Order Hénon Map . . . . . . . . . . . . . . . . . . .

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10.3.3 The Fractional-Order High-Dimensional Chaotic Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Appendix: Program Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

Chapter 1

Introduction

1.1 Overview of Chaos Chaos is an external, seemingly irregular complex motion of a deterministic system caused by internal nonlinearity, and its study has become one of the critical contents of nonlinear science. As a unique form of motion for nonlinear systems, chaos is prevalent in nature, while regular motion exists only within a certain time or a given spatial scale. With the gradual deepening of chaotic systems study, chaos science has rich and profound mathematical connotations and theoretical backgrounds in recent years. And it has been gradually applied to engineering practice and has become an essential part of modern science. With the rapid development of computer technology, chaos theory and applied study are developing rapidly and promoting its intersection and utilization with other fields. For example, chaos has an important theoretical and application status in physics, chemistry, electronics, biology, medicine, engineering, economics, and other scientific and technological fields. As a result, it has contributed significantly to various scientific fields [1]. At the first International Chaos conference, Ford, a famous physicist, argued that chaos, together with quantum mechanics and relativity, was the three major scientific revolutions of the twentieth century. He pointed out: “Quantum mechanics dashed Newton’s dream of controllable measurement processes, Einstein’s theory of relativity eliminated the illusion of absolute time and space, and chaos dispelled the predictable illusion of Laplace”. Understanding and describing the dynamic behaviors of chaotic systems can help people better understand the complex and diverse nature of human beings.

1.1.1 The Origin and Development of Chaos The study of chaos theory began in the early nineteenth century. Jules Henri Poincaré, a French mathematician, was the first scientist to study chaos. In 1903, in his study of the solar system’s stability, he organically combined topology with dynamical © Science Press 2022 K. Sun et al., Solution and Characteristic Analysis of Fractional-order Chaotic Systems, https://doi.org/10.1007/978-981-19-3273-1_1

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systems and discovered the chaos phenomenon in a conserved system. He proposed a new mathematical method to clarify the concepts related to chaos, such as singular point, singular ring, stability, bifurcation, etc. He also proposed a method to study chaos theory, such as the Poincaré section and parameter perturbation. He pointed out in his book that small differences in the initial conditions have produced great differences in later phenomena, and small errors in the former cause great differences in the latter, so the prediction becomes impossible [2]. It actually implies the properties of chaotic phenomena: deterministic systems are inherently stochastic. In 1892, Russian mathematician Lyapunov gave the mathematical definition of the concept of stability and proposed a general method to solve the stability problem. He also did a series of works on the stability theory. Together with the work done by Poincaré, he laid a solid foundation for the stability theory of ordinary differential equations. Until today, when people need to judge whether a dynamic system is chaotic, the Lyapunov exponent is still one of the main indexes. After Poincaré and Lyapunov, many physicists and mathematicians continue to contribute to the study of chaos theory. In 1954, Kolmogorov [3], the probability master of the Soviet Union, proposed the prototype of the famous KAM theorem. Later proved by Arnold, V. I. and Moser, J. in Physical Review Letters [4], the theorem clearly shows that chaos exists in dissipative and conservative systems. It is believed that this theorem marks the creation of chaos theory and the opening of modern chaos study. In the 1960s and 1970s, chaos study began to develop rapidly. In 1963, American meteorologist Lorenz [5] discussed the turbulence phenomena that often occurred in the atmosphere and the difficulties in weather forecast based on the atmospheric circulation model and obtained the famous Lorenz equation. The non-periodic phenomenon is calculated and simulated according to the determined equation, then leading to the view that it is impossible to use stepwise extension methods for longterm weather forecasting. Since then, chaos has been formally studied as a theory, and Lorenz was called the “father of chaos”. In 1964, French astronomer Hénon [6] was inspired by the Lorenz attractor and obtained a simpler Hénon map when studying globular clusters. In 1971, Dutch mathematician Takens and French physicist Ruelle [7] used the chaos to explain and analyze the mechanism of turbulence and introduced the concept of “strange attractor” in dissipative systems for the first time. In 1975, American mathematician York and his Chinese-American student Li [8] gave a mathematical definition for chaos in their paper Period Three Implies Chaos, namely, the definition of Li-York chaos. The specific contents are given as follows: Suppose there is a continuous self-mapping f (x) on a closed interval [a, b], if it is shown to have three periodic points, then it has n periodic points, where n is any positive integer. In 1976, American mathematical biologist May [9] systematically analyzed chaotic dynamic characteristics of the Logistic model in his paper, discussed the chaotic region structure of the model in detail, and revealed for the first time one of the ways to chaos, that is, period-doubling bifurcation. The First International Chaotic Conference in Italy in 1977 marked the official birth of chaos. In 1978, the American mathematician Feigenbaum [10] further studied the Logistic model

1.1 Overview of Chaos

3

and found a certain rule in the period-doubling bifurcation of the model. There is a geometric convergence rate in the bifurcation spacing, which is the famous Feigenbaum constant. This constant indicates a universal theory in the one-dimensional chaotic map system. It marks the beginning of the study of chaos theory from qualitative analysis to the quantitative calculation stage. The study of chaos theory started relatively late in China, but it develops rapidly, and many leading scientific research achievements have emerged. In 1984, Hao Bolin, a famous scientist, wrote a book titled Chaos. Symbolic dynamics and chaos dynamics were deeply expounded, which played a positive role in promoting the development of chaos science [11]. In 1986, the first chaos conference in China was held in Guilin, which further expanded the influence of chaos science in China and made more researchers enter this field. In 1994, a paper by Hao Berlin and Xie Fagan completely solved the problem of the number of periods in a one-dimensional continuous map with multiple critical points [12]. Since the twenty-first century, Professor Chen Guanrong from the City University of Hong Kong and others have made significant contributions to chaos theory, especially bifurcation theory, chaos control theory, and their applications, and proposed the Chen system. Later Lü Jinhu, Yu Simin, and Liu Chongxin et al., also made outstanding contributions to chaos generation and circuit implementation. They successively proposed the Lü system, Liu system, multi-scroll system, multiwing system, and so on. In addition, the International Symposium on Chaos Fractal Theory and Its Application organized by Professor Chen Guanrong and others has been successfully held for more than ten years, which provides a good platform for domestic scholars to investigate chaos and communicate with foreign counterparts. In 2015, the Chaos Secure Communication Professional Committee was established under the Chinese Cryptography Society. The first chaos Secure Communication Conference was successfully held at the Beijing Institute of Science and Technology, which made a solid step for the applied study of chaos.

1.1.2 Definition of Chaos Because of the complexity and singularity of chaotic systems, there is still no unified definition of chaos. Different definitions of chaos reflect different aspects of chaotic motion. Until today, the definition of chaos based on the Li-York theorem is still recognized as a highly influential definition of chaos in nonlinear dynamics theory, which is expressed explicitly as Definition 1.1 [8] f (x) is a continuous self-mapping on the closed interval [a, b]. If f (x) satisfies the following two conditions, there will be chaos: (1) (2)

p, q ∈ S, and the period of f (x) is infinite; There are countless subsets S on [a, b] satisfying the following three points: (a)

For any p, q ∈ S, when p = q,

4

1 Introduction

  lim sup f n ( p) − f n (q) > 0

(1.1)

  lim inf  f n ( p) − f n (q) = 0

(1.2)

n→∞

(b)

For any p, q ∈ S, n→∞

(c)

For any p, q ∈ S, where q is any periodic point of p,   lim sup f n ( p) − f n (q) > 0

n→∞

(1.3)

According to Li-York Theorem, “Period three means chaos”, that is to say, if any continuous function f (x) on the closed interval [a, b] has a periodic point with period three, then chaos must exist, that is, there are n periodic points, n is any positive integer. The above definition shows that in the interval map f (x), any two initial values belong to the set S. After countless iterations, the minimum distance between the two sequences may be zero or any positive number greater than zero. That is, when it approaches infinite iterations, the behavior of the system is uncertain, since the distance between sequences can be dissociated between a positive number and zero. Obviously, this is different from the periodic motion. Li-York called this motion form “chaos”. Since Li-York introduced “chaos” to nonlinear system analysis in 1975, the name chaos has gradually been accepted and known in academia.

1.1.3 Characteristics of Chaos Chaos is a kind of complex phenomenon, which is produced in deterministic nonlinear systems, and only occurs in a limited area. It has a complex motion form and there is no repeated movement trajectory. Unlike the motion state under the general concept, the characteristics of chaotic systems are summarized as follows: (1)

(2)

Extremely sensitive to initial conditions. Sensitivity to the initial value is a typical characteristic of chaotic systems. With a small change in the initial value, the system’s state variables will change significantly after a period of evolution. As the system continues to evolve, it will appear greater changes. This property indicates that the state of a chaotic system is difficult to predict, and its trajectory is challenging to estimate after running for a long time from an initial condition. Therefore, the field of spread spectrum communication is to use the sensitivity of chaotic signals to improve the system’s security. Internal randomness. Chaos is an uncertain behavior produced by a deterministic nonlinear system. It has intrinsic randomness and has nothing to do with external factors of the system. Although the equation of the nonlinear system is deterministic, the dynamic behavior generated by it is difficult to determine. In any region of its attractors, the probability distribution density function is

1.2 Overview of Fractional-Order Chaotic Systems

(3)

(4)

(5)

(6)

(7)

(8)

5

not zero, which is the randomness generated by the deterministic nonlinear system. In fact, the unpredictability of chaos and the extreme sensitivity to initial values lead to the internal randomness of chaotic systems, which also shows that the chaotic system is locally unstable. Determinacy. The determinism of chaotic systems means that the equations describing chaotic systems are deterministic. That is, the chaotic phenomenon is produced by the deterministic nonlinear system, which has the characteristics of randomness and is unpredictable. Hence, the chaotic system is the unity of determinism and randomness. Boundedness. The evolutionary trajectory of chaos is limited to a specific region called the chaotic attraction region. No matter how unstable the chaotic system is, its evolution trajectory will never exceed that region. The region is deterministic and bounded, called boundedness, so the whole chaotic system is stable. Ergodicity. The chaotic variables generated by the chaotic system are ergodic in the domain of attraction. That is to say, the chaotic orbit will experience each state point in the chaotic attraction domain in a limited time, and the orbit will not appear repetitively, which is called ergodicity. Dimensionality. The motion trajectory of chaotic systems is the result of infinite stretching and folding. The motion is carried out in a finite interval, with rich levels of motion state and similar structure. Unlike the motion generated by a general linear system, this infinite number of stretching and folding can only be represented by a fractional dimension. The fractal dimension also depicts the self-similarity of the motion form of chaotic systems. Universality. Universality refers to some common characteristics of different nonlinear systems when they tend to a chaotic state. It does not change according to specific system equations or system parameters, manifested as universal constants of chaotic systems, such as the Feigenbaum constant. Universality is an embodiment of the inherent law of chaos. Positive Lyapunov exponent. It refers to the quantitative description of the overall effect of mutual approximation or separation of motion trajectories generated by nonlinear systems. A positive Lyapunov exponent means that adjacent orbits are separated exponentially. In addition, the positive Lyapunov exponent also indicates the loss of information of adjacent points. The larger the Lyapunov exponent is, the more serious the loss of information and implies a higher degree of chaos.

1.2 Overview of Fractional-Order Chaotic Systems The history of fractional calculus can be traced back to the letters from L’ Hospital to Leibniz more than 300 years ago. It refers to how to solve the problem when the order is 0.5 and even how to understand the differential equation when the order is arbitrary. Because there is no practical engineering application background, and the calculation

6

1 Introduction

is large, the development of fractional calculus theory has been very slow. In recent years, with the progress of computer technology, fractional calculus operators have been widely used in nature, electromagnetic oscillation, system control, material mechanics, and other fields. At the same time, fractional wavelet transform, fractional Fourier transform, fractional image processing, and other technology have gained the attention of researchers in the field of signal processing [13]. The fractional-order chaotic system equation can be obtained by replacing the integer differential operator with the fractional difference operator, such as fractionalorder Lorenz system [14], fractional-order Chen system [15], fractional-order Chua system [16], fractional-order Rössler system [17], fractional-order Lü system [18], fractional-order Duffing system [19], fractional-order simplified Lorenz system [20]. It is found that after the introduction of fractional calculus operators, those chaotic systems have richer dynamic characteristics. For example, Ref. [14] analyzed the dynamic characteristics of the fractional-order Lorenz system. It shows that the fractional-order system has similar stability and system equilibrium point as the integer-order system. Reference [17] used bifurcations and the maximum Lyapunov exponential spectrum algorithm based on the small data algorithm to analyze the dynamic characteristics of fractional-order Rössler system changing with parameters and orders and finds many periodic windows. The above fractional-order chaotic systems are mainly three-dimensional, and the study of four-dimensional fractionalorder hyperchaotic systems has also attracted the attention of scholars [21, 22]. For example, Ref. [21] studies a four-dimensional non-equilibrium system that will not produce chaos in the case of integer order. Still, after introducing the fractional calculus operator, it is found that the system has more decadent chaotic behaviors, and the minimum order of chaos generated by the system is Sect. 1.3.2. In Ref. [22], a new fractional-order hyperchaotic system was designed based on the Lorenz system. The Hopf bifurcation phenomenon of the system changing with parameters and orders was analyzed and verified by numerical simulation. Compared with the three-dimensional fractional-order chaotic systems, there are relatively fewer reports regarding the fractional-order hyperchaotic systems.

1.2.1 Definitions and Properties of Fractional Calculus (1)

Definition of integral calculus

Before defining fractional calculus, the definition of integer-order difference operator and function derivation are introduced. Definition 1.2 The first derivative is defined as D f (t) = lim

t→0

f (t) − f (t − t) , t

(1.4)

1.2 Overview of Fractional-Order Chaotic Systems

7

where D is the differential operator and t is a time interval. For second-order derivative and third-order derivative, the following equations can be used for calculation: D f (t) − D f (t − t) t f (t) − 2 f (t − t) + f (t − 2t) , = lim t→0 t 2

D 2 f (t) = lim

t→0

D 2 f (t) − D 2 f (t − t) t→0 t f (t) − 3 f (t − t) + 3 f (t − 2t) − f (t − 3t) . = lim t→0 t 3

(1.5)

D 3 f (t) = lim

(1.6)

According to the above calculation formula, the general definition of higher-order derivatives can be sorted out. Definition 1.3 n derivative is defined as n  i=0

D f (t) = limt→0 n

(−1)

  n k k t k

f (t − kt) ,

(1.7)

where n is a positive integer, when n = 0, let D 0 f (t) = f (t). For integral calculus, it is defined as ⎧ t ⎪ ⎪ f (x)d x, n = 1 ⎨ −n 0 Dt0 f (t) = t , (1.8) ⎪ −n+1 ⎪ ⎩ Dt0 f (x)d x, n > 1 0

Let’s consider the case where n is a fraction, which is the definition of fractional calculus. (2)

Definition of fractional calculus

At present, there are many definitions of fractional calculus, most of which are Grunwald–Letnikov (G-L) fractional calculus definition, Riemann–Liouville (R-L) fractional calculus operator, and Caputo fractional calculus operator. The definitions and properties of these three fractional calculus operators are presented below. G-L definition is widely used in the engineering field, also known as the series definition. It is derived from the definition of integral calculus of continuous function, which is similar to the definition of the integer-order difference in form. Definition 1.4 [23] The fractional-order G-L differential is defined as

8

1 Introduction t

q Dt0

t 1  (q + 1) f (t) = lim (−1)k f (t − kt), −q t→0 t k!(q − k + 1) k=0

where q > 0, q ∈ R, (x) =

+∞ 0

(1.9)

e−t t x−1 dt is the Gamma function.

Definition 1.5 [23] The fractional-order G-L integral is defined as t

−q Dt0

t 1  (q + k) f (t) = lim (−1)k f (t − kt), t→0 t −q k!(q) k=0

(1.10)

where q > 0, q ∈ R. Definition 1.6 [23, 24] The fractional-order Riemann–Liouville differential is defined as ∗

Dt0 x(t) := ∗ Dtm0 Jt0

m q

m−q

=

d dt m dm dt m

x(t) t 1

 m−q−1 (t − τ ) x(τ )dτ ,m − 1 < q < m t0

(m−q)

x(t), q = m

,

(1.11)

where, q ∈ R + , Jt0 is the integral operator of order q, According to the definition, ∗ q q Dt0 Jt0 = x(t). q

Definition 1.7 [23, 24] The fractional-order R-L integral is defined as q Jt0 x(t)

1 = (q)



t

(t − τ )q−1 x(τ )dτ ,

(1.12)

t0

where q ∈ R + , Jt0 is the integral operator of order q. For t ∈ [t0 , t1 ], q ≥ 0, γ > −1, r ≥ 0 and constant C, the fractional Riemann–Liouville integral satisfies the following basic properties: q

Jt0 (t − t0 )γ = q

q

Jt0 C = q

(γ + 1) (t − t0 )γ +q , (γ + q + 1)

(1.13)

C (t − t0 )q , (q + 1)

(1.14)

q+r

Jt0 Jtr0 x(t) = Jt0

x(t).

Definition 1.8 [23, 24] The fractional-order Caputo differential is defined as

(1.15)

1.2 Overview of Fractional-Order Chaotic Systems

Jt0 (t − t0 )γ = q

q

Jt0 C =

9

(γ + 1) (t − t0 )γ +q , (γ + q + 1)

(1.13)

C (t − t0 )q , (q + 1)

(1.14)

q

q+r

Jt0 Jtr0 x(t) = Jt0

x(t).

(1.15)

where q ∈ R + , m ∈ N , Dt0 is the Caputo differential operator of order q. When t ∈ [t0 , t1 ], m ∈ N , m − 1 < q < m, the Caputo differential has the following properties: q

Dt00 Jt00 x(t) = x(t), q

q

Jt0 Dt0 x(t) = x(t) −

m−1  k=0

x (k) (t0+ )

(1.17) (t − t0 )k . k!

(1.18)

In practical applications, the Caputo definition is generally used for the timefractional derivative, and the R-L or G-L definition is widely used for the spacefractional derivative. The following is a comparison and analysis of the three definitions of fractional calculus: (1)

(2)

(3)

The R-L definition is equivalent to the G-L definition if the function f (t) has continuous derivatives of order m + 1 and m is at least q = m −1; otherwise, if the function f (t) does not satisfy these conditions, then the two definitions are no longer consistent. In general, R-L definition has a wider range of applications than G-L definition. If the function f (t) has a continuous derivative of order m + 1 and m is decided by q = m − 1, let m = n − 1, then n = m + 1, if satisfied f (k) (ε) = 0(k = 0, 1, 2, . . . , n − 1), then the Caputo definition is equivalent to the G-L definition; otherwise, if the above conditions are not met, the two definitions are not equivalent. In fact, both the R-L definition and the Caputo definition are improved forms of the G-L definition. For the definition of fractional R-L differential, it is necessary to specify the value of a fractional derivative of the unknown solution at the initial value point t = t 0 . Because the physical meaning of the fractional derivative is not clear and the size is difficult to measure, it is difficult to use the R-L differential definition in the actual system. On the other hand, when using the Caputo differential definition, it is only necessary to specify the values of x(t 0 ), x (t 0 ), …, x (n−1) (t 0 ), and its physical meaning is obvious. It can be seen that the Caputo definition is more conducive to solve the actual physical system and has better practical engineering application value.

10

(3)

1 Introduction

Properties of fractional calculus

Fractional calculus is derived from integral calculus. When q = n, it is integral calculus. The operation of fractional calculus has the following basic properties: (1)

Linearity q

q

q

q

q

Dt0 [λ1 f (t) + λ2 g(t)] = λ1 Dt0 f (t) + λ2 Dt0 g(t), q

(1.20)

f (t).

(1.21)

Jt0 [λ1 f (t) + λ2 g(t)] = λ1 Jt0 f (t) + λ2 Jt0 g(t). (2)

Exchange law Jt0 [Jtv0 f (t)] = Jtv0 [Jt0 f (t)] = Jt0 q

(3)

q

q+v

Reversibility q

q

Dt0 [Jt0 f (t)] = f (t), q

q

Jt0 (Dt0 ) f (t) = f (t) −

m−1 

f (k) (t0+ )

k=0

(4)

q

(t − t0 )k , k!

(1.23)

C (t − t0 )q . (q + 1)

(1.24)

The fractional-order integral of the time variable is Jt0 (t − t0 )γ = q

(6)

(1.22)

where f (k) (t0+ ) is the initial value. The fractional-order integral of constant C is Jt0 C =

(5)

(1.19)

(γ + 1) (t − t0 )γ +q . (γ + q + 1)

(1.25)

The Laplace transform of the fractional-order difference is

q L[0 Dt0

f (t)] = 0

∞

e−st Dt0 f (t)dt

= s q F(s) −

q

n−1 

q−k−1

s k [Dt0

f (t)]|t=0 , (n − 1 < q < n),

k=0

(1.26) where F(s) = L[f (t)] is the Laplace transform of f (t).

1.2 Overview of Fractional-Order Chaotic Systems

(4)

11

The physical meaning of fractional calculus

At present, there is still no unified physical meaning and geometric explanation for fractional calculus. However, according to the above relationship between fractional-order difference and integer-order difference, and the properties of fractional calculus, it can be found that the fractional-order difference comprehensively considers the influence of history and nonlocal distribution. This long memory characteristic and globality make fractional calculus more accurately describe the physical model in nature.

1.2.2 Overview of Solving Algorithms for Fractional-Order Chaotic Systems In order to solve the fractional calculus equations, many definitions of fractional calculus are proposed. The most commonly used definitions are fractional Grunwald–Letnikov calculus (G-L definition), fractional Riemann–Liouville calculus (RL definition), and Caputo calculus. Although the R-L definition can simplify the computation of fractional calculus operators, its physical meaning is unclear, and its size is more challenging to measure than the algorithm based on the Caputo definition. Based on the above definition of calculus operator, the numerical algorithm of fractional calculus operator mainly has the following categories: (1)

(2)

(3) (4)

Analytical method: Including Adomian decomposition algorithm [25], homotopy-perturbation algorithm [26], differential transformation algorithm [27], and variational iterative algorithm [28]. Finite difference method: Including prediction–correction algorithm [29], display format algorithm, implicit format algorithm, and Crank–Nicolson format algorithm [30]. The frequency-domain algorithm is based on Laplace transform [31]. Other algorithms: meshless algorithm and finite element algorithm.

At present, the most commonly used algorithms for solving fractional-order chaotic systems are frequency-domain algorithm and forecast-correction algorithm, which are widely reported in the literature. The frequency-domain algorithm is based on the R-L definition. The integer-order calculus operator is used to fit the fractional-order operator. The fractional-order calculus operator in the time domain is transformed into the transfer function in the frequency domain. Then the piecewise linear function is used to approximate the frequency domain, and the numerical solution of the system is obtained. The frequency-domain algorithm is the theoretical basis for the current analog circuit design of the fractional-order chaotic system. However, the system function of the integral operator of this algorithm is relatively complex, which is not conducive to analyze the system’s dynamic characteristics when the order changes continuously. Moreover, whether the algorithm can accurately solve the fractional-order chaotic system has been questioned by some scholars

12

1 Introduction

[32, 33]. The prediction–correction algorithm uses the Adams–Bashforth predictor formula and the Adams–Moulton correction formula to obtain the discrete iterative of the system. This algorithm can analyze the continuous change of the system with the differential order, and the accuracy is higher than that of the frequency-domain algorithm. In recent years, Wang and Yu [26] designed the Multistage homotopyperturbation method for the fractional-order chaotic system. The research shows that the algorithm can obtain the analytical solution of the system with high precision. Arena et al. [34] designed the Different Transform Method for the fractional-order chaotic system and compared the algorithm with the Runge–Kutta algorithm in the case of integer order. However, these two algorithms have not been applied in further literature, and the existing literature only uses these two algorithms to solve the system. Still, it does not use the algorithm to analyze the system’s dynamic characteristics. Compared with the Multistage homotopy-perturbation method and different transform methods, the Adomian decomposition algorithm is more widely used in fractional-order chaotic systems. Cafagna and Grassi used the Adomian decomposition algorithm to solve the fractional-order Chen system [35] and fractional-order Rölsser system [36], respectively, and analyzed the system’s dynamic characteristics. The results show that the system can generate chaos in a lower order. For example, the minimum order of chaos generated by the fractional-order Chen system is 0.24 (q = 0.08), which is far smaller than the minimum order of other algorithms. Literature [37] used the Adomian decomposition algorithm to solve fractional-order simplified Lorenz system, analyzed the system dynamics characteristics, and compared the Adomian decomposition algorithm with other algorithms. The results show that the Adomian decomposition algorithm has higher accuracy and consumes fewer resources than the prediction–correction algorithm. Obviously, the solution accuracy of different numerical algorithms is different. In addition, the study by Tavazoei and Haeri [38] also showed that the current numerical solution algorithms for the fractional-order chaotic system all have the problem of solving accuracy, and the dynamic characteristics of the system obtained by different algorithms should also be different. So, which of these numerical solution algorithms is more reliable, and how to choose the algorithm in practical applications is worth studying. However, at present, few papers comparatively study the characteristics of different solving algorithms, such as algorithm speed, calculation accuracy, algorithm time and space complexity, etc. These studies provide a basis for the selection of algorithms for fractional-order chaotic systems. The solution algorithms of the fractional-order chaotic system are required to further research for the establishment of effective discrete iterative and design of DSP/FPGA digital circuit of the fractional-order chaotic system.

1.3 Review on the Analysis Methods of Fractional-Order …

13

1.3 Review on the Analysis Methods of Fractional-Order Chaotic Systems 1.3.1 Study on the Dynamical Behavior of Fractional-Order Chaotic Systems The study of dynamic characteristics of fractional-order chaotic systems is a hot topic in the field of fractional-order chaos. At present, the methods used to study the dynamic characteristics of fractional-order chaotic systems mainly include bifurcation diagram, 0–1 test, the maximum Lyapunov exponents algorithm based on sequence, Poincaré section and attractor phase diagram, etc. Among these methods, only the Poincaré section can determine whether the system is hyperchaotic. However, if only the Poincaré section is used to judge whether the system is hyperchaotic, it is not very convenient. Unlike integer-order chaotic systems, it is challenging to design the Lyapunov exponent spectrum algorithm for fractional-order chaotic systems. Few papers have reported the Lyapunov exponents spectrum calculation method for fractional-order chaotic systems when the parameters and orders change. The prediction–correction algorithm can only get time series, and its iterative is challenging to facilitate the calculation of the Lyapunov exponents spectrum. Li et al. [39] designed the Lyapunov exponents spectrum algorithm for the fractionalorder chaotic system based on the frequency-domain algorithm. Literature [40] used this algorithm to analyze the dynamic characteristics of the fractional-order Lorenz system. However, due to the insufficient accuracy of the frequency-domain algorithm, its effect in the actual analysis is not very good. In addition, Lyapunov exponents spectrum algorithms based on phase space reconstruction technology, such as Wolf algorithm [41], Jacobian algorithm [42], and neural network algorithm [43], are greatly affected by subjective factors in the selection of embedding dimension and delay time. In order to calculate the Lyapunov exponents spectrum of the fractionalorder chaotic system more accurately, Caponetto and Fazzino [44] designed the Lyapunov exponents spectrum algorithm for fractional chaotic systems based on the analytical solution obtained by the Adomian algorithm. However, the paper did not analyze the dynamic characteristics of the fractional-order chaotic system changing with parameters and orders. In addition, the Hopf bifurcation characteristics, topological horseshoes, and stability characteristics of fractional-order chaotic systems have aroused the interest of scholars. For example, Li and Wu [22] studied the Hopf bifurcation characteristics of a new fractional-order Lorenz hyperchaotic system, indicating that Hopf bifurcation exists when the fractional-order chaotic system changes with parameters and orders; Jia et al. [45] studied the topological horseshoe theory of fractional-order Lü systems, and verified the existence of chaos in fractional-order chaotic systems from the theoretical level; Li and Li [46] studied the stability of fractional-order nonlinear systems based on the T-S fuzzy theory and took the fractional-order Rölsser system as an example to verify the effectiveness of the proposed theory.

14

1 Introduction

It can be seen that the study on the dynamical characteristics of the fractional-order chaotic system in the current literature is mainly to verify whether there is chaos in the system after the integer-order chaotic system is converted into a fractional-order chaotic system. The analysis method is mainly inherited from the dynamic characteristic analysis method of integer-order chaotic systems. However, the minimum order of chaos, dynamic characteristics, and the relationship between fractional calculus and chaos in fractional-order chaotic systems lack a systematic and in-depth discussion. Therefore, the analysis of the dynamic characteristics of the fractional-order chaotic system needs to be further studied.

1.3.2 Complexity of Fractional-Order Chaotic System and Its Definition The concept of complexity emerged in the 1980s and has become the research frontier of contemporary science. At present, researchers with different academic backgrounds explore the complexity from different perspectives, so the definition and understanding of the complexity are different. At present, there are at least 45 definitions of complexity. Generally speaking, entropy can be used to describe complexity, because entropy describes the degree of chaos in the system. In fact, many current definitions of complexity are based on the concept of entropy. The earliest definition of entropy comes from physics, proposed by Clausius in 1865. It was used to measure the degree of “chaos” of a thermodynamic system, also known as thermodynamic entropy. Definition 1.9 Thermodynamic entropy: In classical thermodynamics, use increment to define as   dQ ds = , (1.27) T reversible where T is the thermodynamic temperature of the substance, dQ is the heat added to the substance in the process of entropy increase, and the subscript “reversible” means that the change process caused by the heating process is reversible. Its integral form is

x S − S0 =

dQ , T

(1.28)

x0

where S is the entropy of equilibrium x of the system, and S 0 is the entropy of system x 0 state.

1.3 Review on the Analysis Methods of Fractional-Order …

15

The definition of thermodynamic entropy gives the definition prototype of entropy, but its form needs to be further improved in other fields. For example, Shannon entropy in information science [47] is defined as follows. Definition 1.10 Shannon entropy: Suppose the symbol space of a discrete source is U = {u1 , u2 , …, uN }, and the probability space of corresponding symbol occurrence is P = {p1 , p2 , …, pN }, then the source weighted average information is defined as H (U ) = −

N 

pi log( pi ).

(1.29)

i=1

The above equation is also called Shannon entropy. Obviously, the larger value of H, the more informative source and the greater the average uncertainty of all possible events in the source, which means the greater complexity of the source. In recent years, the application range of Shannon entropy has become wider and wider, and many complexity algorithms based on Shannon entropy have been proposed. For example, Permutation Entropy (PE) [48], Spectral Entropy (SE) [49], Wavelet Entropy (WE) [50], Strength Statistical Complexity Measure (SCM) [51], Fuzzy Kernel Entropy (FKE) [52] and Symbol Entropy (SymEn) [53], etc. The complexity of a nonlinear system can also be described by Kolmogorov complexity, which was proposed by Kolmogorov [54] in 1965, and used to characterize the number of bits that can generate a certain (0, 1) sequence with the shortest degree of computer power. The definition of Kolmogorov complexity is shown in Definition 1.11. Definition 1.11 Kolmogorov complexity: For a (0, 1) sequence, calculate the number of forbidden words or forbidden strings in the sequence, and call it absolute complexity c(n); the absolute complexity of any sequence c(n) tends to be a constant b(n) lim c(n) = b(n) =

n→∞

n . log2 n

(1.30)

Kolmogorov complexity is defined as the ratio of absolute complexity c(n) to a constant value b(n), namely Kc =

c(n) log n = c(n) 2 . b(n) n

(1.31)

It can be seen that K c is a physical quantity related to sequence length n, and the K c measure value of a completely random sequence approaches 1; While the K c measurement value of a regular or periodic sequence approaches 0. Lempel and Ziv implemented the algorithm according to the algorithm principle, called the Lempel– Ziv algorithm [55]. Since the Lempel–Ziv algorithm is only applicable to the (0, 1) sequence, that is, the time series needs to be quantified, and the system complexity can

16

1 Introduction

only be counted on a one-dimensional time scale. Pincus [56] proposed the approximate entropy algorithm (ApEn) based on the definition of Kolmogorov complexity in 1991. This algorithm does not require quantification when analyzing the complexity of time series. However, the similarity of the algorithm uses the Heaviside function. The algorithm has a relatively large dependence on the threshold r and the phase space dimension m, limiting its application range. In 2000, Richman and Moorman [57] improved the ApEn algorithm and proposed the Sample entropy algorithm (SampEn), which did not calculate the statistics of its own matching and was an improvement on the ApEn algorithm, but meaningless ln0 would appear in the case of no template matching. Later, Chen et al. [58] based on the SampEn algorithm and adopted the concept of fuzzy membership, proposed the Fuzzy Entropy (FuzzyEn), further improved the SampEn algorithm, and the measurement effect was better. In addition, the G-P correlation dimension complexity [59] also belongs to the complexity measurement algorithm defined by Kolmogorov. In addition, some algorithms are not defined based on the above complexity but can also achieve excellent measurement effects, such as C 0 complexity [60]. The C 0 complexity algorithm first performs the fast Fourier transform on the signal and then removes the spectrum whose energy is lower than a certain value in the power spectrum. That is, the sequence is decomposed into regular and irregular components, and the measured value is the proportion of irregular components in the sequence. The more significant the proportion, the more random the signal. In summary, more complex the sequence is, more random the obtained chaotic sequence is. The more complex the sequence, the more random the chaotic sequence obtained. Therefore, the complexity definition of fractional-order chaotic systems in this paper is shown in Definition 1.12. Definition 1.12 Fractional-order chaotic system complexity: The complexity of fractional-order chaotic systems refers to the degree to which the system generates a sequence close to a random signal. The greater the complexity, the more complex the sequence, the better its anti-interference and anti-interception ability, that is, the better its security. Remark 1.1 When different algorithms measure the complexity of a fractionalorder chaotic system, the measured value must be different due to different algorithm principles. Still, different algorithms represent different aspects of system complexity. Remark 1.2 Regarding whether there is an objective complexity, this is a question worthy of in-depth study. Since there is currently no unified definition of complexity, defining “objective” complexity is challenging. Remark 1.3 In practical applications, a more complex sequence should be selected for encryption applications. The complexity algorithm provides an effective basis for the parameter selection of fractional chaotic systems.

1.3 Review on the Analysis Methods of Fractional-Order …

17

1.3.3 Complexity Analysis of Fractional-Order Chaotic Systems The complexity analysis of chaotic system sequences is an important research content in the field of information security and has attracted extensive attention. The more complex a sequence is, the more random it is, and the more difficult it is to recover it. Practical chaotic sequences should have as much complexity as possible to ensure the anti-interference and anti-interception capabilities of the spread spectrum communication system. People have done a lot of research on the complexity of discrete chaotic systems. Grassberger and Procaccia [59] analyzed the complexity of the Henon map by GP algorithm, and Balasubramanian et al. [61] analyzed the complexity of Logistic map by Lempel–Ziv algorithm and ApEn algorithm, and different distinguished states of the system by complexity measure. Literature [62] uses the PE algorithm to analyze the complexity of chaotic pseudo-random sequences. The TD-ERCS system proposed by Sheng et al. [63] has higher complexity than other discrete chaotic systems (such as Logistic map and Hénon map, etc.), and the calculation result is consistent with the results of the Lempel–Ziv algorithm and the ApEn algorithm. Liang et al. [64] analyzed and compared the complexity of different discrete chaotic systems using the WE algorithm, which also showed that the TD-ERCS system has higher randomness. In this paper, the WE algorithm has better global characteristics than the SE algorithm by determining the energy size of each frequency band of wavelet packet energy. Feng et al. [65] used the symbolic entropy (SymEn) algorithm to analyze the randomness of the discrete chaotic system and showed that the discrete chaotic system could be used as a source of randomness. Although the SymEn algorithm has lower requirements for parameter selection than ApEn, the SymEn algorithm is only for symbol sequences. It needs to know the sequence symbol space, which limits its practical application. In recent years, the complexity analysis of integer-order chaotic systems has gradually gained attention. Micco et al. [66] used the SCM algorithm to analyze the complexity of a class of integer-order continuous chaotic systems. The SCM complexity algorithm has the characteristics of fewer parameters and less sensitivity to parameters, but it cannot measure the complexity of periodic sequences well. Literature [67] used the SCM algorithm and the SE algorithm to analyze the complexity of the multi-wing chaotic system. It showed that the complexity of the multi-wing chaotic system does not increase with the number of wings. Jiang et al. [68] used the PE algorithm to analyze the complexity of the semiconductor laser chaotic system and showed that the complexity of the chaotic light obtained by the dual-optical feedback system is always greater than that of the single-optical feedback system. Yang et al. [69] also used the PE algorithm to analyze the complexity of the erbiumdoped fiber ring chaotic laser. The research results show that the intracavity loss has a more significant impact on the complexity of the chaotic laser. As the intracavity loss increases, its PE complexity has gradually increased.

18

1 Introduction

The above studies have shown that complexity is closely related to the dynamic characteristics of a chaotic system. When the system is in a chaotic state, the complexity measure is relatively high. The complexity measure is relatively low when the system is in a non-chaotic state, such as a periodic state. Unlike discrete chaotic system time series, which fluctuates sharply between adjacent data, the difference between adjacent data of continuous chaotic sequence is not very large. Therefore, it has higher requirements on the algorithm’s complexity, and the required sequence length is also longer. The existing literature shows that the complexity measure of the discrete system is higher than that of the continuous chaotic system. In fact, the current complexity analysis of the fractional-order chaotic system is not very sufficient, and there are few reports about it. Most studies on the complexity of chaotic systems focus on discrete chaotic systems and integer-order chaotic systems. As the application of fractional-order chaotic systems in information security and secure communication has become a new direction of nonlinear scientific application research, the study of its complexity will become more and more critical. This includes two aspects. On the one hand, we should start from the complexity of the fractional-order system itself, which is generally related to the nonlinear term of the system and the fractional calculus operator. The more the nonlinear term in the system is, the greater the complexity is. The system’s complexity is different under different definitions, and the relationship between the complexity of the system and the order needs further research. Of course, this part is more inclined to the dynamic characteristics of the system. On the other hand, it studies the time series complexity analysis of fractionalorder chaotic systems, uses different algorithms to measure, analyzes the change rule of the time series complexity of fractional chaotic systems with system parameters and orders, and provides an effective parameter selection basis for fractional-order chaotic systems in cryptography and secure communication applications.

1.4 Overview of the Application of Fractional Chaotic Systems With the in-depth study of fractional-order chaotic systems, people turn from theoretical study to application study. The circuit implementation of fractional-order chaotic systems has become one of the research focuses. In 2006, Bohannan, G. W. et al. applied for a patent for the invention of fractional devices. Professor Liu Chongxin’s research team conducted circuit simulation studies on multiple fractional-order chaotic systems by designing the equivalent circuit of fractionalorder operators. Based on the time-domain–frequency-domain conversion approximation algorithm and analog electronic technology, the implementation of the analog circuit of the fractional-order chaotic system has been extensively studied. Involving the Lü system, the hyperchaotic system, the multi-wing system, the high-dimensional

1.4 Overview of the Application of Fractional Chaotic Systems

19

chaotic system and based on memristors Circuit implementation, etc. The timedomain analysis method of fractional-order differential equations has certain advantages when studying the characteristics of fractional-order nonlinear systems, but its analog circuit implementation is more complicated. Compared with the analog circuit of fractional-order chaotic systems, the design and implementation of the digital circuit are more meaningful. The implementation of digital circuits has flexibility in choosing the algorithm for solving fractional-order chaotic systems, and the implementation of digital circuits for fractional-order chaotic systems is conducive to its practical application. However, the solution of fractional-order chaotic systems is relatively complex, so the implementation of the digital circuit of fractional-order chaotic systems is worth studying in-depth.

1.4.1 Circuit Implementation of Fractional-Order Chaotic Systems The circuit implementation of chaotic systems is one of the essential bases of chaos application. It is helpful to study the dynamical characteristics of chaotic systems, prove the existence of chaotic attractors, and carry out the research on chaos control and synchronization methods. In fact, the chaos phenomenon is widespread in many circuit systems. The first discovery of chaos in circuits was in 1927. Danish electrical engineer VanderPol, B. discovered an essential phenomenon in the neon lamp relaxation oscillator. He thought it was a kind of “Irregular noise”, but later studies showed that he observed chaos. In 1983, Professor Chua, L. O. of the University of California at Berkeley designed a third-order autonomous chaotic circuit, which people call Chua’s circuit. The simple structure of the circuit has rich nonlinear characteristics and has become a classic example of nonlinear chaotic circuits. In nonlinear dynamical behaviors, chaotic circuits can also verify various chaotic phenomena, chaotic synchronization, control laws, bifurcations, etc. The status and role of chaotic circuits in chaos research are more and more critical. For more than 30 years, domestic and foreign scholars have devoted themselves to various chaotic systems’ circuit design and hardware implementation. In the past, the main methods of chaotic system study were theory and numerical simulation. After the chaotic circuit design method is proposed, hardware can be used to generate actual chaotic signals. In this way, chaos has broad application prospects in the field of signal processing. At present, the circuit implementation of chaos mainly includes analog circuits and digital circuits. Chaos analog circuits can generate real chaotic signals. The design and implementation methods of chaotic analog circuits are mainly divided into personalization, modularization, and improved modularization. For example, many chaotic circuits such as Chua circuits adopt personalized design methods. The advantage of this method is that it uses fewer circuit components. In addition, the circuits of many chaotic equations, such as the Lorenz equation, cannot be realized by this design method. Modular design is a common chaotic circuit design method based on the

20

1 Introduction

dimensionless equation of state. It consists of three parts, namely, variable proportional compression transformation, differential-integral transformation, and time scale transformation. Although this method is universal, it requires more components. On this basis, omitting the differential–integral conversion is an improved modular chaotic circuit design method. It is also universal and uses fewer components. At present, the implementation of analog circuits for integer-order chaotic systems is very common. Suykes, Yalcin, Elwakil, Yu Simin, Lu Jinhu, Wallace K. S. Tang, Liu Chongxin, et al. have done a lot of work. However, chaotic analog circuits still have their limitations. Chaos analog circuit relies on manual construction, the process is complex, not flexible, there are many problems such as large distribution of circuit parameters, components are easy to aging, easy to be affected by temperature, especially poor repeatability, circuit designers need to have high design skills and debugging experience. With the development of Digital Processing technologies of integrated circuits such as FPGA (Field Programmable Gate Array) and DSP (Digital Signal Processing), Using FPGA and DSP to realize chaotic systems is gradually becoming a trend. The design and application study of existing chaotic system circuits shows that there are few technical obstacles to realize integer-order chaotic systems based on FPGA and DSP. The realization of chaotic digital circuits needs to solve the chaotic system first, get the numerical iterative of the system, and then design the software and hardware of the system according to the iterative. Implementing the chaotic system on the DSP platform is less complicated and cost-effective. There is a floating-point number operation unit inside the floating-point DSP chip, which reduces the difficulty of software design and improves program operation speed. The rich peripheral interfaces are convenient for system integration. The software programming uses the C language and has rich library functions, which provides convenience for designers. It is relatively difficult to implement a chaotic system with FPGA. The modular design needs to be combined with the iterative solution obtained. According to the principle and characteristics of FPGA, many modules need to be designed by themselves, including floating-point arithmetic modules and various interfaces, etc., which are generally implemented using hardware description languages. But it also provides enough free space for designers to improve the performance of the system. When the digital circuit is used to realize the chaotic system, the development environment is equipped with strong simulation and debugging functions, which is convenient for solving many design problems. The digital circuit of the chaotic system has the characteristics of flexible implementation, convenient parameter modification, good repeatability, and strong anti-interference ability. Digital circuit realization of chaotic systems laid a solid foundation for the practical application of chaotic systems in other fields.

1.4 Overview of the Application of Fractional Chaotic Systems

21

1.4.2 Synchronization Control of Fractional-Order Chaotic Systems The synchronization control of the fractional-order chaotic system is the key to applying the fractional-order chaotic system in secure communication. So far, for integer-order chaotic systems, people have proposed many synchronization control methods and techniques. However, due to the complexity of fractional-order nonlinear systems and the lack of efficient solution algorithms, there are relatively few studies on chaotic synchronization control of fractional-order chaotic systems, and the control strategies and methods are relatively simple, mainly based on Laplace transform theory and Lyapunov Stability Principle and Stability Theory of Fractional Linear System. But these three synchronization theories are not yet mature, and the controller designed based on the Laplace transform theory lacks flexibility and universality. The synchronization method based on the Lyapunov stability principle is conducive to determining the range of control parameters of the synchronization controller. Still, it needs to construct the Lyapunov function, and the controller is often complex, which is not conducive to engineering implementation. Although the synchronization strategy based on the stability theory of fractional-order linear systems is a common method at present, it depends on calculating the eigenvalues of the control matrix near the equilibrium point. At present, the primary method of studying the synchronization control of fractional-order chaotic systems is to extend the synchronization control method of integer-order chaotic systems to fractionalorder chaotic systems, such as coupling synchronization, feedback control synchronization, sliding mode control synchronization, adaptive control synchronization, projection synchronization, and so on. Due to the complexity of fractional calculus operators, the synchronization and control of fractional chaotic systems is not yet mature, and its research has just begun. Therefore, there are still a lot of problems that need to be further studied. The following three aspects are reviewed. (1)

Fractional-order synchronization control algorithm needs further study, especially the synchronization control algorithm of the fractional-order chaotic network. The choice of the synchronous controller is also worth considering in practical applications. A relatively simple controller is more conducive to synchronous simulation or digital circuit realization. The coupled synchronization controller is very simple, and the actual effect is perfect, but there are very few theoretical proofs about fractional-order coupled synchronization. Therefore, the coupled synchronization of fractional-order chaotic systems is worthy of further discussion. When the time series of the driving system and the response system are in a nonlinear functional relationship, the synchronization system has stronger security. Therefore, the function projection synchronization with complex function relationship and simple controller will be the current research focus. As an essential aspect of the synchronization of fractional-order chaotic systems, network synchronization of fractional-order chaotic systems

22

(2)

(3)

1 Introduction

is a current research hotspot, which is helpful for multi-point communication in actual secure communication. Considering the simplicity and effectiveness of coupled synchronization controllers, this book mainly studies the synchronization of fractional-order chaotic systems with coupled networks. The comprehensive evaluation and analysis of the performance of the fractional-order chaotic synchronization system need to be strengthened. Synchronization performance analysis is an important content of synchronization control research. In addition to stability, there are also robustness, synchronization establishment time, synchronization accuracy, etc. Although some existing synchronization control methods can achieve synchronization, it takes too long to establish synchronization, which is of little significance to applying chaotic secure communication. In addition, when synchronization is realized, it is necessary to study the general rule of the synchronization system performance with system parameters, order and control parameters. Then the methods to improve the performance of the synchronization system are proposed. Therefore, in the process of synchronization control of fractional-order chaotic systems, the goal should be to improve the synchronization performance of the synchronization system and to explore the law of the synchronization performance of the synchronization system with the system fractional-order and controller parameters. Furthermore, the optimal synchronization methods and parameters of fractional-order chaotic systems can be obtained, which lays a theoretical foundation for applying fractional-order chaotic systems in secure communication. The selection of numerical algorithms in synchronous simulation requires further research. After the synchronization controller is designed, the next step is how to better apply the synchronization system in the field of secure communication. At present, the synchronization simulation algorithms of fractional-order chaotic systems mostly use prediction–correction algorithms and frequency-domain algorithms. In literature [70–74], the analog circuit of fractional-order chaotic synchronization system has been realized, and its algorithm is the frequency-domain algorithm. However, the analog circuit is greatly affected by device parameters, and its controllability is poor in practical application compared with digital circuit implementation. Although the prediction–correction algorithm of fractional differential operators has certain advantages in studying the chaotic characteristics and synchronization control of fractional nonlinear systems, it consumes more and more system memory resources. It requires more and more time with the iteration, making it difficult for DSP/FPGA digital circuits to realize synchronization. Therefore, research on new solution algorithms that are beneficial to realizing the digital circuit of the fractional-order chaotic synchronization system has practical application value.

1.4 Overview of the Application of Fractional Chaotic Systems

23

In a word, although some achievements have been made in the synchronization control of fractional-order chaotic systems, there are still many application foundations and technical problems to be studied to realize the practical application of fractional-order chaotic systems.

1.4.3 Application of Fractional-Order Chaotic Systems in Secure Communication In 1992, Heidari-Bateni et al. began to apply chaotic pseudo-random sequences in direct sequence spread spectrum (DSSS) systems. Due to the initial sensitivity and infinite period of chaotic signals, a considerable number of sequences can be generated by chaotic systems as spread spectrum codes, and they have an excellent correlation and balance. The current research results show that compared with the traditional m sequence and Gold sequence, the chaotic sequence has better performance in spread spectrum communication. The chaotic system, a pseudo-random sequence generator, is designed for the spread spectrum communication system under low voltage conditions. The pseudo-random binary sequence generated in a chaotic system has the following advantages when used as a spreading code. (1)

(2)

(3)

Arbitrary length and large quantity. Due to the sensitivity of the initial value of chaos, an infinite number of spread spectrum sequences can be generated by changing the initial value and parameters of the chaotic system. The classical spread spectrum codes have m sequence and Gold sequence. Although their performance meets the requirements, the disadvantage is that the number is limited, and bits are fixed. Especially in the face of large-capacity CDMA, its shortcomings are more apparent, and the ultra-large sequence capacity of chaotic sequences can better meet the requirements. The binary sequence generated by chaos has a very large period and good random performance, so the system is difficult to decipher and has good confidentiality. By changing the initial value or system parameters of the chaotic system, different chaotic sequences can be generated, which are easy to generate and replicate.

It can be seen that pseudo-random sequences based on chaotic systems have greatly improved the performance of spreading codes. Still, with the rapid improvement of computer technology, deciphering techniques have been more intensively studied and used. The traditional spreading sequences generated by integer-order and discrete chaotic maps are slowly showing their limitations. The disadvantages of low complexity and the ease to decipher are its shortcomings as a spread spectrum code, so further research is needed in this area. In recent years, the study of fractional-order chaotic systems has revealed that, unlike integer-order chaotic systems and discrete chaotic systems, they have many advantages in terms of application, in addition to the advantages mentioned above. The performance of the pseudo-random sequence

24

1 Introduction

generated by the fractional-order chaotic system as a spread spectrum code is worth studying. Therefore, applying the fractional-order chaotic system to the spread spectrum communication system and analyzing its influence on the performance of the communication system has practical application significance.

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26. Wang S, Yu Y (2012) Application of multistage homotopy-perturbation method for the solutions of the chaotic fractional order systems. Int J Nonlinear Sci 13(1):3–14 27. Rashidi MM, Erfani E (1991) The modified differential transform method for investigating nano boundary-layers over stretching surfaces. Int J Numer Meth Heat Fluid Flow 21(7):864–883 28. Wu GC, Lee EWM (2010) Fractional variational iteration method and its application. Phys Lett A 374(25):2506–2509 29. Diethelm K (1998) An algorithm for the numerical solution of differential equations of fractional order. Electron Trans Numer Anal 5(3):1–6 30. Sweilam NH, Assiri TA (2015) Non-standard Crank-Nicholson method for solving the variable order fractional cable equation. Appl Math Inf Sci 9(2):943–951 31. Charef A, Sun HH (1992) Fractal system as represented by singularity function. IEEE Trans Autom Control 37(9):1465–1470 32. Tavazoei MS, Haeri M (2007) Unreliability of frequency-domain approximation in recognizing chaos in fractional-order systems. IET Signal Proc 1(4):171–181 33. Tavazoei MS, Haeri M (2008) Limitations of frequency domain approximation for detecting chaos in fractional order systems. Nonlinear Anal 69(4):1299–1320 34. Arena F, Gatti G, Martra G et al (2012) Adaptation of differential transform method for the numeric-analytic solution of fractional-order Rössler chaotic and hyperchaotic systems. Abstr Appl Anal Special Issue (4):305–309 35. Cafagna D, Grassi G (2008) Bifurcation and chaos in the fractional-order Chen system via a time-domain approach. Int J Bifurcat Chaos 18(7):1845–1863 36. Cafagna D, Grassi G (2009) Hyperchaos in the fractional-order Rössler system with lowestorder. Int J Bifurcat Chaos 19(1):339–347 37. He S, Sun K, Wang H (2014) Solution of the fractional-order chaotic system based on Adomian decomposition algorithm and its complexity analysis. Acta Physica Sinica 63(3):030502 38. Tavazoei MS, Haeri M (2009) A proof for non existence of periodic solutions in time invariant fractional order systems. Automatica 45(8):1886–1890 39. Li CP, Gong ZQ, Qian D et al (2010) On the bound of the Lyapunov exponents for the fractional differential systems. Chaos 20(1):261–300 40. Jia H, Chen Z, Xue W (2013) Analysis and circuit implementation for the fractional-order Lorenz system. Acta Physica Sinica 62(14):140503 41. Wolf A, Swift JB, Swinney HL et al (1985) Determining Lyapunov exponents from a time series. Physica D 16(3):285–317 42. Ellner S, Gallant AR, McCaffrey D et al (1991) Convergence rates and data requirements for Jacobian-based estimates of Lyapunov exponents from data. Phys Lett A 153(6):357–363 43. Maus A, Sprott JC (2013) Evaluating Lyapunov exponent spectra with neural networks. Chaos Soliton Fractal 51(1):13–21 44. Caponetto R, Fazzino S (2013) An application of Adomian decomposition for analysis of fractional-order chaotic systems. Int J Bifurcat Chaos 23(3):1350050 45. Jia HY, Chen ZQ, Qi GY (2013) Topological horseshoe analysis and circuit realization for a fractional-order Lü system. Nonlinear Dyn 74(2):203–212 46. Li Y, Li J (2014) Stability analysis of fractional order systems based on T-S fuzzy model with the fractional order α: 0 0 , x(t) = H(t), t ∈ [−τ, 0]

(4.36)

where x(t) is the system variable and g(t) is a constant term in the system. Similarly, the delay system is described in three parts as follows: q

Dt0 x(t) = L(x(t), x(t − τ )) + N (x(t), x(t − τ )) + g(t),

(4.37)

where m ∈ N, m − 1 < q ≤ m. L(x(t), x(t − τ)) and N(x(t), x(t − τ)) represent linear and nonlinear terms of the system, respectively. Applying integrate on both sides of Eq. (4.37), which is derived as q

q

q

x = Jt0 L(x, xτ ) + Jt0 N (x, xτ ) + Jt0 g + ,

(4.38)

where F is calculated according to Eq. (4.3), xτ = x(t − τ ) and bk are the initial conditions. x i is calculated by  x0 =  x = i

q

q

Jt0 g + , t > 0 , H(t), t ≤ 0

q

Jt0 L(xi−1 , xτi−1 ) + Jt0 Ai−1 (x0 , · · · , xi−1 , xτ0 , · · · , xτi−1 ), t > 0 0, t ≤ 0

(4.39) ,

(4.40)

References

59

in which i = 1, 2, …, ∞. Then the system analytic solution is x(t) =

∞ 

xi .

(4.41)

i=0

The calculation formula of the time-delay nonlinear term N(x, xτ ) in the time-delay system is N (x, xτ ) =

∞ 

An−1 (x0 , · · · , xn−1 , xτ0 , · · · , xτn−1 ),

n=0

1 dn A = n! dλn n



n  k=0

λ x

k k

 n 

(4.42)

 λk xτk

k=0

.

(4.43)

λ=0

Similarly, x i can be derived using the following formula: ⎧ 0 x = ⎪ ⎪ ⎪ 1 q q 0 0 ⎪ 0 ⎪ ⎪ ⎪ x = Jt0 Lx + Jt0 A (x ) ⎪ ⎪ q q 1 0 2 1 1 ⎪ ⎪ ⎨ x = Jt0 Lx + Jt0 A (x , x ) . .. ⎪ . ⎪ ⎪ ⎪ ⎪ q q ⎪ ⎪ xi = Jt0 Lxi−1 + Jt0 Ai−1 (x0 , x1 , · · · , xi−1 ) ⎪ ⎪ ⎪ ⎪ ⎩ .. .

(4.44)

The above is the modified Adomian decomposition method for solving fractionalorder time-delay chaotic systems.

References 1. Adomian G (1984) A new approach to nonlinear partial differential equations. J Math Anal Appl 102(2):420–434 2. Cafagna D, Grassi G (2008) Bifurcation and chaos in the fractional-order Chen system via a time-domain approach. Int J Bifurc Chaos 18(7):1845–1863 3. Cafagna D, Grassi G (2009) Hyperchaos in the fractional-order Rössler system with lowestorder. Int J Bifurc Chaos 19(1):339–347 4. He SB, Sun KH, Wang HH (2014) Solution of the fractional-order chaotic system based on Adomian decomposition algorithm and its complexity analysis. Acta Phys Sin 63(3):030502 5. Caponetto R, Fazzino S (2013) An application of Adomian decomposition for analysis of fractional-order chaotic systems. Int J Bifurc Chaos 23(3):1350050 6. Wang HH, Sun KH, He SB (2015) Characteristic analysis and DSP realization of fractionalorder simplified Lorenz system based on Adomian decomposition method. Int J Bifurc Chaos 25(06):1550085

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7. Wang HH, Sun KH, He SB (2015) Dynamic analysis and implementation of a digital signal processor of a fractional-order Lorenz-Stenflo system based on the Adomian decomposition method. Phys Scr 90(1):015206 8. Cherruault Y, Adomian G (1993) Decomposition methods: a new proof of convergence. Math Comput Model 18(12):103–106 9. Adomian G (1984) On the convergence region for decomposition solutions. J Comput Appl Math 11(3):379–380 10. Abbaoui K, Cherruault Y (1994) Convergence of Adomian’s method applied to nonlinear equations. Math Comput Model 20(9):69–73 11. Cherruault Y, Adomian G, Abbaoui K et al (1995) Further remarks on convergence of decomposition method. Int J Biomed Comput 38(1):89–93 12. Singla RK, Das R (2015) Adomian decomposition method for a stepped fin with all temperaturedependent modes of heat transfer. Int J Heat Mass Transf 82:447–459 13. Kang SM, Nazeer W, Tanveer M et al (2015) Improvements in Newton-Rapshon method for nonlinear equations using modified Adomian decomposition method. Int J Math Anal 9(39):1910–1928 14. Mohamed AS, Mahmoud RA (2016) Picard, Adomian and predictor–corrector methods for an initial value problem of arbitrary (fractional) orders differential equation. J Egyptian Math Soc 24(2):165–170 15. Grigorenko I, Grigorenko E (2003) Chaotic dynamics of the fractional Lorenz system. Phys Rev Lett 91(3):034101 16. Grigorenko I, Grigorenko E (2003) Erratum: chaotic dynamics of the fractional Lorenz system. Phys Rev Lett 91:034101. Phys Rev Lett 96(19):199902 17. Jia HY, Chen ZQ, Xue W (2013) Analysis and circuit implementation for the fractional-order Lorenz system. Acta Phys Sin 62(14):140503 18. Sun KH, Sprott JC (2009) Dynamics of a simplified Lorenz system. Int J Bifurc Chaos 19(4):1357–1366

Chapter 5

Performance Comparison of Solution Algorithms

The numerical solution algorithms of fractional-order chaotic system are the basis of theoretical analysis and practical applications of fractional-order chaotic system. At present, exploring and optimizing the numerical solution of fractional-order differential equations are a hot spot in the research of fractional-order chaos theory [1–7]. Because different algorithms have different principles and truncation errors, the characteristics of fractional-order chaotic systems are also different. In fact, the solutions obtained by different algorithms are approximates of fractional-order chaotic systems [8–10]. Therefore, this chapter firstly proves that the fractional-order differential equation has a unique solution and then studies the convergence of different algorithms, the influence on the minimum order of chaos, and the dynamic characteristics when different algorithms are employed.

5.1 Uniqueness of Solutions of Fractional-Order Differential Equations Similar to integer-order differential equations, fractional-order differential equations have a unique solution. If the differential operator in a fractional-order differential equation is of order q, the fractional-order differential equation is called a q-order differential equation. The fractional-order linear differential equation with constant coefficients is q

q

q

an Dt0n y(t) + · · · + a1 Dt01 y(t) + a0 Dt00 y(t) = f (t),

(5.1)

where qn > qn−1 > · · · > q0 > 0, ai (i = 0, 1, · · · , n) is a constant, and the initial conditions are q −k−1

[0 D t n

y(t)]t=0 = bk ,

© Science Press 2022 K. Sun et al., Solution and Characteristic Analysis of Fractional-order Chaotic Systems, https://doi.org/10.1007/978-981-19-3273-1_5

(5.2) 61

62

5 Performance Comparison of Solution Algorithms

where k = 0, 1, 2, …, n − 1. For fractional-order differential equations, especially linear differential equations, the existence and uniqueness of solutions under initial conditions can be proved by the Laplace transform. Theorem 5.1 [11] If the function f (t) is absolutely integrable in (0, T) and satisfies the initial condition (5.2), the equation q

Dt y(t) = f (t)

(5.3)

has a unique solution in this interval. Proof The Laplace transform is performed on both sides of the above formula s qn Y (s) −

n−1 

q −k−1

s0k Dt n

y(t)|t=0 = F(s).

(5.4)

k=0

Substitute the initial conditions into the above formula and obtain the following result: Y (s) = s −qn F(s) +

n−1 

bk s −k−1 .

(5.5)

k=0

The inverse Laplace transform is performed on the above formula and can be obtained y(t) =

1 (qn )



t

(t − τ )qn −1 f (τ )dτ +

0

n−1  k=0

bk tk. (k + 1)

(5.6)

According to the  definition of fractional-order calculus and the properties of Gamma function 1 (−m) = 0, (m = 0, 1, 2, …), it can be obtained ⎧ ⎪ qn ⎪ D ⎪ t ⎨

tk = 0, (k + 1)

(k) ⎪ tk ⎪ ⎪ ⎩ = 1, (k + 1)

k = 0, 1, 2, . . . n − 1.

(5.7)

Substitute Eqs. (5.6) and (5.7) into (5.3), and we can get  q Dt n y(t)

=

q Dt n

1 (qn )

 0

t

qn −1

(t − τ )

f (τ )dτ +

n−1  k=0

bk tk (k + 1)



5.2 Accuracy and Performance q

−qn

= Dt n Dt

63

f (t) +

n−1 

q

bk Dt n

k=0

tk . (k + 1)

= f (t)

(5.8)

It can be seen that Eq. (5.6) is the solution of the original equation, that is, the existence is proved. Theorem 5.2 [11] If the function is absolutely integrable within (0, T) and satisfies the initial condition (5.2), there is a unique solution to Eq. (5.1) in this interval. Proof Assuming that y1 (t) and y2 (t) are the solution of Eq. (5.3), then z(t) = y1 (t) − y2 (t) must also satisfy Eq. (5.3), and the Laplace transform of z(t) is Z(s) = 0, so it is almost true in the interval (0, T ), that is, the uniqueness is proved. Similarly, it can be proved that Eqs. (5.1) and (5.2) have unique solutions y(t) in the interval (0, T ). It can be seen that the fractional-order chaotic system also has a unique solution, that is, the solutions obtained by different algorithms are the approximations of the “unique solution”, but the problem is which of the commonly used solution algorithms for fractional-order chaotic systems is better. In practical application, how to choose a proper solution algorithm?

5.2 Accuracy and Performance Next, the three numerical solution algorithms of fractional-order chaotic system discussed above are compared and analyzed from the calculation accuracy, calculation speed, time complexity, and space complexity of the algorithm, so as to provide the theoretical and experimental basis for the selection of solution algorithms of fractional-order order chaotic system. Firstly, the calculation accuracy of the three algorithms is analyzed. When q = 1, the system is an integer-order chaotic system, which is a special case of fractional-order chaotic system. The Runge–Kutta algorithm and the Euler algorithm are commonly used in solving integer-order chaotic systems, among which the Runge–Kutta algorithm has higher accuracy. Here, the Adomian algorithm, predictor–corrector algorithm, and the Runge–Kutta algorithm are used to solve the following initial value problem:

q

Dt0 y(t) = y y(0) = 1

,

(5.9)

where q = 1, the exact solution of the system is y = et . Three algorithms are used to solve the system (5.9). The results are recorded as, yn(A) , yn(R) , and yn(Y ) . The solution

64

5 Performance Comparison of Solution Algorithms

Fig. 5.1 Error curves under different algorithms. a Adomian algorithm, b Runge–Kutta algorithm, and c predictor–corrector algorithm

process is shown in Eqs. (5.10), (5.11), and (5.12), respectively. Compared with the exact solution, the calculation error is shown in Fig. 5.1. It can be seen that the cumulative errors of the three algorithms increase exponentially with the increase of time. In terms of error size, the error of the predictor–corrector algorithm is the largest, Runge–Kutta is three orders of magnitude smaller, and Adomian is three orders of magnitude smaller than Runge–Kutta. It can be seen that the numerical solution of the Adomian decomposition algorithm is more accurate.

(t − t0 )q (t − t0 )8q , y (A) = y0 1 + + ··· + (q + 1) (8q + 1) ⎧ K 1 = yn(R) ⎪ ⎪ ⎪ ⎪ ⎪ h ⎪ ⎪ ⎪ K 2 = yn(R) + K 1 ⎪ ⎪ 2 ⎪ ⎨ h , K 3 = yn(R) + K 2 ⎪ 2 ⎪ ⎪ ⎪ ⎪ K 4 = yn(R) + h K 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ y (R) = y (R) + h (K 1 + 2K 2 + 2K 3 + K 4 ) n n+1 6

(5.10)

(5.11)

⎧ n  ⎪ hq ⎪ p (Y ) ) ⎪ y = y + (y + α j,n+1 y (Y ⎪ 0 n+1 n+1 j ) ⎪ ⎪ (q + 2) ⎪ j=0 ⎪ ⎪ ⎪

⎪ q ⎪ ⎪ n − (n − 2)(n + 1)q j = 0 ⎪ ⎨ α j,n+1 = (n − j + 2)q+1 + (n − j)q+1 − 2(n − j + 1)q+1 1 ≤ j ≤ n . ⎪ ⎪ n ⎪ ⎪ 1  ⎪ p ) ⎪ = y + b j,n+1 y (Y y ⎪ 0 n+1 j ⎪ ⎪ (q) ⎪ j=0 ⎪ ⎪ ⎪ ⎪ hq ⎩ b j,n+1 = q ((n − j + 1)q − (n − j)q ) 0 ≤ j ≤ n (5.12)

5.2 Accuracy and Performance

65

From the perspective of truncation error, the truncation error of the predictor– corrector algorithm is e=

max

j=0,1,...,N

|x(t j ) − x h (t j )| = o(h p ),

(5.13)

where p = min(2,1 + q), when q = 1, e = o(h2 ), and the truncation error of the Runge–Kutta algorithm is e=

max

j=0,1,...,N

|x(t j ) − x h (t j )| = o(h 5 ).

(5.14)

It can be seen that its solution accuracy is higher. The convergence characteristic of the Adomian algorithm is similar to the Taylor series, and its convergences are fast. Usually, a more accurate solution can be obtained by taking the first several terms, and its error will converge to [12] e≤k

MP , P!

(5.15)

where P is the  number of iteration items. For instance, in Eq. (5.10), P = 8, ∞ ||N (P) (0)|| ≤ k, i=0 |u i | ≤ M. The time complexity and space complexity of the Adomian decomposition algorithm and predictor–corrector algorithm are compared and analyzed below. The performance and characteristics of the three algorithms are shown in Table 5.1. It can be seen that each iteration of the predictor–corrector algorithm requires all previous historical data, so it requires more operation time and memory space than the other two algorithms, and its time and space complexity is O(n2 ). For the Adomian algorithm, when the iterative term is infinite, the obtained solution is an accurate solution. In practical calculation, taking the finite term can achieve high accuracy. Ref. Table 5.1 Performance comparison of three algorithms for fractional-order order chaotic systems Frequency-domain algorithm

Predictor–corrector algorithm

Adomian decomposition algorithm

Algorithm type

Frequency-domain

Time-domain

Time-domain

time complexity

O(n)

O(n2 )

O(n)

space complexity

O(n)

O(n2 )

O(n)

accuracy

≤3 dB

O(hp ), e = min(2,1 + e ≤ k·M P /P! [12] q)

fractional-order derivative definition

R-L definition

Caputo definition

Caputo definition

application

Theoretical analysis and analog circuits

Theoretical analysis

Theoretical analysis and digital circuits

66

5 Performance Comparison of Solution Algorithms

[13] compares the predictor–corrector algorithm with the Adomian algorithm. The research shows that the error of the Adomian decomposition method is smaller and better than that of the predictor–corrector algorithm when taking the finite term. The accuracy of the frequency-domain algorithm can be controlled within 3 dB, which can achieve a good approximation with the actual system within the expected frequency band, but there are large errors in both high- and low-frequency bands. Of course, by improving the transfer function of the fractional-order integral operator, the solution can be more accurately approached to the original system to a certain extent [14]. From the application, the frequency-domain algorithm can be used as the theoretical basis for the implementation of fractional-order order analog circuits; the predictor–corrector algorithm is mainly used for the theoretical analysis of fractional-order order chaotic systems. Because the iteration of the Adomian algorithm is only related to the data of the previous step, the calculation speed is similar to the Runge–Kutta algorithm, and the amount of calculation is slightly larger than the Runge–Kutta algorithm, which is conducive to the implementation of digital circuits, for instance, DSP-based digital circuit design, of fractional-order chaotic systems. Therefore, the digital circuit designs of fractional-order chaotic systems and the pseudo-random sequence generator based on fractional-order chaotic systems deserve further research [15]. The time complexity and space complexity of the Adomian decomposition algorithm and predictor–corrector algorithm are shown in Table 5.2. It can be seen that the Adomian algorithm is superior to the predictor–corrector algorithm in both time complexity and space complexity. Table 5.2 shows the time required to obtain different length sequences when solving fractional-order simplified Lorenz system in the last three lines (the computer CPU frequency is different, and the calculation results will be different. Here, the computer: Intel Dual E2180 2.0 GHz). It can be seen that the time growth of the predictor–corrector algorithm is much faster than that of the Adomian algorithm. Compared with the Adomian algorithm and predictor–corrector algorithm, the Adomian algorithm is a better choice to solve fractional-order chaotic systems. Table 5.2 Comparison of fractional-order chaotic numerical algorithms

Adomian decomposition algorithm

Predictor–corrector algorithm

Time complexity

O(n), faster

O(n2 ), slower

Space complexity

O(1), smaller

O(n), bigger

N = 1000

0.9701 s

2.0154 s

N = 2000

1.8403 s

7.7051 s

N = 5000

4.5162 s

51.3995 s

5.3 Influence of Different Factors on the Minimum Order of Chaos

67

5.3 Influence of Different Factors on the Minimum Order of Chaos 5.3.1 Minimum Order of Chaos with Different Algorithms The minimum order of chaos generated by the system obtained from different algorithms is compared and studied, and the application of the fractional-order stability principle in the fractional-order chaotic system is discussed. The principle of fractional-order stability was first proposed by Matignon [16], and its applicable object is a fractional-order linear system. This theorem gives the stability criterion of the fractional-order linear system. Its main conclusion is shown in Lemma 5.1. When q is less than the critical value in the lemma, the system is stable. On the basis of Lemma 5.1, Tavazoei [17] proposed Lemma 5.2 based on the stability point of fractional-order system and applied it to the stability judgment of fractionalorder chaotic system. When studying fractional-order synchronization, Hu et al. [17] proposed a new stability theorem for fractional-order system based on Lemma 5.1, which is shown in Lemma 5.3. q

Lemma 5.1 [16] For fractional-order linear systemsDt0 X = AX, (0 < q ≤ 1, X ∈ Rn ), where q < (2/π)arg|eig(A)|, the system is stable. q

Lemma 5.2 [17] For fractional-order nonlinear autonomous systems Dt0 X = f (X), (0 < q ≤ 1, X ∈ Rn , f ∈ C 1 ), the necessary condition for chaos is q > (2/π)arg|eig(A(X * ))|, where A(X) is the Jacobian matrix of the system and X * is the equilibrium point of the system. q

Lemma 5.3 [18] For fractional-order nonlinear autonomous systems Dt0 X = f (X) = A(X)X, (0 < q ≤ 1, X ∈ Rn , f ∈ C 1 ), the necessary condition for chaos is q > (2/π)arg|eig(A(X * ))|, where X * is the equilibrium point of the system. The calculation results of the fractional-order simplified Lorenz chaotic system (C = 5) and the fractional-order Chen chaotic system (a = 35, b = 3, c = 28) and the results in the literature are shown in Table 5.3. According to Lemma 5.2, for the simplified Lorenz system, the minimum order q of generating chaos is 0.9302, while the minimum order of the fractional-order Chen system is 0.8244. It can be seen from Table 5.3 that the predictor–corrector algorithm meets this theorem, while the Adomian algorithm and frequency-domain algorithm can generate chaos at a lower order, so it does not meet Lemma 5.2. The equilibrium Lorenz system are (0, 0, 0) and   √ √ points of the fractional-order ± b(c + d), ± b(c + d), c + d , and the Jacobian matrix is shown in Eq. (5.16). When d = 25, the eigenvalues corresponding to the equilibrium point (0, 0, 0) are λ1 = −45.6608, λ2 = −30.6608,  √ λ1 =√−3.0000,and the eigenvalues corresponding to the equilibrium point ± 105, ± 105, 35 are λ1 = −25.2415, λ2 = 3.6207 + 17.8795i, and λ3 = 3.6207 + 17.8795i. According to Lemma 5.2, the minimum order q of chaos in the system is 0.8726. It can be seen from Figs. 5.3 and 5.4

68 Table 5.3 Comparison of minimum order of fractional-order system obtained by different algorithms

5 Performance Comparison of Solution Algorithms Algorithm/method

Fractional-order simplified Lorenz system

Fractional-order Chen system

Lemma 2

0.9302

0.8244

Predictor–corrector algorithm

0.93 [21]

0.83 [22]

Frequency-domain algorithm

Not researched at low level

Chaos under q = 0.2 [18]

Adomian algorithm

0.45 [23]

Chaos under q = 0.08 [22]

Fig. 5.3 Phase diagram of the fractional-order Lorenz system solved by predictor–corrector algorithm (q = 0.87). a x 1 − x 2 plane, b x 1 − x 3 plane, and c x 2 − x 3 plane

Fig. 5.4 Waveform in time domain of the fractional-order Lorenz system solved by predictor– corrector algorithm (q = 0.86). a x 1 sequence, b x 2 sequence, and c x 3 sequence

that the conclusion obtained by using the predictor–corrector algorithm meets the requirements of Lemma 5.2. Grigorenko et al. [19] analyzed its chaotic dynamic characteristics and showed that the system can generate chaos at 2.91 and 2.96. Recently, Jia et al. [20] used the frequency-domain algorithm to analyze the dynamic characteristics when order q is 0.7, 0.8, and 0.9, respectively. Numerical experiments show that the system still has chaos at lower order. In Chap. 4, the minimum order of chaos generated by the system based on the Adomian decomposition algorithm is q = 0.813. Obviously, the minimum order is less than the minimum order obtained by Lemma 5.2.

5.3 Influence of Different Factors on the Minimum Order of Chaos

69

⎡

Jsys

⎤ −a a 0 = ⎣ c − x3 d −x1 ⎦. x2 x1 −b

(5.16)

From the above analysis and calculation results, it can be seen that the minimum order of chaos generated by the system obtained by the predictor–corrector algorithm meets Lemma 5.2, while the Adomian decomposition algorithm and frequencydomain algorithm do not meet the lemma. The reasons are preliminarily analyzed as follows. Lemma 5.1 is only applicable to fractional-order linear systems and has been strictly mathematically proved. At present, this lemma is mainly used in fractionalorder synchronous control since most error systems are fractional-order linear systems; Lemma 5.2 was proposed in reference [23], but it was not proved. The author believes that fractional-order calculus has a memory function. Accordingly, fractional-order chaotic system is more stable than its corresponding integer chaotic system. And Lemma 5.1 can be used after linearizing the system by using an equilibrium point; Lemma 5.3 is proved by the Lyapunov stability theorem, but the research of Li Lixiang et al. [24] shows that the order of A(X) time-varying matrix in Lemma 5.3 is not necessarily stable, that is, Lemma 5.3 is not always true. Similarly, the Jacobian matrix in Lemma 5.2 is also a time-varying matrix, and the matrix is a Jacobian matrix in the case of integer order. Whether it is suitable for the case of fractional order, it is worthy of further study. In fact, the stability analysis of fractional-order chaotic system is much more complex than that of integer-order chaotic system, which needs further research [24]. From the current algorithm analysis results, although the frequency-domain algorithm and Adomian algorithm do not meet Lemma 5.2, these two algorithms have been recognized by the academic community, and the calculation results have reference value; on the other hand, the frequency-domain algorithm and Adomian algorithm can obtain chaos at lower order and further expand the parameter space of fractional-order chaotic system. When the system is applied to secure communication and information encryption, the key space is larger and the security is higher.

5.3.2 Simulation Step Size on the Minimum Order of Chaos The calculation accuracy is closely related to the simulation step size. When analyzing the minimum order of chaos in a fractional-order system, it is necessary to give the corresponding parameters and simulation time step, otherwise, the conclusion has no practical significance. For example, for the Adomian decomposition algorithm, the simulation time step h will affect the dynamic characteristics of the system. For example, for the fractional-order Lorenz system, when d = 25 and h = 0.01, the minimum order of chaos is q = 0.813; when h is 0.001 and 0.0001, respectively, the bifurcation results of the fractional-order Lorenz system are shown

70

5 Performance Comparison of Solution Algorithms

in Fig. 5.5. It can be seen that when h = 0.001, the minimum order of chaos generated by the system is q = 0.505, and when h = 0.0001, the minimum order of chaos generated by the system is q = 0.402. It can be seen that, based on the Adomian decomposition algorithm, the smaller h is, the smaller the minimum order of chaos is. Let the iteration step size be h = 0.001, intercept the first four terms of Adomian polynomial, and use the same method to reconstruct the maximum Lyapunov exponent diagram, as shown in Fig. 5.6. It can be seen that for any q, the chaotic region of the system is larger than that when h = 0.01, and the same phenomenon also exists for any c. It can be seen that, when the Adomian decomposition algorithm is used to solve the fractional-order chaotic system, the iteration step h has an obvious impact on the characteristics of the system. The smaller h, the larger the chaotic range of the system.

Fig. 5.5 Bifurcation diagram of the fractional-order Lorenz system with different h. a h = 0.001 and b h = 0.0001

Fig. 5.6 Maximum Lyapunov exponent of the fractional-order order simplified Lorenz system when h = 0.001

5.3 Influence of Different Factors on the Minimum Order of Chaos

71

Generally, the smaller the time step of the simulation, the more accurate the solution obtained by the system. Of course, in practical application, the value of h should not be too small, otherwise, after a certain number of iterations, the chaotic sequence will not change a lot, which is not conducive to practical application. If h is too large, the display of system dynamic characteristics is incomplete. Therefore, h = 0.01 is more appropriate, which is not only conducive to practical application, but also maintains the characteristics of the system.

5.3.3 Lowest Order of Chaos with Different Derivative Orders In Ref. [25], the fractional-order Liu system is studied based on the predictor– corrector algorithm. The lowest order of the system is 2.76 in case the system equations are under commensurate orders, and 2.60 in the incommensurate case. Obviously, the lowest order is different in the case of whether the systems are under commensurate orders. And the lowest order under the incommensurate case is less than the lowest order under the commensurate case. In order to further prove this result, the fractional-order simplified Lorenz system is solved based on the Adomian decomposition algorithm, and whether the lowest order of the system under incommensurate case still follows this law is analyzed. According to the analysis above, when c = 5 and h = 0.01, the minimum value of q is 0.595 in the case of the commensurate case of the system equation. Therefore, set q1 = q2 = 0.595 to obtain the maximum Lyapunov exponent of the system varying with q3 , as shown in Fig. 5.7. It can be seen from the figure that when q3 > 0.436, the maximum Lyapunov exponent of the system is greater than 0. The (p-s) trajectories at q3 = 0.436 (chaos) and q3 = 0.435 (convergence) are shown in Fig. 5.8. At this time, the lowest order in the case of the incommensurate case is 1.626, which is smaller than the lowest order value of 1.785 in the commensurate case. Therefore, it can be concluded that the lowest order of fractional-order chaotic system under different orders is lower than that under the commensurate case. 10 5 0

MLE

Fig. 5.7 Maximum Lyapunov exponent of the system at incommensurate case where c = 5, h = 0, and q1 = q2 = 0.595 (MLE)

-5

-10 -15

0.3

0.4

0.5

0.6 q3

0.7

0.8

0.9

1

72

5 Performance Comparison of Solution Algorithms

Fig. 5.8 The (p–s) trajectories at q3 = 0.436 (chaos) and q3 = 0.435 (convergence) when the system is under the incommensurate case. a q3 = 0.435 and b q3 = 0. 436

5.3.4 System Parameters on the Lowest Order of Chaos In Ref. [21], it is found that the system parameter c is an important bifurcation factor in the fractional-order simplified Lorenz system, and its small change will sometimes significantly affect the dynamic characteristics of the system. In order to observe the influence of system parameter C on the lowest order, set the step size h = 0.01 here, and the initial value remains unchanged. When the system equation is of the commensurate case, it is solved based on the Adomian decomposition algorithm. The value of c is increased from 3 to 7, and the interval is 1. When c takes different values, the maximum Lyapunov exponent of the system with order q is shown in Fig. 5.9a. The lowest orders of the system corresponding to different c values are shown in detail in Fig. 5.9b. When c = 3, 4, 5, 6, and 7, the lowest orders are 2.001, 1.893, 1.785, 1.667, and 1.531, respectively. It can be seen that with the increase of

Fig. 5.9 Effect of system parameter c on the lowest order of chaos in the system where h = 0.01. a The maximum Lyapunov exponent under the different value of c (MLE) and b The lowest order under the different value of c

5.3 Influence of Different Factors on the Minimum Order of Chaos

73

c, the lowest order will gradually decrease, which means that the larger the value of system parameter c, the larger the chaotic behavior of the fractional-order order system will be.

5.3.5 The Optimal Minimum Order of Chaos Based on the conclusions of the previous analysis, the Adomian decomposition algorithm is selected as the solution algorithm, the value of system parameter c is set as 7, and the iteration step length h = 0.001. Under the condition of the same order, the maximum Lyapunov exponent of the system is obtained, as shown in Fig. 5.10a. At this time, the lowest order of the system is 1.182 (q = 0.394). Then, set q1 = q2 = 0.394 to obtain the maximum Lyapunov exponent of the system under different orders, as shown in Fig. 5.10b. It can be seen from the figure that when the system is in chaos, the minimum value of q3 is 0.348, so the minimum order of the system is 1.136 (0.394 + 0.394 + 0.348). This result is lower than the results of some previous analyses and lower than the lowest order that the fractional-order simplified Lorenz system can produce chaos in the reported literature. In Fig. 5.11, the correctness of the lowest order is further verified by the 0–1 test. Three algorithms for solving fractional-order chaotic systems have their own advantages. In contrast, the Adomian algorithm converges faster and is suitable for digital circuit implementation. The characteristics of fractional-order chaotic system, or the lowest order of the system in chaos, are closely related to the solution method, the order of the system equation, the system parameters, and the size of the iteration step. The simulation results show that using the Adomian decomposition algorithm to solve the system will produce a lower minimum order than the predictor–corrector algorithm. The chaotic system with incommensurate order is smaller than the lowest order with commensurate order. For the fractional-order simplified Lorenz system,

Fig. 5.10 The maximum Lyapunov exponent of the fractional-order simplified Lorenz system (MLE) where c = 7 and h = 0. 001. a In case of commensurate orders and b q1 = q2 = 0.394, In case of incommensurate orders

74

5 Performance Comparison of Solution Algorithms

Fig. 5.11 The (p–s) trajectories when q3 at different values where c = 7, h = 0.001, and q1 = q2 = 0.394. a q3 = 0.347 and b q3 = 0.348

within the optional range of system parameter c, a larger c value and smaller iteration step size will lower the lowest order of chaos. By synthesizing the previous analysis results, it is obtained that the minimum order of chaos generated by the fractionalorder simplified Lorenz system is 1.136, which is lower than the reported minimum order of chaos generated by the fractional-order simplified Lorenz system. According to the analysis results, a conjecture is proposed: the lowest order of fractional-order chaotic system under the condition of exact solution is lower than that under numerical analysis. Therefore, when reporting the minimum order of chaos in fractionalorder order system, the calculation conditions must be elaborated clearly, otherwise, it has no practical significance.

References 1. Adomian G (1984) A new approach to nonlinear partial differential equations. J Math Anal Appl 102(2):420–434 2. Wang S, Yu Y (2012) Application of multistage homotopy-perturbation method for the solutions of the chaotic fractional order systems. Int J Nonlinear Sci 13(1):3–14 3. Rashidi MM, Erfani E (1991) The modified differential transformation method for investigating nano boundary-layers over stretching surfaces. Int J Numer Meth Heat Fluid Flow 21(7):864– 883 4. Wu GC, Lee EWM (2010) fractional-order variational iteration method and its application. Phys Lett A 374(25):2506–2509 5. Diethelm K (1998) An algorithm for the numerical solution of differential equations of fractional order. Electron Trans Numer Anal Etna 5(3):1–6 6. Sweilam NH, Assiri TA (2015) Non-standard Crank-Nicholson method for solving the variable order fractional cable equation. Appl Math Inf Sci 9(2):943–951 7. Charef A, Sun HH (1992) Fractal system as represented by singularity function. IEEE Trans Autom Control 37(9):1465–1470 8. Wang HH, Sun KH, He SB (2015) Characteristic analysis and DSP realization of fractionalorder simplified Lorenz system based on Adomian decomposition method. Int J Bifurc Chaos 25(06):1550085

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9. Wang Y, Sun K, He S et al (2014) Dynamics of fractional-order sinusoidally forced simplified Lorenz system and its synchronization. Eur Phys J Spec Topics 223(8):1591–1600 10. Sun K, Yang J, Ding J et al (2010) Circuit design and implementation of Lorenz chaotic system with one parameter. Acta Phys Sin 59(12):527–533 11. Zhao CN (2011) Analysis and design of fractional order systems. National Defense Industry Press, Beijing 12. Abbaoui K, Cherruault Y (1994) Convergence of Adomian’s method applied to differential equations. Comput Math Appl 28(5):103–109 13. He SB, Sun KH, Wang HH (2014) Solution of the fractional-order chaotic system based on Adomian decomposition algorithm and its complexity analysis. Acta Phys Sin 63(3):030502 14. Wang M, Sun GH, Wei YL (2011) Limitations of frequency domain approximation in the calculation of fractional order chaotic systems. J Harbin Inst Technol 43(5):8–12 15. Runkin A, Soto J, Nechvatal, J (2001) A statistical test suite for random and pseudorandom number generators for cryptographic applications [OL]. NIST: Special Publication 800-22. http://nvlpubs.nist.gov/nistpubs/Legacy/SP/nistspecialpublication800-22.pdf 16. Matignon D (1997) Stability results for fractional-order differential equations with applications to control processing. Comput Eng Syst Appl 2(1):963–968 17. Tavazoei MS, Haeri M (2007) Unreliability of frequency-domain approximation in recognizing chaos in fractional-order systems. IET Signal Proc 1(4):171–181 18. Hu JB, Han Y, Zhao LD (2009) Adaptive synchronization between different fractional-order hyperchaotic systems with uncertain parameters. Acta Phys Sin 58(3):1441–1445 19. Grigorenko I, Grigorenko E (2003) Chaotic dynamics of the fractional Lorenz system. Phys Rev Lett 91(3):034101 20. Jia HY, Chen ZQ, Xue W (2013) Analysis and circuit implementation for the fractional-order Lorenz system. Acta Phys Sin 62(14):140503 21. Sun KH, Wang X, Sprott JC (2010) Bifurcations and chaos in fractional-order simplified Lorenz system. Int J Bifurc Chaos 20(4):1209–1219 22. Cafagna D, Grassi G (2008) Bifurcation and chaos in the fractional-order Chen system via a time-domain approach. Int J Bifurc Chaos 18(7):1845–1863 23. Xie T, Liu Y, Tang J (2014) Breaking a novel image fusion encryption algorithm based on DNA sequence operation and hyper-chaotic system. Optik 125(24):7166–7169 24. Li LX, Peng HM, Luo Q et al (2013) Problem and analysis of stability decidable theory for a class of fractional order nonlinear system. Acta Phys Sin 62(2):020502 25. Daftardar-Gejji V, Bhalekar S (2010) Chaos in fractional ordered Liu system. Comput Math Appl 59(3):1117–1127

Chapter 6

Dynamics of Fractional-Order Chaotic Systems

6.1 Lyapunov Exponent Spectrum Calculation Algorithm The Lyapunov exponent is the average change rate of the exponential separation or convergence of two adjacent orbits in phase space over time, and is used to characterize the extreme sensitivity of the system to initial values as it evolves over time. A positive Lyapunov exponent indicates that the phase volume of the system is expanding and flooding, resulting in the original adjacent orbits in the attractor becoming more and more irrelevant, and the initial value sensitivity of the system is high. Definition 6.1 [1] The Lyapunov exponent is the average change rate of the exponential separation or coalescence of two adjacent orbits in phase space over time, and is calculated as Ly =

M 1  D(tk ) ln tm − t0 k=1 D(tk−1 )

(6.1)

where M is the total iteration number, and D(t k ) denotes the distance between the two closest points at moment t k . The Lyapunov exponent is a quantitative indicator used to distinguish the complexity of sequences in chaotic attractors, and a positive Lyapunov exponent is a necessary condition for the existence of chaos in a nonlinear system. When the dimension of the system is greater than one, the system is generally described by the set of Lyapunov exponents, i.e., the Lyapunov exponent spectrum. For a multidimensional nonlinear dissipative system, since the system is stable overall, the sum of the Lyapunov exponent spectrum of the system should be less than zero, indicating that the motion of the system in phase space is contracting and stable overall. There are positive and negative values in the Lyapunov exponent spectrum. The negative ones indicate that the phase volume of the system in this direction is contracting and the evolution in this direction is stable, while the positive © Science Press 2022 K. Sun et al., Solution and Characteristic Analysis of Fractional-order Chaotic Systems, https://doi.org/10.1007/978-981-19-3273-1_6

77

78

6 Dynamics of Fractional-Order Chaotic Systems

ones indicate that the motion of the system in this direction is unstable and the phase volume is expanding and flooding. It results in that the original adjacent orbits in the attractor become uncorrelated after evolution, so that it is difficult to predict the state of the small deviation of the initial state after a long evolution, which reveals the initial value sensitivity of chaos. So the system is chaotic if there is a positive value in the Lyapunov exponent spectrum of the multidimensional system. In applications, the maximum Lyapunov exponent Ly1 becomes an important indicator to determine whether a system is chaotic or not. For a continuous chaotic system, let the Lyapunov exponent spectra of the system be Ly1 , Ly2 ,…, Lyn (in descending order), and the necessary conditions for the system to have chaotic attractors are as follows. (1) (2) (3)

There is at least one positive Lyapunov exponent Lyi > 0. There is at least one Lyapunov exponent Lyi = 0. Ly1 + Ly2 + … + Lyn < 0.

According to Chap. 4, the Adomian decomposition algorithm is used to obtain the iterative equation for a fractional-order chaotic system as [2] ⎧ F1Ado (x(m)) ⎪ ⎪ ⎪ ⎪ ⎨ F2 Ado (x(m)) x(m + 1) = F Ado (x(m)) = .. ⎪ ⎪ . ⎪ ⎪ ⎩ Fn Ado (x(m))

(6.2)

The result of each step of the equation is related to the result of the previous step only, so its Jacobian matrix is ⎡ ∂F Jsys

⎢ ⎢ =⎢ ⎢ ⎣

1Ado ∂F1Ado ∂ x1 ∂ x2 ∂F2 Ado ∂F2 Ado ∂ x1 ∂ x2

.. .

.. .

∂Fn Ado ∂Fn Ado ∂ x1 ∂ xn

··· ··· .. . ···

∂F1Ado ∂ xn ∂F2 Ado ∂ xn

.. .

∂Fn Ado ∂ xn

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

(6.3)

Generally, the above Jacobian matrix is obtained using computer symbolic arithmetic functions, such as MATLAB’s Jacobian (·) function and Maple’s Jacobian (·) function. The more the number of terms using Adomian decomposition algorithm, the more complex the obtained Jacobian matrix in form. Here, to analyze the dynamics of fractional-order chaotic systems, the Lyapunov exponent spectrum algorithm is designed based on the above discrete iterative equations, Jacobian matrices, and QR decomposition method. The process of calculating Lyapunov exponent based on QR algorithm is [3] qr [J M J M−1 . . . J1 ] = qr [J M J M−1 . . . J2 (J1 Q0 )] = qr [J M J M−1 · · · J3 (J2 Q1 )][R1 ]

6.1 Lyapunov Exponent Spectrum Calculation Algorithm

79

= qr [J M J M−1 . . . Ji (Ji−1 Qi−2 )][Ri−1 . . . R2 R1 ] = ··· = Q M [R M . . . R2 R1 ] = Q M R

(6.4)

where qr[·] represents the QR decomposition function, and J is the Jacobian matrix of the discrete iterative system, and the Lyapunov exponent spectrum of the system is L yk =

N 1  ln|Ri (k, k)| N h i=1

(6.5)

where k = 1, 2,…, n (system dimension). N is the maximum number of iterations, and h is the iteration time step. Its calculation flowchart of is shown in Fig. 6.1. Taking the fractional-order Lorenz chaotic system as an example, we analyzed the convergence, accuracy, and parameter selection of the algorithm. When q = 1, the sum of Lyapunov exponent spectrum of the system is sys = −a + d − b = −18. Define the global error of the Lyapunov exponent spectrum of the fractional-order Lorenz system as

3



L yer = L yi + 18

i=1

Fig. 6.1 Flowchart of Lyapunov exponent spectrum algorithm for fractional-order chaotic systems

(6.6)

80

6 Dynamics of Fractional-Order Chaotic Systems

Fig. 6.2 Lyapunov exponent spectrum based on Adomian decomposition algorithm. a Global error; b the Lyapunov exponent spectrum of the fractional-order Lorenz system

The simulation results are shown in Fig. 6.2a, and it shows that this Lyapunov exponent spectrum calculation algorithm has high accuracy. When q = 0.96, the convergences of the three Lyapunov exponents are shown in Fig. 6.2b, which shows that the Lyapunov exponent spectra algorithm for fractional-order chaotic systems has good convergence and accuracy.

6.2 Dynamical Characterization of a Family of Fractional-Order Lorenz Systems 6.2.1 Fractional-Order Lorenz Chaotic System q

q

Fractional-order Lorenz chaotic system (Dt0 x1 = a(x2 − x1 ),Dt0 x2 = cx1 − x1 x3 + q d x2 ,Dt0 x3 = x1 x2 − bx2 , a = 40, b = 3, c = 10) has two parameters, d and the order q. Here, we analyze its dynamics as the parameters change. (1)

Dynamical properties for d = 25 and q varying

Let the step size of the fractional order q be 0.0005 and the range of variation be in [0.75, 1.0]. The corresponding Lyapunov exponent spectrum and bifurcation diagram are shown in Fig. 6.3. It can be seen that when 0.813 ≤ q < 1, the system is mostly chaotic and there are also some periodic windows. When q = 0.813, the attractor of the system is shown in Fig. 6.4. When q < 0.813, the system is divergent. So the minimum order of the system to produce chaos is 2.439. The maximum Lyapunov exponent can characterize the system complexity to some extent, and the larger the maximum Lyapunov exponent, the higher the complexity of the system. From Fig. 6.3a, we found that the maximum Lyapunov exponent of the fractional-order Lorenz system decreases with the order q increasing, i.e., the system complexity tends to decrease with the order q increasing. It indicates that the fractional-order Lorenz system has better practical application than the integer-order Lorenz system. (2)

Dynamical properties for q = 0.96 and d varying

6.2 Dynamical Characterization of a Family of Fractional-Order …

81

Fig. 6.3 Dynamics of fractional-order Lorenz systems with order q. a Lyapunov exponent spectrum; b bifurcation diagram

Fig. 6.4 Attractor of the fractional-order Lorenz system (d = 25, q = 0.813)

The dynamics of the fractional-order Lorenz system with parameter d is shown in Fig. 6.5, where the parameter d varies from 0 to 38 in steps of 0.1. From Fig. 6.5, as the parameter d decreases, the system enters into chaos at d = 32.1 from the periodic state by period-doubling bifurcation. When d ∈ [9.8, 32.1], the system is basically chaotic except for a few period windows, such as d ∈ [14.5, 16.3] ∪ [21.1, 21.5]. Finally, the system is convergent at d = 9.8 after going through a tangent bifurcation. To observe the system state further, the phase diagrams of the system when d = 15, 20, 21.5, and 37 are shown in Fig. 6.6. When d = 15, 21.5, and 37, the system is

Fig. 6.5 Dynamics of the fractional-order Lorenz system with parameter d (q = 0.96). a Lyapunov exponent; b Bifurcation diagram

82

6 Dynamics of Fractional-Order Chaotic Systems

Fig. 6.6 Phase diagram of the fractional-order Lorenz system with parameter d. a d = 15; b d = 20; c d = 21.5; d d = 37

periodic. While for d = 20, the system is chaotic and the attractor phase diagram is consistent with the bifurcation diagram and Lyapunov exponent. It can be seen that the fractional-order Lorenz system has rich dynamical properties when the system parameter d varies. (3)

Chaos diagram of q and d varying

The chaos diagram of the fractional-order Lorenz system based on the maximum Lyapunov exponent with the parameter d and the order q varying simultaneously is shown in Fig. 6.7, where q ∈ [0.75, 1.0] with a step size of 0.0025 and d ∈ [0, Fig. 6.7 Chaos diagram of the fractional-order Lorenz system based on maximum Lyapunov exponent

6.2 Dynamical Characterization of a Family of Fractional-Order …

83

38] with a change step size of 0.38. It can be seen that the chaos region is mainly concentrated at d ∈ [10, 32]. When d ∈ [25, 30] and q ∈ [0.8, 0.97], there exists a region where the system has a large value of the maximum Lyapunov exponent, indicating that the system is relatively more complex in this case and more beneficial for practical applications. Compared with the simulation results based on the frequency-domain approximation algorithm in the literature [4], the results of the dynamical characterization based on the Adomian decomposition algorithm are more accurate and can reflect more detailed dynamical behaviors of the system.

6.2.2 Fractional-Order Simplified Lorenz Chaotic System Setting the step size h = 0.01, the Lyapunov exponent spectra of the fractionalq q order simplified Lorenz system (Dt0 x = 10(y − x), Dt0 y = (24 − 4c)x − x z + cy, q Dt0 z = x y −8z/3) and the corresponding bifurcation diagrams are shown in Fig. 6.8. From Fig. 6.8a, the system enters into chaos at c = −1.1 when q = 0.9 and c changes with the step size of 0.1. As c increases, the maximum Lyapunov exponent (LEmax ) gradually decreases, and at about c = 7.2, the system degenerates to a periodic state by reverse period-doubling bifurcation. The result is consistent with the corresponding bifurcation diagram in Fig. 6.8b. In Fig. 6.8c, the system enters into chaos at q = 0.595 when c = 5 and q changes with the step size of 0.005. As q increases until q = 1, the system is chaotic except for a periodic window at approximately q = 0.88, but LEmax is also decreasing. Setting h = 0.01, when the system parameter c varies, the LEmax with different q is shown in Fig. 6.9. It can be seen that for any q, the LEmax decreases as c increases, and becomes negative after passing through the value 0. That is, the system goes from the chaotic state to the convergent state via the periodic state. Also, the larger q is, the wider the chaotic region is. When the parameter c is a constant, the smaller q is, the larger the LEmax is in the chaotic region. The size of LEmax reflects the complexity of the chaotic system, so the fractional-order simplified Lorenz system is more complex than the integer-order simplified Lorenz system and has a broader application prospect. Setting h = 0.01, when c ∈ [−2, 8] with a change step size of 0.1 and q ∈ [0.5, 1] with a change step size of 0.005, the change of the LEmax of fractional-order simplified Lorenz system is shown in Fig. 6.10. At the bottom of the figure, the LEmax is maximum when c = 5.3 and q = 0.535, and it equals 4.443. It is worth noting that there is a dividing line in the figure for the abrupt change of the LEmax , along the upper right of the boundary, where LEmax decreases and approaches a value of 0 as both q and c increase in the direction, while along the opposite direction, LEmax abruptly changes from a relatively large positive value to a negative value, i.e. in which direction the system abruptly changes from a chaotic state to a convergent state. When applying the fractional-order simplified Lorenz system, we should select a larger LEmax corresponding to the region near the upper part of boundary, but the

84

6 Dynamics of Fractional-Order Chaotic Systems

Fig. 6.8 Lyapunov exponent spectra and bifurcation diagrams for the fractional-order simplified Lorenz system. a Lyapunov exponent spectrum for q = 0.9, c varying. b Bifurcation diagram for q = 0.9, c varying. c Lyapunov exponential spectrum for c = 5, q varying. d Bifurcation diagram for c = 5, q varying Fig. 6.9 Variation of the maximum Lyapunov exponent with c for different q

6.2 Dynamical Characterization of a Family of Fractional-Order …

85

Fig. 6.10 Maximum Lyapunov exponent of the fractional-order simplified Lorenz system

points on the boundary should be avoided, because at these points, although the LEmax is larger, the system is easily affected by instabilities and exits the chaotic state. Also, the minimum order of the fractional-order simplified Lorenz system to generate chaos can be obtained from Fig. 6.10 when c ∈ [6.7, 7.2] and q = 0.535, i.e., the minimum order of the system to produce chaos is 3 × q = 1.605. In order to more clearly illustrate the change of the system state with the order q and parameter c, especially for the determination of the periodic state, the chaos diagram of the system is plotted according to the magnitude of the LEmax value. Considering the effects of computational errors and data truncation, etc., experimental studies show that the system approximates the periodic state for LEmax = 0.03, so the system is set to be chaotic for LEmax > 0.03, periodic for 0 ≤ LEmax ≤ 0.03, and convergent for LEmax < 0. The chaos diagram of the system is shown in Fig. 6.11, where red, blue, black, and white represent the chaotic, periodic, convergent, and divergent states, respectively. Obviously, most of the region of the diagram is chaotic. When c is small, for most of q, the system is mostly convergent, except for a small portion that behaves as a periodic state. As c increases, the system gradually enters into chaos, and the chaotic region gradually increases. At c ∈ [6.1, 7.7], the system starts Fig. 6.11 Chaos diagram of the fractional-order simplified Lorenz system

86

6 Dynamics of Fractional-Order Chaotic Systems

to appear as a periodic state again, and as c increases again, the system starts to converge, and this result is consistent with the results shown in Fig. 6.9. In order to further confirm the existence of chaos in the fractional-order simplified Lorenz system obtained in Fig. 6.11, the “0–1 test” algorithm proposed by Gottwald G A and Melbourne I [5, 6] is used. This method is a quantitative analysis method to test whether a chaotic state exists in a system. The basic principle is to create a stochastic dynamic process for a chaotic sequence, and then, during the operation of the system, to study the dynamic changes in the scale of the stochastic process. The specific principle is as follows. Suppose that a certain one-dimensional sequence obtained in the evolution of the system is φ(n) (n = 1, 2, 3, …). Define an arbitrary constant c, which is greater than zero, and we obtain the two formulas as follows based on φ(n) the definition. p(n) =

n 

φ( j) cos(θ ( j)), n = 1, 2, 3, . . .

(6.7)

φ( j) sin(θ ( j)), n = 1, 2, 3, . . .

(6.8)

j=1

s(n) =

n  j=1

where θ ( j) = jc +

j 

φ(i), j = 1, 2, 3, . . . , n

(6.9)

i=1

From the above definition, the following two conclusions are drawn [7]. (1) (2)

If the system is not chaotic, then the dynamical system has bounded trajectories in the plane (P, S). If the system is chaotic, the dynamical system has a trajectory that is similar to Brownian motion in the plane (P, S).

The “0–1 test” is a new method to determine whether a system is chaotic or not: the system can be visually determined to be chaotic or not by observing the trajectory of the dynamical system in (P, S) plane. If the trajectory of the system in the (P, S) plane is regular and bounded, the system is non-chaotic. If the trajectory is unbounded and resembles Brownian motion, the system is chaotic. The 0–1 test is used below to determine the minimum order of the fractional-order simplified Lorenz system to produce chaos. Setting h = 0.01 and initial values (x 0 , y0 , z0 ) = (0.1, 0.2, 0.3), 40,000 chaotic sequence values in the x-direction are removed and the (p, s)-plane trajectories are plotted for q ∈ [0.5, 0.6] and c = 7 by taking n = 100 according to Eqs. (6.7)–(6.9). The results show that when q ≥ 0.535, the obtained (p, s)-plane trajectories are all Brownian-like, while when q ≤ 0.530, the (p, s)-plane trajectories are all regularly boundedly distributed. Figure 6.12 shows the (p, s) plane trajectories for q = 0.535 and q = 0.53. It can be judged that the results

6.2 Dynamical Characterization of a Family of Fractional-Order …

87

Fig. 6.12 The (p, s)-plane of the fractional-order simplified Lorenz system. a q = 0.535 b q = 0.53

are consistent with those of Fig. 6.10 when h = 0.01, the step size of q is 0.005, and the minimum order of the fractional-order simplified Lorenz system to produce chaos is q = 0.535. The 0–1 test is still used to analyze the minimum order at which the system produces chaos for h = 0.001. Figure 6.13a and b show the (p, s) plane for h = 0.001, q = 0.395 and q = 0.390 for c = 6.950, respectively. q = 0.395 corresponds to a (p, s) plane trajectory similar to Brownian motion, so the system is chaotic. q = 0.390 corresponds to a (p, s) plane trajectory that is regular and bounded, so the system is non-chaotic. It is shown that when h = 0.001, the minimum order of the system generating chaos is 3 × q = 1.185 at q = 0.395. It is seen that the minimum order of the fractional-order simplified Lorenz system generating chaos corresponding to h = 0.01 and h = 0.001 are different, and the minimum order at h = 0.001 is significantly smaller than the minimum order at h = 0.01. When c = 5, taking different h, the minimum order of the system that produces chaos, respectively, is shown in Table 6.1. As h decreases, the minimum order of the system that appears chaotic also decreases, but the decrease becomes gradually

Fig. 6.13 The (p, s)-plane of the fractional-order simplified Lorenz system. a q = 0.395, b q = 0.39

88 Table 6.1 Minimum order for generating chaos corresponding to different h for c = 5

Table 6.2 Comparison of different solution algorithms for solving the fractional-order simplified Lorenz systems

6 Dynamics of Fractional-Order Chaotic Systems h

q

Minimum order

10–1

1

divergent

10–2

0.595

1.785

10–3

0.45

1.35

10–4

0.355

1.065

10–5

0.29

0.87

10–6

0.245

0.735

10–7

0.21

0.63

10–8

0.185

0.555

Solution algorithm

Chaotic regions q = 0.9

c=5

Adomian decomposition algorithm

c ∈ [−1.1, 7.2]

q ∈ [0.595, 1]

Prediction-correction algorithm

c ∈ [2.6, 7.4]

q ∈ [0.93, 1.07]

smaller. It is shown that the minimum order of the system producing chaos obtained, when the Adomian decomposition algorithm is applied to solve the fractional-order chaotic system, is conditional and the iteration step size h has an effect on the minimum order of the system where chaos exists. The smaller h is, the smaller the minimum order is. Ref. [8] analyzed the fractional-order simplified Lorenz system using the Adams–Bashforth–Moulton prediction-correction algorithm. Table 6.2 compares the Adomian decomposition algorithm (without considering the case of q > 1) with the prediction-correction algorithm for solving the fractional-order simplified Lorenz system under the same conditions. As seen from the table, the chaotic region of the system is wider when Adomian decomposition algorithm is used, which indicates that the Adomian decomposition algorithm is more accurate in solving the system.

6.2.3 Fractional-Order Lorenz Hyperchaotic System When the 2nd term of the Lorenz system is introduced with a nonlinear feedback controller u, the original Lorenz system becomes a Lorenz hyperchaotic system [9] as defined by

6.2 Dynamical Characterization of a Family of Fractional-Order …

⎧ x˙ = 10(y − x) ⎪ ⎪ ⎪ ⎨ y˙ = 28x − x z + y − u ⎪ z˙ = x y − 8z/3 ⎪ ⎪ ⎩ u˙ = Ryz

89

(6.10)

where R is the control parameter and takes values at the range 0 < R ≤ 1. The system is hyperchaotic for R ∈ (0, 0.152), chaotic for R ∈ [0.152, 0.210) ∪ [0.34, 0.49], and periodic when R takes other values [8]. Introducing the fractional-order Caputo differential operator, the equations of the fractional-order Lorenz hyperchaotic system are written as ⎧ q Dt0 x = 10(y − x) ⎪ ⎪ ⎪ q ⎨ Dt0 y = 28x − x z + y − u (6.11) q ⎪ ⎪ Dt0 z = x y − 8z/3 ⎪ ⎩ q Dt0 u = Ryz The Adomian decomposition algorithm is used to solve the fractional-order Lorenz hyperchaotic system, and the numerical solution obtained is expressed as ⎧ 6  j ⎪ ⎪ xn+1 = c1 h jq ( jq + 1) ⎪ j=0 ⎪ ⎪ ⎪ 6 ⎪  ⎪ j ⎪ c2 h jq ( jq + 1) ⎨ yn+1 = j=0 6  ⎪ j ⎪ c3 h jq ( jq + 1) z n+1 = ⎪ ⎪ j=0 ⎪ ⎪ ⎪ 6 ⎪  ⎪ j ⎩ u n+1 = c4 h jq ( jq + 1)

(6.12)

j=0

j

where ci (i = 1, 2, 3, 4. j = 1, 2,…, 6) is calculated as follows. c10 = xn , c20 = yn , c30 = z n , c40 = u n ⎧ 1 c ⎪ ⎪ ⎪ 1 ⎪ ⎨ c1 2 ⎪ c31 ⎪ ⎪ ⎪ ⎩ 1 c4

⎧ 2 c1 ⎪ ⎪ ⎪ ⎪ ⎨ c2 2 2 ⎪ c ⎪ 3 ⎪ ⎪ ⎩ 2 c4

(6.13)

= 10(c20 − c10 ) = 28c10 − c10 c30 + c20 − c40 = c10 c20 − 8c30 /3

(6.14)

= Rc20 c30

= 10(c21 − c11 ) = 28c11 − c10 c31 − c11 c30 + c21 − c41 = c11 c20 + c10 c21 − 8c31 /3   = R c21 c30 + c20 c31

(6.15)

90

6 Dynamics of Fractional-Order Chaotic Systems

⎧ 3 c1 = 10(c22 − c12 ) ⎪ ⎪ ⎪ ⎪ ⎪ (2q + 1) ⎪ ⎪ c23 = 28c12 − c10 c32 − c11 c31 2 − c12 c30 + c22 − c42 ⎪ ⎪  (q + 1) ⎨ (2q + 1) ⎪ c33 = c10 c22 + c11 c21 2 + c12 c20 − 8c32 /3 ⎪ ⎪  (q + 1) ⎪ ⎪   ⎪ ⎪ ⎪ (2q + 1) ⎪ 2 0 ⎩ c43 = R c20 c32 + c21 c31 + c c 2 3  2 (q + 1)

⎧ 4 c = 10(c23 − c13 ) ⎪ ⎪ ⎪ 1 ⎪ ⎪ (3q + 1) ⎪ ⎪ c24 = 28c13 − c10 c33 − (c12 c31 + c11 c32 ) − ⎪ ⎪ (q + 1)(2q + 1) ⎪ ⎪ ⎪ ⎨ c13 c30 + c23 − c43 ⎪ (3q + 1) ⎪ ⎪ ⎪ c34 = c10 c23 + (c12 c21 + c11 c22 ) + c13 c20 + 8c33 /3 ⎪ ⎪ (q + 1)(2q + 1) ⎪ ⎪   ⎪ ⎪ (3q + 1) ⎪ 3 0 ⎪ c4 = R c0 c3 + (c2 c1 + c1 c2 ) ⎩ + c2 c3 4 2 3 2 3 2 3 (q + 1)(2q + 1) ⎧ 5 c1 = 10(c24 − c14 ) ⎪ ⎪ ⎪ ⎪ ⎪ (4q + 1) ⎪ ⎪ ⎪ c25 = 28c14 − c10 c34 − (c13 c31 + c11 c33 ) ⎪ ⎪ (q + 1)(3q + 1) ⎪ ⎪ ⎪ ⎪ (4q + 1) ⎪ ⎪ ⎪ −c12 c32 2 − c14 c30 + c24 − c44 ⎪ ⎪  (2q + 1) ⎪ ⎪ ⎪ ⎪ (4q + 1) ⎨ 5 c3 = c10 c24 + (c13 c21 + c11 c23 ) (q + 1)(3q + 1) ⎪ ⎪ ⎪ (4q + 1) ⎪ ⎪ ⎪ +c12 c22 2 + c14 c20 − 8c34 /3 ⎪ ⎪  (2q + 1) ⎪ ⎪ ⎪ ⎪ (4q + 1) ⎪ ⎪ ⎪ c45 = R(c20 c34 + (c23 c31 + c21 c33 ) ⎪ ⎪ (q + 1)(3q + 1) ⎪ ⎪ ⎪ ⎪ (4q + 1) ⎪ ⎪ ⎩ +c22 c32 2 + c24 c30 )  (2q + 1)

(6.16)

(6.17)

(6.18)

6.2 Dynamical Characterization of a Family of Fractional-Order …

⎧ 6 c1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ c26 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 6 c3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ c46 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

91

= 10(c25 − c15 ) (5q + 1) − (q + 1)(4q + 1) (5q + 1) − c15 c30 + c25 − c45 (c12 c33 + c13 c32 ) (2q + 1)(3q + 1) (5q + 1) = c10 c25 + (c11 c24 + c14 c21 ) + (q + 1)(4q + 1) (5q + 1) + c15 c20 − 8c35 /3 (c12 c23 + c13 c22 ) (2q + 1)(3q + 1) (5q + 1) = R(c20 c35 + (c21 c34 + c24 c31 ) + (q + 1)(4q + 1) (5q + 1) + c25 c30 ) (c22 c33 + c23 c32 ) (2q + 1)(3q + 1) = 28c15 − c10 c35 − (c11 c34 + c14 c31 )

(6.19)

According to Eqs. (6.12) to (6.19), the numerical solution of the fractional-order Lorenz hyperchaotic system can be obtained using MATLAB programming. (1)

Let R = 0.21 and the dynamics for q variations

Let the fractional order q vary at [0.60, 1.00] and the step value is q = 0.001. The Lyapunov exponent spectrum and the bifurcation diagram of the fractionalorder Lorenz hyperchaotic system with order q varying are shown in Fig. 6.14. From the figure, it can be seen that when q ∈ [0.635, 0.662] ∪ (0.999, 1], the maximum Lyapunov exponent is 0, i.e., the system is periodic. When q ∈ (0.662, 0.920], the system has two positive Lyapunov exponents, i.e., the system is hyperchaotic in this case. When q ∈ (0.920, 0.999], the system has only one positive Lyapunov exponent, indicating that the system is chaotic. It can be seen that the fractional-order Lorenz hyperchaotic system exhibits rich dynamics as the order q increases, so the fractional order q can be used as the bifurcation parameter of the system. For R = 0.21, the attractors of the system for q taking different parameters are shown in Fig. 6.15. It is worth pointing out that the system is periodic at integer order, while the system

Fig. 6.14 Dynamics of the fractional-order Lorenz hyperchaotic system with order q. a Lyapunov exponent spectrum. b Bifurcation diagram

92

6 Dynamics of Fractional-Order Chaotic Systems

Fig. 6.15 Attractors for the fractional-order Lorenz hyperchaotic system for different q (R = 0.21). a q = 1.00; b q = 0.92; c q = 0.90; d q = 0.82; e q = 0.72; f q = 0.65

becomes more complex, with hyperchaotic and chaotic states observed at fractionalorder case. It can be seen that with the introduction of the fractional calculus operator, the system is more complex and more beneficial for practical applications. (2)

Set q = 0.96 and dynamics with R varying

The Lyapunov exponential spectrum and bifurcation diagram of the fractional-order hyperchaotic system with parameter R varying are shown in Fig. 6.16, where R varies in the range [0, 1] and the step size ΔR = 0.0025. From Fig. 6.16, it can be seen that the system is hyperchaotic for R ∈ [0, 0.1875], chaotic for R ∈ (0.1875, 0.2275] ∪ (0.3400, 0.4975], and periodic for R ∈ (0.2275, 0.3400] ∪ (0.4975, 1]. As seen in Fig. 6.16, the Lyapunov exponent spectrum results remain consistent with the bifurcation diagram results. To further observe the system state, the attractors of the system when taking different values of R are shown in Fig. 6.17. When R = 0.05, the

Fig. 6.16 Dynamics of the fractional-order Lorenz hyperchaotic system with parameter R. a Lyapunov exponent spectrum; b bifurcation diagram

6.2 Dynamical Characterization of a Family of Fractional-Order …

93

Fig. 6.17 Attractors for the fractional-order Lorenz hyperchaotic system for R taking different values (q = 0.96). a R = 0.05; b R = 0.20; c R = 0.25; d R = 0.30; e R = 0.40; f R = 0.50; g R = 0.80; h R = 1.00

hyperchaotic attractor is in a double-wing state, while when R = 0.20 and 0.40, the chaotic attractor shows an asymmetric state. When R = 0.25, 0.30, 0.80, and 1.00, the system is periodic, while when R = 0.50 the system is quasi-periodic.

6.2.4 Fractional-Order Rössler Chaotic System According to the principle of Adomian decomposition algorithm for solving fractional-order differential equations, the algorithm treats the constant terms differently, so the fractional-order Rössler system containing constant terms is chosen for analysis. Since Rössler O E proposed the Rössler system in 1976, the system has been widely studied as one of the typical chaotic systems, and the integer-order chaotic system is defined as [10] ⎧ ⎪ ⎨ x˙ = −y − z y˙ = x + ay ⎪ ⎩ z˙ = b + z(x − c)

(6.20)

94

6 Dynamics of Fractional-Order Chaotic Systems

where a, b, and c are the system control parameters that determine the dynamics of the system. The corresponding fractional-order Rössler system is written as ⎧ q ⎪ ⎨ Dt0 x = −y − z q Dt0 y = x + ay ⎪ ⎩ q Dt0 z = b + z(x − c)

(6.21)

Solving the fractional-order Rössler system according to the Adomian decomposition method and the properties of fractional calculus, the iterative form of the numerical solution is obtained, and according to the analysis in Chap. 2, its first five terms are taken and expressed as ⎡

⎤ ⎡ ⎤ xm+1 x10 + x11 + x12 + x13 + x14 ⎣ ym+1 ⎦ = ⎣ y10 + y11 + y12 + y13 + y14 ⎦ z m+1 z 10 + z 11 + z 12 + z 13 + z 14

(6.22)

⎧ x10 = x0 ; y10 = y0 ⎪ ⎪ ⎪ ⎨ hq z 10 = z 0 + b ⎪ (q + 1) ⎪ ⎪ ⎩ C10 = x10 ; C20 = y10 ; C30 = z 0

(6.23)

where

⎧ C11 = −C20 − C30 ; C110 = −b ⎪ ⎪ ⎪ ⎪ ⎪ hq h 2q ⎪ ⎪ ⎪ x11 = C11 + C110 ⎪ ⎪ (q + 1) (2q + 1) ⎪ ⎪ ⎪ ⎪ ⎪ C = C + aC 10 20 ⎨ 21 hq ⎪ y11 = C21 ⎪ ⎪ (q + 1) ⎪ ⎪ ⎪ ⎪ ⎪ C31 = C10 C30 − cC30 ; C310 = bC10 − bc ⎪ ⎪ ⎪ ⎪ ⎪ hq h 2q ⎪ ⎪ ⎩ z 11 = C31 + C310 (q + 1) (2q + 1)

(6.24)

6.2 Dynamical Characterization of a Family of Fractional-Order …

⎧ C12 = −C21 − C31 ; ⎪ ⎪ ⎪ ⎪ ⎪ h 2q ⎪ ⎪ ⎪ x12 = C12 ⎪ ⎪ (2q + 1) ⎪ ⎪ ⎪ ⎪ ⎪ C22 = C11 + aC21 ; ⎪ ⎪ ⎪ ⎪ ⎪ h 2q ⎪ ⎪ y = C ⎪ 12 22 ⎪ ⎨ (2q + 1)

95

C120 = −C310 h 3q (3q + 1) = C110

+ C120 C220

h 3q (3q + 1) C32 = C11 C30 + C10 C31 − cC31 ⎪ ⎪ ⎪ ⎪ (2q + 1) ⎪ ⎪ ⎪ C320 = bC11 2 + C30 C10 + C10 C310 − cC310 ⎪ ⎪  (q + 1) ⎪ ⎪ ⎪ ⎪ (3q + 1) ⎪ ⎪ ⎪ C321 = bC110 ⎪ ⎪ (q + 1)(2q + 1) ⎪ ⎪ ⎪ 2q ⎪ ⎪ h h 3q h 4q ⎪ ⎩ z 12 = C32 + C320 + C321 (2q + 1) (3q + 1) (4q + 1) ⎧ C13 = −C22 − C32 , C130 = −C120 − C320 , ⎪ ⎪ ⎪ ⎪ ⎪ C131 = −C321 ⎪ ⎪ ⎪ ⎪ ⎪ h 3q h 4q h 5q ⎪ ⎪ x = C + C + C ⎪ 13 13 130 131 ⎪ ⎪ (3q + 1) (4q + 1) (5q + 1) ⎪ ⎪ ⎪ ⎪ ⎪ C = C + aC 23 12 22 ⎪ ⎪ ⎪ ⎪ ⎪ C = C + aC 230 120 220 ⎪ ⎪ ⎪ ⎪ 3q ⎪ h h 4q ⎪ ⎪ y = C + C ⎪ 13 23 230 ⎪ ⎪ (3q + 1) (4q + 1) ⎪ ⎪ ⎪ ⎨ (2q + 1) C33 = C12 C30 + C10 C32 − cC32 + C11 C31 2  (q + 1) ⎪ ⎪ ⎪ ⎪ C330 = C30 C120 + C10 C320 + (bC12 + C11 C310 + C31 C110 ) ⎪ ⎪ ⎪ ⎪ ⎪ (3q + 1) ⎪ ⎪ ⎪ − cC320 ⎪ ⎪ (2q + 1)(q + 1) ⎪ ⎪ ⎪ ⎪ (4q + 1) ⎪ ⎪ ⎪ C331 = bC120 ⎪ ⎪ (3q + 1)(q + 1) ⎪ ⎪ ⎪ ⎪ (4q + 1) ⎪ ⎪ ⎪ +C10 C321 + C110 C310 2 − cC321 ⎪ ⎪  (2q + 1) ⎪ ⎪ ⎪ ⎪ ⎪ h 3q h 4q h 5q ⎪ ⎩ z 13 = C33 + C330 + C331 (3q + 1) (4q + 1) (5q + 1) + C220

(6.25)

(6.26)

96

6 Dynamics of Fractional-Order Chaotic Systems

⎧ C14 = −C23 − C33 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ C140 = −C230 − C330 ⎪ ⎪ ⎪ ⎪ C141 = −C331 ⎪ ⎪ ⎪ ⎪ h 4q h 5q h 6q ⎪ ⎪ ⎪ x = C + C + C 14 14 140 141 ⎪ ⎪ (4q + 1) (5q + 1) (6q + 1) ⎪ ⎪ ⎪ ⎪ ⎪ C = C + aC 24 13 23 ⎪ ⎪ ⎪ ⎪ ⎪ C = C + aC 240 130 230 ⎪ ⎪ ⎪ ⎪ 4q ⎪ h h 5q h 6q ⎪ ⎪ ⎪ y14 = C24 + C240 + C131 ⎪ ⎪ (4q + 1) (5q + 1) (6q + 1) ⎪ ⎪ ⎪ ⎪ ⎪ C = C C + C C + (C C + C C ) 340 13 30 10 33 11 32 12 31 ⎪ ⎪ ⎪ ⎪ ⎪ (3q + 1) ⎪ ⎨ − cC33 (2q + 1)(q + 1) ⎪ ⎪ C341 = C30 C130 + C10 C330 − cC330 + (bC13 + C11 C320 + C31 C120 ) ⎪ ⎪ ⎪ ⎪ ⎪ (4q + 1) (4q + 1) ⎪ ⎪ + (C110 C32 + C12 C310 ) 2 ⎪ ⎪ ⎪ (3q + 1)(q + 1)  (2q + 1) ⎪ ⎪ ⎪ ⎪ ⎪ C342 = C30 C331 + C10 C331 − cC331 + (bC130 + C11 C321 ) ⎪ ⎪ ⎪ ⎪ (5q + 1) (5q + 1) ⎪ ⎪ ⎪ + (C110 C320 + C120 C310 ) ⎪ ⎪ (4q + 1)(q + 1) (3q + 1)(2q + 1) ⎪ ⎪ ⎪ ⎪ (6q + 1) (6q + 1) ⎪ ⎪ ⎪ C343 = bC131 + C110 C321 ⎪ ⎪ (5q + 1)(q + 1) (2q + 1)(4q + 1) ⎪ ⎪ ⎪ ⎪ 4q 5q ⎪ h h ⎪ ⎪ z 14 = C340 + C341 ⎪ ⎪ ⎪ (4q + 1) (5q + 1) ⎪ ⎪ ⎪ 6q ⎪ h h 7q ⎪ ⎪ ⎩ +C342 + C343 (6q + 1) (7q + 1)

(6.27)

According to Eqs. (6.22)–(6.27), taking a = 0.55, b = 2, c = 4, q = 0.8, and h = 0.01, the attractor of the fractional-order Rössler system is obtained as shown in Fig. 6.18, which is similar to the attractor of the corresponding integer-order system. Dynamics of the fractional-order Rössler system is analyzed in the following two cases. (1)

The order q varying

The bifurcation diagram and the corresponding Lyapunov exponent spectrum are shown in Fig. 6.19 with q ∈ [0.2, 1], a = 0.55, b = 2, and c = 4, and it shows that the bifurcation diagram and the corresponding Lyapunov exponent spectrum are consistent. Starting from q = 0.268, the system behaves as a periodic state. As q increases, the system starts to period-doubling bifurcation at q = 0.346, and gradually enters into chaos. The minimum order for the existence of chaos in the system is approximately 3 (system dimension) × q = 1.119 at q = 0.373, and the corresponding attractor is shown in Fig. 6.20a. For q ∈ [0.346, 1], the system is mostly

6.2 Dynamical Characterization of a Family of Fractional-Order …

97

Fig. 6.18 Attractor of the fractional-order simplified Rössler system

Fig. 6.19 Bifurcation diagram and Lyapunov exponent spectrum of the fractional-order Rössler system with order q. a Bifurcation diagram, b Lyapunov exponent spectrum

Fig. 6.20 Attractors of the fractional-order Rössler system in the x–y plane. a q = 0.373, b q = 0.564

98

6 Dynamics of Fractional-Order Chaotic Systems

chaotic, but also includes periodic windows, period-doubling bifurcations, tangent bifurcations, etc. For example, there exists a periodic window at q = 0.564, and the corresponding attractor is shown in Fig. 6.20b. Figures 6.19 and 6.20 illustrate that in the fractional-order Rössler system, the differential order q can be viewed as another bifurcation parameter in addition to the system control parameters a, b, and c. Compared to the integer-order Rössler system, when the differential order is fractional, the system has richer dynamical behaviors, including chaotic states, periodic states, period-doubling bifurcation, and tangent bifurcation. In Fig. 6.19, the step size of q is 0.001 when analyzing the fractional-order Rössler system, and the minimum order of the system to produce chaos is obtained as q = 0.373. The step size of q can be smaller according to the obtained iterative equation. However, in Ref. [11], the time domain–frequency domain approximation algorithm was used to analyze the fractional-order Rössler system with an analytical step size of q of only 0.1, and the minimum order of the system to produce chaos was obtained to be approximately between q = 0.7~0.8. So it is more accurate to solve and analyze the fractional-order Rössler system using Adomian decomposition algorithm. (2)

Parameter a varying

Setting the parameters b = 2, c = 4, the step size of a = 0.001, we analyze the dynamics of the system for a ∈ [0.2, 0.6], as shown in Fig. 6.21. Figure 6.21a–c shows the bifurcations for q = 1, q = 0.8, and q = 0.6, respectively, and the bifurcations are similar for the three cases. As a increases, the system enters into chaos from the periodic state by period-doubling bifurcation. When a reaches a certain value, the system immediately appears degradation. However, as a varies, for different values of q, the location of the first bifurcation of the system appears different, with q = 1, q = 0.8, and q = 0.6 starting to bifurcate at a = 0.326, a = 0.339, and a = 0.366, respectively. Also, the maximum value of a for which chaos exists in the system is different from the maximum value of a for which chaos exists at q = 1, q = 0.8, and q = 0.6 being a = 0.556, a = 0.560, and a = 0.572, respectively. Obviously, the chaotic region of the system produces a shift to the right as q decreases. Figure 6.21d shows the maximum Lyapunov exponent of the system with a for different q. It can be seen that the smaller q is, the larger the maximum Lyapunov exponent is. That is, the smaller q is, the more complex the fractional-order Rössler system is, which indicates that in the fractional-order Rössler system, the differential order q affects the dynamics of the system, and the fractional-order Rössler system has a broader application prospect than the integer-order Rössler system.

6.2.5 Fractional-Order Lorenz–Stenflo Chaotic System The Lorenz–Stenflo system was proposed by Stenflo in 1996, and it consists of four nonlinear ordinary differential equations. It describes the evolution of finite amplitude and gravity waves in a rotating atmosphere. The equations of the Lorenz–Stenflo system are written as [12]

6.2 Dynamical Characterization of a Family of Fractional-Order …

99

Fig. 6.21 Bifurcation diagram and maximum Lyapunov exponent of the fractional-order Rössler system with a. a Bifurcation (q = 1). b Bifurcation (q = 0.8). c Bifurcation (q = 0.6). d Maximum Lyapunov exponent

⎧ x˙ = σ (y − x) + sv ⎪ ⎪ ⎪ ⎨ y˙ = r x − x z − y ⎪ z˙ = x y − bz ⎪ ⎪ ⎩ v˙ = −x − σ v

(6.28)

where σ, s, r, b are the system parameters. Many scholars have studied the dynamical behavior of the Lorenz–Stenflo system [13–15]. In the following, the solution and analysis are carried out for the fractional-order case of this system. The equations of the fractional-order Lorenz–Stenflo system are written as ⎧ q Dt0 x = σ (y − x) + sv ⎪ ⎪ ⎪ ⎨ Dq y = r x − x z − y t0 q ⎪ Dt0 z = x y − bz ⎪ ⎪ ⎩ q Dt0 v = −x − σ v (1)

Numerical solution of the fractional-order Lorenz–Stenflo system

(6.29)

100

6 Dynamics of Fractional-Order Chaotic Systems

By employing Adomian decomposition algorithm, the iterative equation for the numerical solution of this four-dimensional fractional-order chaotic system is obtained as ⎧ hq ⎪ ⎪ x = x + [σ (y − x ) + sv ] m+1 m m m m ⎪ ⎪ (q + 1) ⎪ ⎪  ⎪ ⎪ ⎪ + σ [(r x − y − x z ) − σ (y m m m m m − x m ) − svm ] ⎪ ⎪ ⎪ ⎪ 2q ⎪  h ⎪ ⎪ +s(−xm − σ vm ) + ··· ⎪ ⎪ ⎪ (2q + 1) ⎪ ⎪ ⎪ ⎪ hq ⎪ ⎪ = y + (r x − y − x z ) + {r [σ (ym − xm ) + svm ] y ⎪ m+1 m m m m m ⎪ ⎪ (q + 1) ⎪ ⎪ ⎪ ⎪ ⎪ − (r xm − ym − xm z m ) − [σ (ym − xm ) + svm ]z m ⎪ ⎪ ⎨ h 2q (6.30) −x (−bz + x y )} + ··· m m m m ⎪ ⎪ (2q + 1) ⎪ ⎪ ⎪ ⎪ hq ⎪ ⎪ + {−b(−bz m + xm ym ) z m+1 = z m + (−bz m + xm ym ) ⎪ ⎪ (q + 1) ⎪ ⎪ ⎪ ⎪ ⎪ h 2q ⎪ ⎪ ⎪ +[σ (ym − xm ) + svm ]ym +xm (r xm − ym − xm z m )} + ··· ⎪ ⎪ (2q + 1) ⎪ ⎪ ⎪ ⎪ hq ⎪ ⎪ ⎪ vm+1 = vm + (−xm − σ vm ) ⎪ ⎪ (q + 1) ⎪ ⎪ ⎪ ⎪ ⎪ h 2q ⎪ ⎩ +{ −[σ (ym − xm ) + svm ] − σ (−xm − σ vm )} + ··· (2q + 1) Based on the analysis in Chap. 4, selecting the first seven terms to analyze the fractional-order Lorenz–Stenflo system, the numerical solution of the system is expressed as ⎡

⎤ ⎡ ⎤ xm+1 C10 C11 C12 C13 C14 C15 C16 ⎢ ym+1 ⎥ ⎢ C20 C21 C22 C23 C24 C25 C26 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ z m+1 ⎦ = ⎣ C30 C31 C32 C33 C34 C35 C36 ⎦ vm+1 C40 C41 C42 C43 C44 C45 C46  T hq h 2q h 3q h 4q h 5q h 6q 1 (q+1) (2q+1) (3q+1) (4q+1) (5q+1) (6q+1)

(6.31)

where ⎧ C10 ⎪ ⎪ ⎪ ⎨C 20 ⎪ C 30 ⎪ ⎪ ⎩ C40

= xm = ym = zm = vm

(6.32)

6.2 Dynamical Characterization of a Family of Fractional-Order …

⎧ C11 ⎪ ⎪ ⎪ ⎨C 21 ⎪ C31 ⎪ ⎪ ⎩ C41 ⎧ C12 ⎪ ⎪ ⎪ ⎨C 22 ⎪ C32 ⎪ ⎪ ⎩ C42

= σ (C20 − C10 ) + sC40 = rC10 − C20 − C10 C30 = −bC30 + C10 C20

101

(6.33)

= −C10 − σ C40

= σ (C21 − C11 ) + sC41 = rC11 − C21 − C11 C30 − C10 C31 = −bC31 + C11 C20 + C10 C21 = −C11 − σ C41

⎧ C13 = σ (C22 − C12 ) + sC42 ⎪ ⎪ ⎪ ⎪ (2q + 1) ⎪ ⎪ ⎪ ⎨ C23 = rC12 − C22 − C12 C30 − C11 C31  2 (q + 1) − C10 C32 (2q + 1) ⎪ ⎪ ⎪ C33 = −bC32 + C12 C20 + C11 C21 2 + C10 C22 ⎪ ⎪  (q + 1) ⎪ ⎪ ⎩ C43 = −C12 − σ C42 ⎧ C14 = σ (C23 − C13 ) + sC43 ⎪ ⎪ ⎪ ⎪ C24 = rC13 − C23 − C13 C30 − (C12 C31 + C11 C32 ) ⎪ ⎪ ⎪ ⎪ ⎪ (3q + 1) ⎪ ⎪ ⎪ ⎨ (q + 1)(2q + 1) − C10 C33 ⎪ C34 = −bC33 + C13 C20 + (C12 C21 + C11 C22 ) ⎪ ⎪ ⎪ ⎪ ⎪ (3q + 1) ⎪ ⎪ + C10 C23 ⎪ ⎪ (q + 1)(2q + 1) ⎪ ⎪ ⎩ C44 = −C13 − σ C43

(6.34)

(6.35)

(6.36)

⎧ C15 = σ (C24 − C14 ) + sC44 ⎪ ⎪ ⎪ ⎪ (4q + 1) ⎪ ⎪ ⎪ C25 = rC14 − C24 − C14 C30 − (C13 C31 + C11 C33 ) ⎪ ⎪ (q + 1)(3q + 1) ⎪ ⎪ ⎪ ⎪ (4q + 1) ⎪ ⎪ ⎪ ⎨ −C12 C32  2 (2q + 1) − C10 C34 (6.37) (4q + 1) ⎪ ⎪ ⎪ C35 = −bC34 + C14 C20 + (C13 C21 + C11 C23 ) ⎪ ⎪ (q + 1)(3q + 1) ⎪ ⎪ ⎪ ⎪ (4q + 1) ⎪ ⎪ +C C ⎪ + C10 C24 12 22 2 ⎪ ⎪  (2q + 1) ⎪ ⎪ ⎩ C45 = −C14 − σ C44

102

6 Dynamics of Fractional-Order Chaotic Systems

⎧ C16 = σ (C25 − C15 ) + sC45 ⎪ ⎪ ⎪ ⎪ (5q + 1) ⎪ ⎪ ⎪ C26 = rC15 − C25 − C15 C30 − (C14 C31 + C11 C34 ) ⎪ ⎪ (q + 1)(4q + 1) ⎪ ⎪ ⎪ ⎪ (5q + 1) ⎪ ⎪ ⎪ ⎨ −(C13 C32 + C12 C33 ) (2q + 1)(3q + 1) − C10 C35 (6.38) (5q + 1) ⎪ ⎪ ⎪ C36 = −bC35 + C15 C20 + (C14 C21 + C11 C24 ) ⎪ ⎪ (q + 1)(4q + 1) ⎪ ⎪ ⎪ ⎪ (5q + 1) ⎪ ⎪ ⎪ +(C13 C22 + C12 C23 ) + C10 C25 ⎪ ⎪ (2q + 1)(3q + 1) ⎪ ⎪ ⎩ C46 = −C15 − σ C45 According to Eqs. (6.29)–(6.8), let σ = 1.0, s = 1.5, b = 0.7, r = 26, q = 0.7, and h = 0.001, the attractor phase diagram of the fractional-order Lorenz–Stenflo system is obtained as shown in Fig. 6.22. It can be seen that the attractor of the fractionalorder Lorenz–Stenflo system is similar to that of the integer-order Lorenz–Stenflo system. (2)

Analysis and comparison of the dynamical properties of the fractional-order Lorenz–Stenflo system

Next, the dynamics of the fractional-order Lorenz–Stenflo system is analyzed in four cases. (1)

The system is chaotic when σ = 1.0, s = 1.5, b = 0.7, r = 26 and q = 1 [12]. We analyze the case of q ∈ [0.3, 1] and the step size is 0.001. Setting the initial states [x 0 , y0 , z0 , v0 ] = [2.5652, 6.4129, 23.5000, -2.5652], and the number of iterations m = 2 × 104 , the Lyapunov exponent spectrum is obtained as shown in Fig. 6.18a. Clearly, the minimum order at which the system appears chaotic is 1.584 at q = 0.396, and the LEmax is maximum. When q ∈ [0.396, 1], the LEmax decreases as q increases, but remains positive, i.e., the smaller q is, the more complex the system is. This result is consistent with the corresponding bifurcation diagram shown in Fig. 6.23b.

The 0–1 test is used to prove the minimum order for the existence of chaos described above, and the results are shown in Fig. 6.24. Figure 6.24 illustrates that the system is chaotic with h = 0.001 and q = 0.396, while the system is non-chaotic at q = 0.395. It shows that the results in Fig. 6.23 are correct. (2)

The system is periodic when σ = 10, s = 30, b = 8/3, r = 340, and q = 1 [14, 15]. The corresponding fractional-order case is studied below. Under the same conditions as above, the Lyapunov exponent spectrum and the bifurcation diagram of the system are obtained as shown in Fig. 6.25. In Fig. 6.25a and b, the system enters into chaos at q = 0.617 and the corresponding LEmax is maximum. As q increases, LEmax gradually decreases and the system degenerates to a periodic state by the reverse period-doubling bifurcation. Until q = 0.753, the system enters into chaos again by tangent bifurcation. Observed

6.2 Dynamical Characterization of a Family of Fractional-Order …

103

Fig. 6.22 Attractors of the fractional-order Lorenz–Stenflo system. a x–y plane, b x–z plane, c x-v plane, d y–z plane, e y-v plane, f z-v plane

from the direction of gradual decrease of q from 1, the system enters into chaos through period-doubling bifurcation with the bifurcation position q = 0.824. To observe the dynamics of the system in more detail, the range of q ∈ [0.65, 0.70] is expanded and the step size of q is reduced from 0.001 to 0.0001, and the corresponding Lyapunov exponent spectrum and bifurcation diagram are

104

6 Dynamics of Fractional-Order Chaotic Systems

Fig. 6.23 Lyapunov exponent spectrum and bifurcation diagram of the fractional-order Lorenz– Stenflo system. a Lyapunov exponent spectrum. b Bifurcation diagram

Fig. 6.24 The (p, s)-plane of fractional-order Lorenz–Stenflo system. a q = 0.396. b q = 0.395

shown in Fig. 6.25c and d. The figure indicates that there exist many periodic windows for the system, such as q ∈ [0.6517, 0.6522], q ∈ [0.6788, 0.6801], and q ∈ [0.6885, 0.6889]. In addition, both the Lyapunov exponent spectrum and the bifurcation diagram in Fig. 6.25 are consistent, indicating that the Lyapunov exponent spectrum analysis yields the same results as the bifurcation diagram analysis. (3)

Discuss the bifurcation with r when σ = 10, b = 8/3, s = 30, and q equals 1 and 0.8, respectively. Figure 6.26a shows the bifurcation diagram obtained by solving the system using the Adomian decomposition for q = 1. The first saddle-node bifurcation appearing from the direction of decreasing r is located at r ≈ 430 and the first period-doubling bifurcation is located at r ≈ 315. The above results are consistent with that in Ref. [14]. Figure 6.26b shows the bifurcation diagram for q = 0.8, and it shows that the structure of the bifurcation diagrams corresponding to q = 1 and q = 0.8 is similar. The difference is that the first saddle-node bifurcation and the first period-doubling bifurcation at q

6.2 Dynamical Characterization of a Family of Fractional-Order …

105

Fig. 6.25 Lyapunov exponent spectrum and bifurcation diagram of the fractional-order Lorenz– Stenflo system. a Lyapunov exponent spectrum for q ∈ [0.6, 1]. b Bifurcation for q ∈ [0.6, 1]. c Lyapunov exponent spectrum for q ∈ [0.65, 0.7]. d Bifurcation for q ∈ [0.65, 0.7]

Fig. 6.26 Bifurcation diagrams of the fractional-order Lorenz–Stenflo system with r. a q = 1, b q = 0.8

106

6 Dynamics of Fractional-Order Chaotic Systems

Fig. 6.27 Bifurcation diagrams of the fractional-order Lorenz–Stenflo system with s. a q = 1, b q = 0.9

(4)

= 0.8 appear at r ≈ 489 and r ≈ 352, respectively. The results indicate that for lower q, the first backward saddle-node bifurcation and the first backward period-doubling bifurcation appear at higher r. Similarly, fixing σ = 10, b = 8/3, r = 340, Fig. 6.27 shows the bifurcation diagrams of the fractional-order Lorenz–Stenflo system corresponding to q = 1 and q = 0.9 with s varying, and the comparison shows that the structures of the bifurcation diagrams are similar for both, but the smaller q appears to have a smaller value of s corresponding to the first bifurcation.

Comparison shows that the fractional-order Lorenz–Stenflo system has different dynamics from the integer-order Lorenz–Stenflo system. When σ = 1.0, s = 1.5, b = 0.7, r = 26, the corresponding fractional-order Lorenz–Stenflo system has a larger LEmax when the integer-order Lorenz–Stenflo system is chaotic. The smaller the q, the larger the LEmax , i.e., the system dynamics are more complex. When σ = 10, s = 30, b = 8/3, r = 340, the integer-order Lorenz–Stenflo system is periodic, and the corresponding fractional-order Lorenz–Stenflo system with different q has rich dynamical behaviors, such as chaotic states, periodic states, period-doubling bifurcations, and tangent bifurcations. When σ = 10, b = 8/3, s = 30 and σ = 10, b = 8/3, r = 340, respectively, we plot the bifurcation diagrams with r and s varying respectively. The bifurcation structures corresponding to different values of q are similar, but the bifurcation locations change.

6.2.6 Fractional-Order Simplified Lorenz Hyperchaotic System The simplified Lorenz system with one parameter was proposed in Ref. [16], and the mathematical model is expressed as

6.2 Dynamical Characterization of a Family of Fractional-Order …

⎧ ⎪ ⎨ x˙ = 10(y − x) y˙ = (24 − 4c)x − x z + cy ⎪ ⎩ z˙ = x y − 8z/3

107

(6.39)

where c is a system parameter. When c ∈ [−1.59, 7.75], the system is chaotic. Ref. [17] introduces a linear state feedback controller u into the first-order differential equation of this system with respect to y to obtain a simplified Lorenz hyperchaotic system ⎧ x˙ = 10(y − x) ⎪ ⎪ ⎪ ⎨ y˙ = (24 − 4c)x − x z + cy + u ⎪ z˙ = x y − 8z/3 ⎪ ⎪ ⎩ u˙ = −kx

(6.40)

where x, y, z, and u are the system state variables, and c and k are the system parameters. When c = −1 and k = 5, the Lyapunov exponent spectrum of the system λi (i = 1, 2, 3, 4) = (0.4259, 0.3001, 0, −14.3926). It can be seen that the system has two positive Lyapunov exponents in this case, i.e., the system is hyperchaotic. By replacing the integer-order differential operator with the fractional-order differential operator, the fractional-order simplified Lorenz hyperchaotic system is obtained, which can be written as ⎧ q D x = 10(y − x) ⎪ ⎪ tq0 ⎪ ⎨ D y = (24 − 4c)x − x z + cy + u t0 (6.41) q ⎪ Dt0 z = x y − 8z/3 ⎪ ⎪ ⎩ q Dt0 u = −kx Using the Adomian decomposition algorithm, the numerical solution of the fractional-order simplified Lorenz hyperchaotic system is expressed as ⎧ 6  j ⎪ ⎪ xn+1 = c1 h jq ( jq + 1) ⎪ j=0 ⎪ ⎪ ⎪ 6 ⎪  ⎪ j ⎪ c h jq ( jq + 1) ⎨ yn+1 = j=0 2 6  ⎪ j ⎪ c3 h jq ( jq + 1) z n+1 = ⎪ ⎪ j=0 ⎪ ⎪ ⎪ 6 ⎪  ⎪ j ⎩ c4 h jq ( jq + 1) u n+1 =

(6.42)

j=0

j

where the intermediate variables ci (i = 1, 2, 3, 4, j = 1, 2,…, 6) are calculated as c10 = xm , c20 = ym , c30 = z m , c40 = u m

(6.43)

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6 Dynamics of Fractional-Order Chaotic Systems

⎧ 1 c1 ⎪ ⎪ ⎪ ⎪ ⎨ c1 2 1 ⎪ c ⎪ 3 ⎪ ⎪ ⎩ 1 c4

= 10(c20 − c10 ) = (24 − 4c)c10 − c10 c30 + c · c20 + c40 = c10 c20 − 8c30 /3

(6.44)

= −kc10

⎧ 2 c1 = 10(c21 − c11 ) ⎪ ⎪ ⎪ ⎪ ⎨ c2 = (24 − 4c)c1 − c0 c1 − c1 c0 + c · c1 + c1 2 1 1 3 1 3 2 4 2 1 0 0 1 1 ⎪ c3 = c1 c2 + c1 c2 − 8c3 /3 ⎪ ⎪ ⎪ ⎩ 2 c4 = −kc11 ⎧ 3 c1 = 10(c20 − c10 ) ⎪ ⎪ ⎪ ⎪ ⎪ (2q + 1) ⎪ ⎪ c23 = (24 − 4c)c12 − c10 c32 − c11 c31 ⎪ ⎪  2 (q + 1) ⎪ ⎨ 2 0 2 2 −c1 c3 + c · c2 + c4 ⎪ ⎪ ⎪ 3 (2q + 1) ⎪ ⎪ ⎪ c3 = c10 c22 + c11 c21 2 + c12 c20 − 8c32 /3 ⎪ ⎪  (q + 1) ⎪ ⎪ ⎩ 3 c4 = −kc12

⎧ 4 c1 = 10(c23 − c13 ) ⎪ ⎪ ⎪ ⎪ ⎪ (3q + 1) ⎪ ⎪ c24 = (24 − 4c)c13 − (c12 c31 + c11 c32 ) ⎪ ⎪ ⎪ (q + 1)(2q + 1) ⎪ ⎪ ⎪ ⎨ −c13 c30 − c10 c33 + c · c23 + c43 (3q + 1) ⎪ ⎪ c4 = c0 c3 + (c2 c1 + c1 c2 ) ⎪ ⎪ 3 1 2 1 2 1 2 ⎪ (q + 1)(2q + 1) ⎪ ⎪ ⎪ ⎪ 3 0 3 ⎪ +c1 c2 + 8c3 /3 ⎪ ⎪ ⎪ ⎩ 4 c4 = −kc13

⎧ ⎪ c15 = 10(c24 − c14 ) ⎪ ⎪ ⎪ ⎪ (4q + 1) ⎪ ⎪ c5 = (24 − 4c)c4 − c0 c4 − (c3 c1 + c1 c3 ) ⎪ ⎪ 2 1 1 3 1 3 1 3 ⎪ (q + 1)(3q + 1) ⎪ ⎪ ⎪ ⎪ (4q + 1) ⎪ ⎪ ⎪ −c12 c32 2 − c14 c30 + c · c24 + c44 ⎨  (2q + 1) (4q + 1) ⎪ ⎪ ⎪ c35 = c10 c24 + (c13 c21 + c11 c23 ) ⎪ ⎪ (q + 1)(3q + 1) ⎪ ⎪ ⎪ ⎪ (4q + 1) ⎪ ⎪ ⎪ +c12 c22 2 + c14 c20 − 8c34 /3 ⎪ ⎪  (2q + 1) ⎪ ⎪ ⎪ ⎩ c5 = −kc4 4 1

(6.45)

(6.46)

(6.47)

(6.48)

6.2 Dynamical Characterization of a Family of Fractional-Order …

⎧ 6 5 5 ⎪ ⎪ c1 = 10(c2 − c1 ) ⎪ ⎪ ⎪ (5q + 1) ⎪ ⎪ ⎪ c26 = (24 − 4c)c15 − c10 c35 − (c11 c34 + c14 c31 ) ⎪ ⎪ (q + 1)(4q + 1) ⎪ ⎪ ⎪ ⎪ (5q + 1) ⎪ ⎪ ⎪ (c12 c33 + c13 c32 ) − c15 c30 + c · c25 + c45 ⎨− (2q + 1)(3q + 1) (5q + 1) ⎪ ⎪ ⎪ c36 = c10 c25 + (c11 c24 + c14 c21 ) ⎪ ⎪ (q + 1)(4q + 1) ⎪ ⎪ ⎪ ⎪ (5q + 1) ⎪ ⎪ ⎪ +(c12 c23 + c13 c22 ) + c15 c20 − 8c35 /3 ⎪ ⎪ (2q + 1)(3q + 1) ⎪ ⎪ ⎪ ⎩ c6 = −kc5 4 1

109

(6.49)

According to Eqs. (6.42) to (6.49), the numerical solution of the fractional-order simplified Lorenz hyperchaotic system can be obtained by using MATLAB programming. From Eq. (6.41), the system has three parameters, i.e., order q and parameters c and k. In the following paper, we will analyze the dynamics of the system when the order q, parameters c and k vary. (1)

Dynamics with c = -1, k = 5 and order q varying

The fractional order q varies at [0.6, 1] with the step size of 0.001. The Lyapunov exponent spectrum and bifurcation diagram of the fractional-order simplified Lorenz hyperchaotic system with order q are shown in Fig. 6.28. It can be seen that when q ∈ [0.68, 0.748), the maximum Lyapunov exponent of the system is 0, i.e., the system is periodic. When q = 0.748, according to Fig. 6.28b, the system has an internal crisis bifurcation and enters into chaos from the periodic state until q = 1. From Fig. 6.28a, when q > 0.748, the system has two positive Lyapunov exponents, which indicates it is hyperchaotic. The two positive Lyapunov exponents show a decreasing trend with order q increasing, indicating that the system complexity tends to decrease with increasing order q. The attractor phase diagrams at q = 0.747 and q = 0.748 are shown in Fig. 6.29a and b, respectively, which shows that the system is periodic for q = 0.747 and hyperchaotic for q = 0.748. When c = -1, k = 5, and h = 0.01,

Fig. 6.28 Dynamics of the fractional-order simplified hyperchaotic Lorenz system with order q. a Lyapunov exponent spectrum. b Bifurcation diagram

110

6 Dynamics of Fractional-Order Chaotic Systems

Fig. 6.29 Attractors of the fractional-order simplified hyperchaotic Lorenz system. a q = 0.747, b q = 0.748

Fig. 6.30 Dynamics of the fractional-order simplified hyperchaotic Lorenz system with parameter k. a Lyapunov exponent spectrum. b Bifurcation diagram

the minimum order of the fractional-order simplified Lorenz hyperchaotic system producing chaos is 2.992. (2)

Dynamics with q = 0.97, c = −1 and parameter k varying

The parameter k varies from 0.1 to 50 with the step size of 0.1, and its Lyapunov exponent spectrum and bifurcation diagram are shown in Fig. 6.30. From Fig. 6.30a, it can be seen that when k ∈ (0.1, 22.1], the system has two positive Lyapunov exponents and is hyperchaotic. When k ∈ (1.9, 17.4], the system is chaotic. When k takes other values, the system is periodic. From the results of the bifurcation diagram in Fig. 6.30b, it shows that as k decreases, the system undergoes period-doubling bifurcation and then enters into chaos at k = 22.1. The Lyapunov exponent spectrum results are consistent with that of the bifurcation diagram. When k takes different values and the system is in different states, the attractor phase diagrams are shown in Fig. 6.31. It can be seen that the system exhibits rich dynamics as k increases. (3)

Dynamics with q = 0.97, k = 5, and parameter c varying

The parameter c varies at [−6, 5] with the step size of 0.01. The Lyapunov exponent spectrum with the parameter c is shown in Fig. 6.32a, and the corresponding bifurcation diagram is shown in Fig. 6.32b. From Fig. 6.32a, the system is essentially chaotic when c ∈ (−4.0, 3.4), where the hyperchaotic interval is (−4.2, 1.5).

6.2 Dynamical Characterization of a Family of Fractional-Order …

111

Fig. 6.31 Attractors of fractional-order simplified hyperchaotic Lorenz system with different k. a k = 1, b k = 20, c k = 40

Fig. 6.32 Dynamics of the fractional-order simplified hyperchaotic Lorenz system with parameter c. a Lyapunov exponent spectrum. b Bifurcation diagram

While it is periodic when c takes other values. From Fig. 6.32b, it shows that as c increases and decreases, the system does not enter into chaos in the same way. When c increases from -6, the system goes out of chaos at c = −4.2 by tangent bifurcation, while when c decreases from 5, the system enters the multi-periodic state from the monocyclic state and finally enters into chaos at c = 3.4. When c takes different values, the attractor phase diagrams of the fractional-order simplified Lorenz hyperchaotic system are shown in Fig. 6.33, with the periodic attractors (c = −6 and 5), the hyperchaotic attractor with double wings (c = −-1), and the chaotic attractor with asymmetric attractor wings (c = 2). (4)

Chaos diagram with the order q varying

Chaos diagram is an effective method for reflecting the complex dynamics of a system in parameter space. The c-k chaos diagram of the fractional-order simplified Lorenz hyperchaotic system is shown in Fig. 6.34. Its calculation method is described as dividing the c-k plane into 101 × 101 points, calculating the Lyapunov exponent spectrum of each point, and drawing a red dot on the parameter plane if the system is hyperchaotic, a blue dot on the parameter plane if the system is chaotic, a green dot on the parameter plane if the system is quasi-periodic state, and the periodic and convergent states are indicated by white dots. In Fig. 6.34, c takes values at [−25, 5], k takes values at [0, 50], and the values of q in Fig. 6.34a–c take their values as q = 0.87, q = 0.95, and q = 1.0, respectively. From Fig. 6.34, it can be seen that when c ∈ [−5,

112

6 Dynamics of Fractional-Order Chaotic Systems

Fig. 6.33 Attractors of the fractional-order simplified Lorenz hyperchaotic system with different c. a c = −6. b c = −1. c c = 2. d c = 5

Fig. 6.34 Chaos diagram based on Lyapunov exponent spectrum with different q. a q = 0.87. b q = 0.95. c q = 1

5] and k ∈ [0, 40], chaotic and hyperchaotic states basically occupy the parameter space of half of it. Comparing Fig. 6.34a–c, it shows that the chaos diagram of the system based on the maximum Lyapunov exponent shows a similar range of system states when q takes different values, but the chaotic and hyperchaotic intervals have a tendency to shrink as q decreases. As shown in Fig. 6.34, compared to the fixing parameter method, the bifurcation diagram or Lyapunov exponent spectrum with only one parameter varying, the chaos diagram provides more information about the system state and provides a better basis for parameter selection for the practical application of fractional-order chaotic systems. From the previous analysis, it is clear that the minimum order of the system to produce chaos is obtained differently using different algorithms, and it is evident that the dynamics of the system will be different when solved using different algorithms. The results of using the prediction-correction algorithm to analyze the dynamics of the fractional-order simplified Lorenz hyperchaotic system with order q, parameters c and k are shown in Figs. 6.35, 6.36 and 6.37 respectively. The prediction-correction

6.2 Dynamical Characterization of a Family of Fractional-Order …

113

Fig. 6.35 Dynamics of the fractional-order simplified Lorenz hyperchaotic system with q varying (prediction-correction algorithm). a Bifurcation diagram. b Maximum Lyapunov exponent

Fig. 6.36 Bifurcation diagram of the fractional-order simplified Lorenz hyperchaotic system with k variation (prediction-correction algorithm). a Bifurcation diagram of the system for k ∈ [3, 34]. b Bifurcation diagram of the system for k ∈ [3, 5]

Fig. 6.37 Dynamics of the fractional-order simplified Lorenz hyperchaotic system with c (prediction-correction algorithm). a Bifurcation diagram. b Maximum Lyapunov exponent

algorithm iteration must use all historical data at each time, and using it to calculate the Lyapunov exponent spectrum of the system has difficulties, so the main analysis here is the bifurcation diagram and the corresponding maximum Lyapunov exponent. The maximum Lyapunov exponent was calculated by using Wolf’s algorithm [1].

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6 Dynamics of Fractional-Order Chaotic Systems

Figure 6.35 corresponds to Fig. 6.28. From Fig. 6.35, the system is periodic when q ∈ [0.96, 0.9644], while it is chaotic when q > 0.9644. For q = 0.9644, the system generates an internal crisis bifurcation. Its corresponding maximum Lyapunov exponent results agree well with the bifurcation diagram results. Comparing the Adomian decomposition algorithm (see Fig. 6.28), the prediction-correction algorithm has a larger minimum order, i.e., a smaller range of chaos generation. Figure 6.36 corresponds to Fig. 6.30. In Fig. 6.36a, the range of the system parameter k is [2, 34]. The bifurcation diagram of the system with k ∈ [3, 5] containing the period window is enlarged for easier and clearer observation of the evolution of the system as shown in Fig. 6.36b. Compared to Fig. 6.30, there are more period windows in the system when it is solved by using prediction-correction algorithm. The period windows of the system can be obtained from Fig. 6.36 as k ∈ [2, 3.5] ∪ [3.81, 4.35] ∪ [11.09, 15.48] ∪ [28.6, 34], and the system is chaotic when k takes the remaining values. Similarly, when only the interval k ∈ [2, 34] in Fig. 6.29 is observed, the system is chaotic with continuous intervals when it is solved by using Adomian decomposition method. When it is solved by Adomian decomposition algorithm, the system has a larger parameter interval, which means that the system has a larger key space when it applies in practical applications. Figure 6.37 corresponds to Fig. 6.32. In Fig. 6.37a, the system parameter c varies at [−3, 5], and the corresponding maximum Lyapunov exponent is shown in Fig. 6.37b. Its chaotic region is mainly c ∈ [−1.72, 3.47] ∪ [3.66, 3.70], while the system is periodic when c takes other values. From Fig. 6.37a, it can be seen that when c decreases from 5, the system enters into chaos from the periodic state by perioddoubling bifurcation, while at c = −1.72, the system enters into chaos by the internal crisis bifurcation. Similar to the above two cases, the chaotic region of the system is wider when it is solved by using the Adomian decomposition algorithm. Comparing the above three sets of diagrams, it can be seen that the bifurcation diagrams obtained when solving fractional-order chaotic systems by using the Adomian decomposition algorithm and the prediction-correction algorithm have some similarities in shape, but the details of the changes are different. In addition, the range of parameters for which the system is chaotic by using the Adomian decomposition algorithm is wider and has a larger key space in practical applications.

References 1. Wolf A, Swift JB, Swinney HL et al (1985) Determining Lyapunov exponents from a time series. Physica D 16(3):285–317 2. Caponetto R, Fazzino S (2013) An application of Adomian decomposition for analysis of fractional-order chaotic systems. Int J Bifurcat Chaos 23(3):1350050 3. Bremen HFV, Udwadia FE, Proskurowski W (1997) An efficient QR based method for the computation of Lyapunov exponents. Physica D 101(2):1–16 4. Jia H, Chen Q, Xue W (2013) Analysis and circuit implementation of fractional-order Lorenz systems. J Phys 62(14):140503 5. Melbourne I (2004) A new test for chaos in deterministic systems. Proc R Soc Lond 460(2042):603–611

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6. Gottwald GA, Melbourne I (2005) Testing for chaos in deterministic systems with noise. Physica D 212(212):100–110 7. Sun KH, Wang X, Zhu CX (2010) The 0–1 test algorithm for chaos and its applications. Chin Phys B 19(11):110510 8. Sun KH, Wang X, Sprott JC (2010) Bifurcations and chaos in fractional-order simplified Lorenz system. Int J Bifurcat Chaos 20(4):1209–1219 9. Gao T, Chen G, Chen Z et al (2007) The generation and circuit implementation of a new hyper-chaos based upon Lorenz system. Phys Lett A 361(1):78–86 10. Rössler OE (1976) An equation for continuous chaos. Phys Lett A 57(5):397–398 11. Li C, Chen G (2004) Chaos and hyperchaos in the fractional-order Rössler equations. Physica A 341(1–4):55–61 12. Stenflo L (1996) Generalized Lorenz equations for acoustic-gravity waves in the atmosphere. Phys Scr 53(53):83–84 13. Mukherjee P, Banerjee S, Mukherjee P et al (2012) Projective and hybrid projective synchronization for the Lorenz-Stenflo system with estimation of unknown parameters. Phys Scr 45(82):9–11 14. Yu MY (1999) Some chaotic aspects of the Lorenz-Stenflo equations. Phys Scr 82(82):10–11 15. Pal S, Sahoo B, Poria S (2014) Multistable behavior of coupled Lorenz-Stenflo systems. Phys Scr 89(4):39–50 16. Sun KH, Sprott JC (2009) Dynamics of a simplified Lorenz system. Int J Bifurcat Chaos 19(4):1357–1366 17. Sun KH, Liu X, Zhu CX (2014) Dynamics of a strengthened chaotic system and its circuit implementation. Chin J Electron 23(2):353–356

Chapter 7

Complexity Analysis of Fractional-Order Chaotic System

Complexity analysis of nonlinear time series is currently a research hotspot of nonlinear science. Nonlinear time series include electrocardiogram signal (ECG) [1], electroencephalogram signal (EEG) [2], electromyography signal [3] and traffic flow signal [4] and other measured signals and time series signals generated by chaotic systems. Among them, the chaotic time series has a high application value in the field of spread spectrum communication and information encryption. The complexity research of chaotic systems mentioned in this chapter is an important aspect of the dynamics research of chaotic systems, including phase diagram observation, Lyapunov exponent, fractal characteristics, spectrum structure, etc. In practical applications, any research involving chaotic dynamics can be regarded as the study of the complexity of chaotic dynamics. The complexity of chaotic systems uses related algorithms to measure the degree to which the chaotic sequence is close to a random sequence. The closer the sequence is to a random sequence, the higher the complexity, the higher the corresponding security. In essence, the complexity of chaotic systems belongs to the research category of chaotic dynamics characteristics. The complexity of chaotic systems mainly includes two aspects: behavioral complexity and structural complexity. Behavioral complexity refers to the degree of similarity between a sequence and a random sequence, and it is also the main direction of complexity research of chaotic systems. There are many measurement algorithms proposed, including some classic algorithms. Most of the current algorithms are based on the Komologrov complexity algorithm and Shannon Entropy. Structural complexity refers to the complexity of the sequence frequency components. This method mainly analyzes the energy distribution of the signal at each frequency. Using the concept of Shannon entropy, it can be known that the more balanced the energy distribution, the greater the entropy value, the closer the signal is to the noise signal, and the greater the complexity. At present, the research on the structural complexity of chaotic sequences is still in early stage, and there are few relative articles, so there is a large research space. The behavioral complexity measurement algorithm in this chapter mainly studies the permutation entropy and its

© Science Press 2022 K. Sun et al., Solution and Characteristic Analysis of Fractional-order Chaotic Systems, https://doi.org/10.1007/978-981-19-3273-1_7

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7 Complexity Analysis of Fractional-Order Chaotic System

improved algorithm, while the structural complexity measurement algorithm mainly studies the spectral entropy and C 0 complexity algorithm.

7.1 Behavioral Complexity Algorithm Recently, due to the faster calculation speed and higher accuracy of the permutation entropy algorithm, Fan et al. [5] and Li et al. [6] designed the Multiscale Permutation Entropy (MPE) algorithm and used it to analyze the complexity of time series. However, the study by Wu et al. [7] shows that this type of multi-scale algorithm cannot describe the system complexity well at higher scales when measuring the complexity of short time series. In addition, most of the actual chaotic systems are multi-dimensional systems, i.e., the system can generate multiple time series at the same time. But permutation entropy algorithm can only describe the complexity of one of the time series, and cannot analyze the complexity of multiple time series at the same time. In order to analyze the complexity of the phase space of the system, people proposed the Multivariate Sample Entropy (MvSampEn) algorithm [8], the Multivariate Neighborhood Sample Entropy (MN-SampEn) algorithm [9], and the Multivariate Fuzzy entropy (MvFuzzyEn) algorithm [10]. However, these algorithms are computation-intensive and time-consuming. Therefore, this chapter will improve the multi-scale coarse-grained technique, design a Modified MPE algorithm (MMPE), analyze the multi-scale complexity of fractional-order chaotic systems, and design a multivariate permutation entropy (MvPE) algorithm. These algorithms will be used to measure the phase space complexity of a fractional-order chaotic system.

7.1.1 Research on Multivariate Permutation Entropy Algorithm The permutation entropy algorithm can only measure the complexity of a single sequence, and the actual systems are mostly multi-dimensional systems, and the system can generate multiple time sequences at the same time. Here, the MvPE algorithm is proposed on the basis of the PE algorithm to measure the complexity of a multivariable system. (1)

Permutation entropy (PE) algorithm

For a given time series {x(n), n = 1, 2, 3, …, N} and the embedding dimension d, the reconstruction sequence can be defined as X(i) = {x(i), x(i + 1), . . . , x(i + d − 1)}

(7.1)

7.1 Behavioral Complexity Algorithm

119

Fig. 7.1 The 6 permutations of the reconstruction vector when d = 3 (π6 )

where i = 1, 2, …, N-d + 1. Sort the reconstructed vector X(i) in ascending order π = (r 0 , r 1 , …, r d-1 ), the value of the reconstructed vector after sorting is expressed as xi+r0 ≤ xi+r1 ≤ . . . ≤ xi+rd−1

(7.2)

Obviously, there is d! kind of arrangement pattern π for the embedded dimension d. Taking d = 3 as an example, as shown in Fig. 7.1, there are 6 arrangement modes: {π 1 , x 1 ≤ x 2 ≤ x 3 }, {π 2 , x 1 ≤ x 3 ≤ x 2 }, {π 3 , x 2 ≤ x 1 ≤ x 3 }, {π 4 , x 3 ≤ x 1 ≤ x 2 }, {π 5 , x 2 ≤ x 3 ≤ x 1 }and {π 6 , x 3 ≤ x 2 ≤ x 1 }. Let π j = j, j = 1, 2, …, d!, if the pattern of X(i) is π j , and let s(i) = j, then a sequence of permutation patterns {s(i), i = 1, 2, …, N-d + 1} is obtained. According to the obtained permutation pattern sequence {s(i), i = 1, 2, …, N-d + 1}, the Bandt–Pompe probability distribution can be defined as p(π j ) =

#{s|i ≤ N − d + 1; s = j } N −d +1

(7.3)

where the symbol # denotes the number of modes. According to the definition of Shannon entropy and the probability distribution obtained by formula (7.3), the normalized permutation entropy can be defined as [11] PE(x, d) = S[ p]/Smax = [−

d! 

p(π j ) ln p(π j )]/Smax

(7.4)

j=1

where S max = S[Pe ] = ln(d!), Pe = {1/d!, …, 1/ d!} represents the maximum entropy value obtained when the probability of each arrangement pattern is uniformly distributed under random conditions. Obviously, the greater the PE value, the higher the complexity of the time series. The value range of the embedded dimension d is generally {2, 3, …, 7} [11]. (2)

Multivariate Permutation Entropy (MPE) algorithm

For a given number of time series, denoted as {x j (n), n = 1, 2, 3, …, N, j = 1, 2, …, d}, where d is the dimension of the system or the number of the selected time series. Due to the inconsistency of the amplitude of the number, it needs to be normalized. The normalization function is x j (n) =

x j (n) − min(x j (n)) max(x j (n)) − min(x j (n))

(7.5)

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7 Complexity Analysis of Fractional-Order Chaotic System

Fig. 7.2 Principles of obtaining the reconstruction vector X(n) by PE algorithm and MvPE algorithm

Differ from the PE algorithm, the vector of the MvPE algorithm is defined by all the sequences consisting of d groups of data, and its definition is X(n) = {x1 (n), x2 (n), . . . , xd (n)}

(7.6)

Therefore, the d groups of data generated by the system are involved in the calculation of complexity. Figure 7.2 further describes the difference between permutation entropy algorithm and multivariate permutation entropy algorithm in obtaining reconstruction vectors. It can be seen from Fig. 7.2 that the reconstruction sequence acquisition methods and objects of the two measurement algorithms are different. The PE algorithm is for a single sequence, and each reconstruction vector can be obtained by moving a small window with a length d backward, while the MvPE algorithm is for multiple time series, the reconstruction vector is composed of the values of time series at each moment. Similar to the PE algorithm, for {X(n), n = 1, 2, 3 …, N}, we can calculate the corresponding pattern sequence {s(n), n = 1, 2, …, N}. In this way, the probability distribution p(π ) of the model can be defined as p(π j ) =

1 #{s|i ≤ N ; s = j } N

(7.7)

therefore, MvPE is defined as MvPE(x) = S[ p]/Smax = [−

M 

p(π j ) ln p(π j )]/Smax

(7.8)

j=1

where M is the number of types of reconstruction vector arrangement patterns. When d = 3, as shown in Fig. 7.3, the sequence has only 6 possible permutation patterns, which cannot always describe the complexity of the sequence well [12], so 3 sub-modes are defined for each pattern on this basis, and the specific results are shown in Fig. 7.3. In this case, M = 18, and each one can get more information about the sequence to measure the complexity more accurately. The calculation formulas for the values of Bot n and Topn are respectively

7.1 Behavioral Complexity Algorithm

121

Fig. 7.3 The 18 permutations of the reconstructed vector when d = 3 in the sub-mode (π18 )

⎧ 2 1 ⎪ ⎨ Botn = min(X(n)) + max(X(n)) 3 3 1 ⎪ ⎩ T op = min(X(n)) + 2 max(X(n)) n 3 3

(7.9)

PE algorithm and MvPE algorithm can obtain relatively accurate measured values when measuring complex nonlinear time series, reflecting the complex characteristics of the series. Unlike the PE algorithm, the MvPE algorithm focuses more on describing the complex relationship between multiple variables. For a chaotic system, it can be considered to describe the complexity of the phase space, i.e., the complexity of the attractor. The two algorithms have some problems when describing the complexity of periodic sequences, and the subsequent measurement results of this article will also show the existence of these problems. The following will analyze the reasons why PE algorithm and MvPE algorithm cannot obtain ideal results when analyzing periodic sequences. Theorem 7.1: When the period of the measured periodic time series is T, the period of the pattern sequence s(n) is less than or equal to T; when the periods of the measured d time series are T 1 , T 2 , …, T d , the period of the pattern sequence s(n) is less than or equal to the least common multiple of T 1 , T 2 , …, T d . Proof: According to formula (7.1) and Fig. 7.2, the reconstruction vector sequence can be expressed as.

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7 Complexity Analysis of Fractional-Order Chaotic System

⎧ X(1) = [x(1), x(2), . . . , x(d)] ⎪ ⎪ ⎪ ⎪ ⎪ X(2) = [x(2), x(3), . . . , x(d + 1)] ⎪ ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ . ⎪ ⎨ X(T ) = [x(T ), x(T + 1), . . . , x(T + d − 1)] ⎪ ⎪ ⎪ X(T + 1) = [x(T + 1), x(T + 2), . . . , x(T + d)] ⎪ ⎪ ⎪ ⎪ ⎪ X(T + 2) = [x(T + 2), x(T + 3), . . . , x(T + d + 1)] ⎪ ⎪ ⎪ ⎪ ⎪ .. ⎩ .

(7.10)

Obviously, because the sequence {x(n), n = 1, 2, 3, …, N} is periodic, the reconstruction sequence also has periodic characteristics. The pattern sequence s(n) obtained from the above reconstruction sequence is also periodic, and its period is T. If the pattern sequence is periodic in one period, the period of the pattern sequence is less than T. Let T LCM be the least common multiple of T 1 , T 2 , …, T d , because T 1 , T 2 , …, T d are all positive integers, then there is a set of positive integers {κ 1 , κ 2 , …, κ d } such that T LCM = κ 1 T 1 = κ 2 T 2 = … = κ d T d . According to formula (7.6) and Fig. 7.2, the reconstruction sequence of the multivariate permutation entropy algorithm can be expressed as ⎧ X(1) = [x1 (1), x2 (1), . . . , xd (1)] ⎪ ⎪ ⎪ ⎪ ⎪ X(2) = [x1 (2), x2 (2), . . . , xd (2)] ⎪ ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ . ⎪ ⎨ X(TLC M ) = [x1 (κ1 T1 ), x2 (κ2 T2 ), . . . , xd (κd Td )] ⎪ ⎪ ⎪ ⎪ X(TLC M + 1) = [x1 (κ1 T1 + 1), x2 (κ2 T2 + 1), . . . , xd (κd Td + 1)] ⎪ ⎪ ⎪ ⎪ X(TLC M + 2) = [x1 (κ1 T1 + 2), x2 (κ2 T2 + 2), . . . , xd (κd Td + 2)] ⎪ ⎪ ⎪ ⎪ ⎪ .. ⎩ .

(7.11)

Therefore, the period of the pattern sequence s(n) is T LCM . If the pattern sequence is also periodic in one period, the period of the pattern sequence T LCM will be less than T LCM . Theorem 7.1 is proved. Therefore, the PE algorithm and the MvPE entropy algorithm cannot describe the complexity of the periodic sequence well. Taking the PE algorithm as an example, suppose there is a periodic sequence {1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, …}. When d = 3, the pattern sequence can be obtained as {1, 1, 1, 4, 5, 1, 1, 1, 1, 4, 5, 1, ….}, obviously it is also a periodic sequence, according to the formula (7.3), the Bandt–Pompe probability distribution is P = [0.6, 0.2, 0.2]. Similarly, when d = 4, the Bandt–Pompe probability distribution is P = [0.4 0.2 0.2 0.2], and when d ≥ 5, the Bandt–Pompe probability distribution is P = [0.2 0.2 0.2 0.2 0.2]. Obviously, for the case of d ≥ 5, the measured value reaches the maximum value 1, indicating that the sequence is a completely random sequence at this case, which does not conform

7.1 Behavioral Complexity Algorithm

123

Fig. 7.4 The Lyapunov exponent spectrum of the hyperchaotic Hénon map with the parameter b varying

to the actual situation. For the MvPE algorithm, when it measures multiple periodic sequences, similar results can be obtained. (3)

Characteristic of Multivariate Permutation Entropy Algorithm

The hyperchaotic Hénon map equation is [13] ⎧ 2 ⎪ ⎨ xn+1 = a − yn − bz n yn+1 = xn ⎪ ⎩ z n+1 = yn

(7.12)

where a and b are parameters of the system. Fix the parameter a = 1.42, the parameter b varies from −0.2 to 0.2, and the step size of b is 0.001. The Lyapunov exponent spectrum of the system is shown in Fig. 7.4. When b ∈ [−0.2, −0.183) ∪ (−0.139, − 0.039) ∪ (−0.027, 0.0148), the system is in a hyperchaotic, when b ∈ [0.0148, 0.2], the system is in a chaotic, and when b ∈ [−0.183, −0.139], the system is a periodic. It can be seen that with the change of parameter b, the system exhibits rich dynamic characteristics. The PE algorithm and the multivariate permutation entropy algorithm are used to analyze the complexity of sequence x of the system with the parameter b varying, and the sequence length is 105 . It can be seen from Fig. 7.5a that when the PE algorithm is used to analyze the complexity, and d = 5, the complexity curve is in better agreement with the maximum Lyapunov exponent. When the MvPE algorithm is used to measure the system complexity, since the system is a three-dimensional system, the complexity measurement uses two permutation modes (π6 and π18 ). From Fig. 7.5b, we can see that when π18 is used, the multivariate complexity measurement result is more accurate and better than the result of permutation entropy algorithm when d = 5. Comparing Figs. 7.4 and 7.5, we can see that the complexity result does not necessarily distinguish between the chaotic state and the hyperchaotic state. When the system is chaotic or hyperchaotic, the complexity is relatively high. The effect of sequence sampling interval τ on the measured value of PE algorithm and MvPE algorithm is different. Take the fractional-order Lorenz system as an example. Set the order q = 0.96, the parameter d = 25, and the simulation time

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7 Complexity Analysis of Fractional-Order Chaotic System

Fig. 7.5 The complexity of hyperchaotic Hénon map with the parameter b varying a PE algorithm result; b MvPE algorithm result

Fig. 7.6 The influence of sampling interval τ on the analysis results of PE algorithm and MvPE algorithm

step size is h = 0.01. For the generated sequence, the sampling interval τ means that take a value per τ digits to form a new sequence. As shown in Fig. 7.6, the probability distribution of each mode of the MvPE algorithm does not change with the increase of the sampling interval τ, i.e., its measured value remains unchanged. For the PE algorithm, as the sampling interval τ increases, its Bandt–Pompe probability distribution is getting more and more uniform, and the measured value is getting larger and larger. It can be seen that the measured value of MvPE is basically not affected by the sampling interval τ, while the PE algorithm is greatly affected by the sampling interval. The literature [14] shows that for the PE algorithm, when the sampling interval τ increases to a certain extent, the measured value will remain unchanged. For PE and MvPE algorithms, the sequence used for calculation should be as long as possible, such as 105 . It can be seen from Fig. 7.5a that for the PE algorithm, when the embedding dimension d = 5, the complexity can be analyzed more accurately. Therefore, unless otherwise stated, the embedding dimension d of the PE algorithm in this chapter is 5, and MvPE is used. When the MvPE algorithm

7.1 Behavioral Complexity Algorithm

125

is used to analyze the three-dimensional system, the arrangement mode under the sub-modes will be selected for the mode probability statistics. Next, we will study the influence of different degrees of noise on the measured values of PE and MvPE. Similarly, analyze the chaotic sequence generated by the fractional-order Lorenz system, and its parameters remain the same as above. The noise signals η˜ 1 , η˜ 2 , and η˜ 3 are added to the chaotic signals x 1 , x 2 , and x 3 respectively, where the noise signal is obtained by combining the normalized noise signal (ηi ) with the amplitude of the chaotic sequence (x i ), and its calculation formula is η˜ i = Pnoise · (max(xi ) − min(xi )) · (ηi − mean(ηi ))

(7.13)

where Pnoise is the scale factor, and the value range is [0, 1]. The chaotic signal contaminated by the noise signal can be defined as x˜i = xi + η˜ i

(7.14)

Figure 7.7 shows the complexity results of the fractional-order Lorenz system after being polluted by different degrees of noise signals. It can be seen that for the PE algorithm, when the proportion of the noise signal reaches 30% of the amplitude of the chaotic signal, the measured value is very close to 1, i.e., the PE algorithm has failed in this case. For the MvPE algorithm, as the proportion of the noise signal increases, the measured value shows a slow increase trend, which is relatively less affected by noise. The reason is that the MvPE algorithm measures the complexity of the phase space, when the magnitude of the noise is not large enough to affect the relative relation between different sequences, it will have little effect on the measured value. For the PE algorithm, appropriate noise helps to better measure the complexity, but when the noise amplitude continues to increase, the algorithm will fail [11]. According to the above analysis, compared with the PE algorithm, the MvPE algorithm has the following advantages or characteristics. (1)

MvPE algorithm has better noise robustness, and its measured value is less affected by noise;

Fig. 7.7 The influence of noise signal on the measured values of PE and MvPE

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7 Complexity Analysis of Fractional-Order Chaotic System

(2)

MvPE algorithm measures the complexity of multiple time series and reflects the complexity of the phase space of the system; For a three-dimensional system, using the arrangement mode under the submode, the MvPE algorithm can obtain more time series information and better measure the system complexity.

(3)

Compared with the MvFuzzyEn algorithm and the MvSampEn algorithm, the MvPE algorithm has a faster calculation speed and is more suitable for online detection of the complexity of multivariable systems.

7.1.2 Complexity Analysis of Fractional-Order Chaotic Systems ➀

The multivariate complexity varies with the order q varying

The MvPE algorithm is used to measure the complexity of fractional-order Lorenz q q q system (Dt0 x1 = 10(x2 − x1 ), Dt0 x2 = cx1 −x1 x3 +d x2 , Dt0 x3 = x1 x2 −bx2 , a = 40, q b = 3, c = 10), the fractional-order Lorenz hyperchaotic system (Dt0 x = 10(y − x), q q q Dt0 y = 28x − x z + y − u, Dt0 z = x y − 8z/3, Dt0 u = Ryz) and the fractional-order q q simplified Lorenz hyperchaotic system (Dt0 x = 10(y − x), Dt0 z = x y − 8z/3, q q Dt0 y = (24 − 4c)x − x z + cy + u, Dt0 u = −kx), where sequence length N = 105 , time step size h = 0.01, fractional-order Lorenz system parameter d = 25, fractional-order Lorenz hyperchaotic system parameter R = 0.21, and the fractionalorder simplified Lorenz hyperchaotic system parameter c = −1, k = 5. The MvPE complexity measurement results of each system with the order q varying are shown in Fig. 7.8. It can be seen from Fig. 7.8a that the MvPE complexity value of the fractionalorder Lorenz system fluctuates between 0.65 and 0.75 with the order q varying. According to Fig. 6.3, the fractional-order Lorenz system is mainly chaotic when the order q varies. Therefore, when the order of the fractional-order Lorenz system q varies, the phase space complexity of the system changes relatively little. It can be seen from Fig. 7.8b, c that the MvPE measured values of the fractional-order

Fig. 7.8 The MvPE complexity of the fractional-order chaotic system with the order q varying (a) Fractional-order Lorenz system; (b) Fractional-order Lorenz hyperchaotic system; (c) Fractional-order simplified Lorenz hyperchaotic system

7.1 Behavioral Complexity Algorithm

127

Lorenz hyperchaotic system and the fractional-order simplified Lorenz hyperchaotic system with the order q varying and the corresponding Lyapunov exponent spectrum results are consistent, but among them the MvPE value of the fractional-order Lorenz hyperchaotic system remains basically unchanged when the order q varies, while the multivariate complexity of the fractional-order simplified Lorenz hyperchaotic system decreases with the increase of the order q. In short, the multivariate complexity of each system remains basically unchanged when the order q varies, which shows that the complexity of the attractor of the system does not increase in the case of fractional order. In fact, the literature [15] shows that most of fractional-order chaotic systems, including Lorenz family systems, have no more complex attractors in the case of fractional-order compared to the corresponding integer-order systems. ➁

Multivariate complexity varies with parameters varying

The MvPE algorithm is used to analyze the complexity of the fractional-order Lorenz system, the fractional-order Lorenz hyperchaotic system, and the fractional-order simplified Lorenz hyperchaotic system with parameters varying. The results are shown in Fig. 7.9, where the sequence length is N = 105 , the time step is h = 0.01,

Fig. 7.9 The MvPE complexity of the fractional-order chaotic system with parameters varying (a) Fractional-order Lorenz system parameter d varying; (b) Fractional-order Lorenz hyperchaotic system parameter R varying; (c) Fractional-order simplified Lorenz hyperchaotic system parameter k varying; (d) Fractional-order simplified Lorenz hyperchaotic system parameter c varying

128

7 Complexity Analysis of Fractional-Order Chaotic System

the order of the fractional-order Lorenz system is q = 0.96, the order of the fractionalorder Lorenz hyperchaotic system is q = 0.96, and the order of the fractional-order simplified Lorenz hyperchaotic system is q = 0.97. In Fig. 7.9a, the change range of the parameter d of the fractional-order Lorenz system is [0, 38], and the parameter change step size is 0.076. Comparing with the system dynamics analysis results shown in Fig. 6.5, the MvPE complexity is basically consistent with it, i.e., the complexity measurement result of the system in chaotic state is significantly higher than the complexity measurement result of the system in periodic state, but the MvPE algorithm fails to find the several smaller periodic windows of the system. When the system is chaotic, the MvPE complexity basically remains unchanged. Therefore, the attractor of the system in chaotic state does not become more complicated. In Fig. 6.46b, the variation range of the parameter R of the fractional-order Lorenz hyperchaotic system is 0 to 1, and the variation step size is 0.002. It can be seen that when R < 0.5, the multivariate complexity of the fractional-order Lorenz hyperchaotic system firstly decreases and then increases with the increase of R. This corresponds to the analysis results of the dynamic characteristics; when R > 0.5, the system is periodic, the MvPE complexity change mode is relatively simple, and the measured value is smaller than the chaotic state. In Fig. 7.9c, d, the variation ranges of the parameters k and c of the fractional-order simplified Lorenz hyperchaotic system are [0, 50] and [−6, 4], respectively, and the step size is 0.1 and 0.02, respectively. Figure 7.9c shows that the multivariate complexity of the system at k ∈ (0, 22] is consistent with the corresponding maximum Lyapunov exponent spectrum, and then the complexity decreases, and the system is non-chaotic. In Fig. 7.9d, the high-complexity area of the system is mainly concentrated between c ∈ (−4, 3). Therefore, the third-order chaotic system has good complexity performance, and the MvPE complexity curve agrees well with the dynamic characteristics. ➂

Multivariable complexity-based chaos diagrams of fractional-order chaotic systems

The chaos diagrams of the fractional-order Lorenz system, the fractional-order Lorenz hyperchaotic system, and the fractional-order simplified Lorenz hyperchaotic system based on MvPE complexity are shown in Fig. 7.10. In Fig. 7.10, the ordinate is the order q, and the abscissa is the system parameters (d, R, and c). The drawing method is depicted as divide the parameter and order plane into 101 × 101 dot array, the MvPE complexity at each point is calculated and expressed in the contour map. Compared with the parameter-fixed method to analyze the system complexity, drawing a chaos diagram requires more calculations, and it can get more information about the system complexity, and can better grasp the system complexity from a macro perspective. The q-d plane multivariate complexity of the fractional-order Lorenz system is shown in Fig. 7.10a. It can be seen that the high-complexity area is mainly concentrated between d ∈ (10, 30). As the value of q and d increase, the chaotic area gradually decreases. As shown in Fig. 7.10a, the high-complexity region of the q-R plane of the fractional-order Lorenz hyperchaotic system is mainly between R ∈ (0, 0.4). When q takes a value of about 0.8, the system has a wider high-complexity

7.1 Behavioral Complexity Algorithm

129

Fig. 7.10 The multivariable permutation entropy-based chaos diagrams of fractional-order chaotic systems (a) Fractional-order Lorenz system parameter q-d plane; (b) Fractional-order Lorenz hyperchaotic system parameter q-R plane; (c) Fractional-order simplified Lorenz hyperchaotic system parameter q-c plane

area. As the value of q increases, the high-complexity area gradually shrinks. The high-complexity region of fractional-order simplified Lorenz hyperchaotic system tends to decrease as the value of q increases in the middle of the parameter plane. The chaos diagram of multivariate complexity provides an effective basis for the selection of system parameters.

7.1.3 Research on Improved Multi-Scale Permutation Entropy Algorithm In fact, the existing multi-scale coarse-graining process still has shortcomings. Firstly, the coarse-grained process is irreversible, i.e., the coarse-grained sequence cannot be restored to the original sequence. Secondly, as the scale factor increases, the more detailed information about sequence changes will be lost in a multi-scale coarse-grained sequence. Here, we improve the coarse-graining process and propose an improved multi-scale permutation entropy algorithm. The improved multi-scale permutation entropy algorithm is described as follows. Step 1: Coarse-graining process. For a given one-dimensional discrete time sequence {x(i): i = 1, 2, …, N}, through sampling, the following reconstructed “coarse-grained” sequence can be obtained Y s (i, j) = xi+( j−1)s

(7.15)

where i = 1, 2, …,s, 1 ≤ j ≤ [N/s]. Obviously, when s = 1, the sequence Y s is the original sequence {x(i): i = 1, 2, …, N}, when s > 1, s coarse-grained sequences can be obtained. Step 2: Improve the entropy complexity of multi-scale permutation. The entropy complexity of the improved multi-scale permutation is defined as

130

7 Complexity Analysis of Fractional-Order Chaotic System

1 PE(Y s (i, :), d) s i=1 s

MMPE(x, s, d) =

(7.16)

Obviously, the improved multi-scale permutation entropy complexity is the average value of the permutation entropy of s coarse-grained sequences. The multi-scale permutation entropy algorithm can obtain s complexity at different scales, and a given value is often needed when actually analyzing the complexity of the system. Therefore, the mean value of the improved multi-scale permutation entropy algorithm is defined as E MMPE =

Smax 1 

Smax

MMPE(x, s, d)

(7.17)

s=1

Among them, S max is the maximum value of the scale factor. This definition can also be applied to the multi-scale permutation entropy algorithm, E MPE =

Smax 1 

Smax

MPE(x, s, d)

(7.18)

s=1

The difference between MPE and MMPE is mainly reflected in the following two aspects. (1)

Coarse-graining methods are different, but there is a connection between the two, 1 s Y (i, j) s i=1 s

y s ( j) =

(2)

(7.19)

The coarse-graining process of MMPE is reversible, i.e., the original sequence can be restored from the coarse-grained sequence, while the coarse-graining process of MPE is an irreversible process. The MMPE measured value is the average of the PE values of s sequences, and the result is more stable than MPE. In order to compare the different performances of MMPE and MPE in measuring the multi-scale complexity of chaotic sequences, the following experiments are carried out in this chapter. The MPE and MMPE measurement results of the Logistic map time series and the fractional-order Lorenz system time series are shown in Fig. 7.11, where the length of the sequence is N = 104 , d = 5, the Logistic map equation is μ(n+1) = 4μ(n)(1−μ(n)), and the fractional-order Lorenz system parameters are q = 1, a = 10, b = 8/3, c = 28. It can be seen from Fig. 7.11 that as the scale factor increases, the MMPE measured value becomes more stable. The multi-scale complexity of the Logistic map remains basically unchanged, while the multi-scale complexity

7.1 Behavioral Complexity Algorithm

131

Fig. 7.11 Comparison of MMPE and MPE analysis results (a) MPE algorithm result; (b) MMPE algorithm result

of the continuous chaotic system increases first and then remains stable. The complexity measurement results of the MPE algorithm and the MMPE algorithm applied to the fractional-order Lorenz system under different orders are shown in Fig. 7.12, where the system orders are q = 0.95 and q = 0.90, respectively. It can be seen from the figure that when the order q = 0.90, the system complexity is higher, which is consistent with the previous Lyapunov exponent spectrum analysis results. According to Fig. 7.12b, d, it can be seen that the MMPE algorithm measures more accurately under higher scale factors.

Fig. 7.12 MMPE and MPE complexity of fractional-order Lorenz system with different orders (a) MPE algorithm result; (b) MPE algorithm result with large scale factor; (c) MMPE algorithm result; (d) MMPE algorithm result with large scale factor

132

7 Complexity Analysis of Fractional-Order Chaotic System

Therefore, the designed MMPE algorithm has better performance than the MPE algorithm.

7.1.4 Complexity Analysis of Fractional-Order Chaotic System (1)

Complexity analysis with the order q

The improved multi-scale permutation entropy algorithm (MMPE) is used to calculate the complexity of the fractional-order Lorenz system, the fractional-order Lorenz hyperchaotic system, and the fractional-order simplified Lorenz hyperchaotic system. The system parameters are the same as the previous section. When calculating MMPE, the embedding dimension of the permutation entropy algorithm (PE) is d = 5. The MMPE measurement results of the three fractional-order chaotic systems with the order q varying are shown in Fig. 7.13. It can be seen from the figure that the MMPE complexity of a fractional-order chaotic system tends to decrease as the order increases. In Fig. 7.13a, the MMPE measurement result is consistent with the change trend of the maximum Lyapunov exponent of the fractional-order Lorenz system, indicating that the system has higher complexity when the order q is a small value, and MMPE can detect the periodic windows of the system. In Fig. 7.13b, when the order q is greater than 0.66, the complexity of the system is greater than that of the periodic state, and the change trend of the complexity curve is basically consistent with the maximum Lyapunov exponent spectrum of the system. In Fig. 7.13c, the complexity of the system in periodic state is smaller. When the order of the system is close to 0.6, the complexity of the system is close to zero, and the Lyapunov exponent spectrum shows a rapid upward trend, which corresponds to the result of the bifurcation diagram. In this case, the system is multi-periodic, i.e., compared to the system’s Lyapunov exponent spectrum result, the complexity can better describe the system complexity from other angles. (2)

Improved multi-scale permutation entropy characteristics with parameters varying

Fig. 7.13 The improved multi-scale permutation entropy complexity of fractional-order chaotic system with the order q varying (a) Fractional-order Lorenz system; (b) Fractional-order Lorenz hyperchaotic system; (c) Fractional-order simplified Lorenz hyperchaotic system

7.1 Behavioral Complexity Algorithm

133

Fig. 7.14 The improved multi-scale permutation entropy complexity of the fractional-order chaotic system with parameters varying a Fractional-order Lorenz system parameter d varying; b Fractionalorder Lorenz hyperchaotic system parameter R varying; c Fractional-order simplified Lorenz hyperchaotic system parameter k varying; d Fractional-order simplified Lorenz hyperchaotic system parameter c varying

The MMPE algorithm is used to analyze the complexity of the fractional-order Lorenz system, the fractional-order Lorenz hyperchaotic system, and the fractional-order simplified Lorenz hyperchaotic system with parameters varying. The results are shown in Fig. 7.14. It can be seen from Fig. 7.14 that the complexity change law of the system with parameters varying agrees well with the dynamic characteristics of the corresponding system. Compared with the MvPE measurement result, the MMPE measurement result can better reflect the dynamic characteristics of the system. In Fig. 7.14a, the depression of the complexity curve indicates that the system is periodic in this case. In Fig. 7.14b, at k = 0.3, a window with a relatively small complexity measured value is observed, but in fact the system is periodic in this case. Figure 7.14c, d is the improved multi-scale permutation entropy results of the fractional-order simplified Lorenz hyperchaotic system with parameters varying. It can be seen from the figures that the complexity is higher when the system is chaotic. In short, the MMPE algorithm can effectively analyze the system dynamics, but the change trend of MMPE complexity between different states is not as obvious as the Lyapunov exponent spectrum or bifurcation diagram.

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7 Complexity Analysis of Fractional-Order Chaotic System

Fig. 7.15 The improved multi-scale permutation entropy-based chaos diagrams of fractionalorder chaotic systems (a) Fractional-order Lorenz system parameter q-d plane; (b) Fractionalorder Lorenz hyperchaotic system parameter q-R plane; (c) Fractional-order simplified Lorenz hyperchaotic system parameter q-c plane

(3)

The multi-scale permutation entropy-based chaos diagrams

The chaos diagrams of the fractional-order Lorenz system, the fractional-order Lorenz hyperchaotic system, and the fractional-order simplified Lorenz hyperchaotic system based on the improved multi-scale permutation entropy are shown in Fig. 7.15. The high-complexity region is the chaotic region, similar to Fig. 7.10. The chaotic region of the fractional-order Lorenz system is mainly concentrated between d ∈ (10, 30). When the value of q is small and the value of d is between 20 and 30, the MMPE complexity of the system is higher; the chaotic area of the fractional-order Lorenz hyperchaotic system is mainly in the R < 0.5, and the high-complexity area is clearer than that in Fig. 7.10b; the chaotic area of the fractional-order simplified hyperchaotic Lorenz system is mainly on the right side of the parameter plane, and the larger the value of order q and parameter c, the wider the chaotic area.

7.2 Structural Complexity Algorithm 7.2.1 SE and C0 Complexity Spectral Entropy (SE) [16] uses Fourier transform to obtain the corresponding spectral entropy value through the energy distribution in the Fourier transform domain, combined with Shannon entropy; the main calculation principle of C 0 complexity is to decompose the sequence into regular and irregular components. Its measured value is the proportion of irregular components in the sequence [17]. The SE algorithm and the C 0 algorithm are described as follows. (1)

Spectrum entropy complexity calculation algorithm

Step 1: Eliminate DC. For the chaotic pseudo-random sequence {x(n), n = 0, 1, 2, …, N-1} of length N, the following formula is used to remove the DC part, so that the frequency spectrum can reflect the signal energy information more effectively. x(n) = x(n) − x

(7.20)

7.2 Structural Complexity Algorithm

where x =

1 N

N −1

135

x(n).

n=0

Step 2: Fourier transform. Discrete Fourier Transform of the sequence X (k) =

N −1 

x(n)e− j

2π N

nk

=

n=0

N −1 

x(n)W Nnk

(7.21)

n=0

where k = 0, 1, 2, …, N−1. Step 3: Calculate se. For the transformed X(k) sequence, take the first half of the calculation. According to Paserval’s theorem, calculate the power spectrum value of a certain frequency point p(k) =

1 |X (k)|2 N

(7.22)

where k = 0, 1, 2, …, N/2–1, so the total power of the sequence is defined as ptot =

N /2−1 1  |X (k)|2 N n=0

(7.23)

Then the relative power spectrum probability Pk of the sequence is Pk =

obviously,

N /2−1

p(k) = ptot

1 N

1 |X (k)|2 N N /2−1

|X (k)|

= 2

|X (k)|2 N /2−1 |X (k)|2

k=0

(7.24)

k=0

Pk = 1. Using the relative power spectral density Pk , combined

n=0

with the concept of Shannon entropy, the spectral entropy se of the signal is obtained as N /2−1

se = −



Pk ln Pk

(7.25)

k=0

in the formula, if Pk is 0, then Pk ln Pk is defined as 0. Step 4: Calculate the normalized SE. It can be proved that the size of the spectral entropy converges to ln(N/2). In order to facilitate comparative analysis, the spectral entropy is normalized, and the normalized spectral entropy SE is

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7 Complexity Analysis of Fractional-Order Chaotic System N /2

SE(N ) =

 1 Pk ln Pk ln(N /2) k=0

(7.26)

It can be seen that when the sequence power spectrum distribution is more unbalanced, the sequence spectrum structure is simpler, and the signal has obvious oscillation laws. The smaller the SE measured value obtained, i.e., the smaller the complexity, otherwise the greater the complexity. (2)

C 0 complexity calculation algorithm

Step 1: First remove the irregular part, set the mean square value of {X(k), k = 0,1, 2, …, N-1} as GN =

N −1 1  |X (k)|2 N k=0

(7.27)

Introduce the parameter r, keep the spectrum more than r times the mean square value, and set the rest to zero [17], i.e.,  X˜ (k) =

X (k) i f |X (k)|2 > ξ G N 0 i f |X (k)|2 ≤ ξ G N

(7.28)

Step 2: Inverse Fourier transform. For the X˜ (k), we have inverse Fourier transform x(n) ˜ =

N −1 N −1 1  ˜ 1  ˜ 2π X (k)e j N nk = X (k)W N−nk , N k=0 N k=0

(7.29)

where n = 0, 1, …, N−1. Step 3: Calculate C 0 complexity. Define C 0 complexity as [17] C0 (r, N ) =

N −1  n=0

2 |x(n) − x(n)| ˜ /

N −1 

|x(n)|2

(7.30)

n=0

The C 0 complexity algorithm based on the FFT transform removes the regular part of the signal transformation domain, leaving the irregular part. The larger the proportion of the irregular part of the sequence energy, the closer the corresponding time domain signal is to the random sequence, the greater the complexity. And the C 0 complexity algorithm has many excellent properties [18, 19]. The influence of the time sequence length N on the structure complexity is shown in Fig. 7.16. It can be seen that when the sequence length is greater than 2 × 104 , the measurement is stable. In the subsequent complexity analysis of the fractional-order chaotic system, unless otherwise stated, the sequence length value is taken as N = 105 .

7.2 Structural Complexity Algorithm

137

Fig. 7.16 The influence of time series length N on structure complexity (a) C 0 complexity; (b) SE complexity

7.2.2 Structural Complexity Analysis of Fractional-Order Chaotic System (1)

Structure complexity varies with order q varying

The structural complexity measurement results of the fractional-order Lorenz system, the fractional-order Lorenz hyperchaotic system, and the fractional-order simplified Lorenz hyperchaotic system with the order q are shown in Figs. 7.17 and 7.18.

Fig. 7.17 The C 0 complexity of the fractional-order chaotic system with the order q varying (a) Fractional-order Lorenz system; (b) Fractional-order Lorenz hyperchaotic system; (c) Fractional-order simplified Lorenz hyperchaotic system

Fig. 7.18 SE complexity of the fractional-order chaotic system with the order q varying (a) Fractional-order Lorenz system; (b) Fractional-order Lorenz hyperchaotic system; (c) Fractional-order simplified Lorenz hyperchaotic system

138

7 Complexity Analysis of Fractional-Order Chaotic System

The two figures are calculated using C 0 complexity and SE algorithm respectively. It can be seen from Figs. 7.17 and 7.18 that based on the two algorithms, basically consistent conclusions can be obtained. The complexity of the fractionalorder chaotic system shows a decreasing trend with the increase of the order q, and the decreasing trend of C 0 complexity is more obvious. The frequency-domain structure complexity of the sequence shows that the fractional-order chaotic system has higher complexity than the integer-order system, and is more conducive to practical applications. (2)

Structural complexity with parameters varying

The C 0 complexity and SE complexity measurement results of the fractional-order Lorenz system, the fractional-order Lorenz hyperchaotic system, and the fractionalorder simplified Lorenz hyperchaotic system with parameters varying are shown in Figs. 7.19 and 7.20, respectively. It can be seen from the figure that the change trend of C 0 algorithm and SE algorithm are basically the same, and they can describe the complexity of the system well, and comparing the bifurcation diagram of the corresponding system with the Lyapunov exponent spectrum results, it can be seen that the structural complexity and dynamic characteristics have good consistency. When analyzing the complexity of the fractional-order Lorenz system, the SE complexity is measured when the system

Fig. 7.19 The C 0 complexity of the fractional-order chaotic system with parameters varying (a) Fractional-order Lorenz system parameter d varying; (b) Fractional-order Lorenz hyperchaotic system parameter R varying; (c) Fractional-order simplified Lorenz hyperchaotic system parameter k varying; (d) Fractional-order simplified Lorenz hyperchaotic system parameter c varying

7.2 Structural Complexity Algorithm

139

Fig. 7.20 The spectral entropy complexity of the fractional-order chaotic system with parameters varying (a) Fractional-order Lorenz system parameter d varying; (b) Fractional-order Lorenz hyperchaotic system parameter R varying; (c) Fractional-order simplified Lorenz hyperchaotic system parameter k varying; (d) Fractional-order simplified Lorenz hyperchaotic system parameter c varying

is convergent, and the measurement performance of the algorithm is not ideal in this case. Compared with the SE algorithm, the C 0 algorithm measures better. When the system is periodic or convergent, the complexity of C 0 tends to zero, but the measured value of the C 0 algorithm is more volatile. Compared with the MvPE algorithm and the MMPE behavioral complexity algorithm, the C 0 algorithm and the SE algorithm have a better distinguishing performance. (3)

The structural complexity-based chaos diagrams of fractional-order chaotic systems

The chaos diagrams of the fractional-order Lorenz system, the fractional-order Lorenz hyperchaotic system, and the fractional-order simplified Lorenz hyperchaotic system based on the C 0 algorithm and the SE algorithm are shown in Figs. 7.21 and Fig. 7.22, respectively. The system parameter setting and measurement sequence are consistent with the previous ones. It can be seen from the figure that in the parameter plane, the chaotic region and periodic region of the system are similar to the behavioral complexity algorithm-based chaos diagram results. Because the structure complexity algorithm result is more consistent with the Lyapunov exponent spectrum result, the boundary of high-complexity area and the low-complexity area of

140

7 Complexity Analysis of Fractional-Order Chaotic System

Fig. 7.21 C 0 complexity-based chaos diagrams of fractional-order chaotic systems (a) Fractionalorder Lorenz system parameter q-d plane; (b) Fractional-order Lorenz hyperchaotic system parameter q-R plane; (c) Fractional-order simplified Lorenz hyperchaotic system parameter q-c plane

Fig. 7.22 Spectral entropy complexity-based chaos diagrams of fractional-order chaotic systems (a) Fractional-order Lorenz system parameter q-d plane; (b) Fractional-order Lorenz hyperchaotic system parameter q-R plane; (c) Fractional-order simplified Lorenz hyperchaotic system parameter q-c plane

the structural complexity-based chaos diagram is more obvious. Comparing the C 0 algorithm and the SE algorithm, the C 0 algorithm has a better performance.

References 1. Villazana S, Seijas C, Caralli A (2015) Lempel-Ziv complexity and Shannon entropy-based support vector clustering of ECG signals [J]. Revista Ingeniería Uc 22(1):7–15 2. Raghu S, Sriraam N, Kumar GP (2015) Effect of wavelet packet log energy entropy on electroencephalogram (EEG) signals [J]. Int J Biomed Clin Eng 44(1):32–43 3. Topcu C, Akgul A, Bedeloglu M et al. (2015) Entropy analysis of surface EMG for classification of face movements [C]. In: Signal Processing and Communications Applications Conference (SIU), 2015 23th International Congress IEEE, pp 1–4 4. Xiang ZT, Chen YF, Li YJ et al (2014) Complexity analysis of traffic flow based on multi-scale entropy [J]. Acta Phys Sin 63(3):038903 5. Fan CL, Jin ND, Chen XT (2013) Multi-scale permutation entropy: a complexity measure for discriminating two-phase flow dynamics [J]. Chinese Phys Lett 30(9):090501 6. Li D, Li X, Liang Z et al (2010) Multiscale permutation entropy analysis of EEG recordings during sevoflurane anesthesia. J Neural Eng 7(4):371–371 7. Wu SD, Wu CW, Lee KY et al (2013) Modified multiscale entropy for short-term time series analysis. Phys A 392(392):5865–5873 8. Mosabber Uddin A, Mandic DP (2011) Multivariate multiscale entropy: a tool for complexity analysis of multichannel data. Phys Rev E 84(6):3067–3076

References

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9. Richman JS (2011) Multivariate neighborhood sample entropy: a method for data reduction and prediction of complex data. Methods Enzymol 487:397–408 10. Li P, Liu CY, Li LP et al (2013) Multiscale multivariate fuzzy entropy analysis. Acta Phys Sin 62(12):120512 11. Bant C, Pompe B (2002) Permutation entropy: a natural complexity measure for time series. Phys Rev Lett 88(17):1741–1743 12. Sun KH, He SB, Sheng LY (2011) Complexity analysis of chaotic sequence based on the intensive statistical complexity algorithm [J]. Acta Phys Sin 60(2):020505 13. Baier G, Klein M (1990) Maximum hyperchaos in generalized Hénon maps. Phys Lett A 151(6):281–284 14. Micco LD, Fernández JG, Larrondo HA et al (2012) Sampling period, statistical complexity, and chaotic attractors. Phys A 391(8):2564–2575 15. Letellier C, Aguirre LA (2013) Dynamical analysis of fractional-order Rössler and modified Lorenz systems. Phys Lett A 377(28):1707–1719 16. Phillip PA, Chiu FL, Nick SJ (2009) Rapidly detecting disorder in rhythmic biological signals: a spectral entropy measure to identify cardiac arrhythmias. Phys Rev E 79(1):011915 17. Shen EH, Cai ZJ, Gu FJ (2005) Mathematical foundation of a new complexity measure. Appl Math Mech 26(9):1188–1196 18. Sweilam NH, Assiri TA (2015) Non-standard Crank-Nicholson method for solving the variable order fractional cable equation. Appl Math Inf Sci 9(2):943–951 19. Charef A, Sun HH (1992) Fractal system as represented by singularity function. IEEE Trans Autom Control 37(9):1465–1470

Chapter 8

Circuit Design and Realization of Fractional-Order Chaotic System

The circuit realization of the fractional-order chaotic system is the basis of the application research of the fractional-order chaotic system, and it can be divided into analog circuit realization and digital circuit realization. The realization of the analog circuit of the fractional-order chaotic system is based on the time–frequency domain approximation algorithm. Generally, the approximated fractional operator calculated by Ahmad [1] is directly used to realize the analog circuit of the fractional-order chaotic system in the form of equivalent circuit. And the digital circuit implementation of the fractional-order chaotic system is mainly based on the Adomian decomposition algorithm. In this chapter, some fractional-order chaotic systems analyzed above are realized by analog circuits, DSP and FPGA. Firstly, based on the time– frequency domain approximation algorithm, the fractional-order simplified Lorenz chaotic system at q = 0.965 is realized by using the analog circuit. Through circuit simulation, the minimum order at which the system is chaotic under the algorithm is verified. Then, the digital circuit realization method of the fractional-order chaotic system is studied, and the ideas and steps of the digital circuit realization are given, and the DSP and FPGA experimental verification are completed. Finally, the circuit implementation methods of the fractional-order chaotic system are compared.

8.1 Research on Analog Circuit of Fractional Chaos System The analog circuit realization of the fractional-order chaotic system is based on the time–frequency domain approximate solution algorithm [2], and the design of the equivalent circuit of the fractional integral operation is the key. According to the Laplace transform of fractional calculus [3], the time domain equation is converted to the frequency domain, and the fractional integral operator with order q in the frequency domain is described by a transfer function H(s) = 1/sq . Then, integerorder operators are used to approximate H(s) = 1/sq . Here, the equivalent circuit of © Science Press 2022 K. Sun et al., Solution and Characteristic Analysis of Fractional-order Chaotic Systems, https://doi.org/10.1007/978-981-19-3273-1_8

143

144

8 Circuit Design and Realization of Fractional-Order … C

vi

1/sq

R vo

vi

R vo

(a)

(b)

Fig. 8.1 Integral circuit. a Integer-order integral circuit. b Fractional-order integral circuit

the fractional integral operator whose order is q is designed by using series circuit unit, tree circuit unit, or hybrid circuit unit. Comparing the equations of the fractional-order chaotic system and the integerorder chaotic system, it can be found that the difference between them is mainly the differential operator. The fractional-order chaotic system uses fractional-order differential operators, while the integer-order chaotic system uses integer-order differential operators. Therefore, the circuit design of a fractional-order chaotic system has many similarities with the circuit design of the corresponding integer-order chaotic system. The main difference lies in the integral circuit. Figure 8.1a and b show the integral circuits of the integer-order system and the fractional-order system, respectively. Taking q = 0.965 and q = 0.945 as examples, and according to the time– frequency domain approximation algorithm, the fractional-order simplified Lorenz chaotic system with an order step of 0.001 is realized by using analog circuit. First, set the approximate error y = 1 dB, and wmax = 100 rad/s, and PT = 0.01. Based on the time–frequency domain conversion method in Chap. 2, the integral operators with q = 0.965 and q = 0.945 can be obtained as follows: H (s) = H (s) =

1 s 0.945

≈

1 s 0.965

≈

1.5455(s + 8.1088)(s + 7408.4) , (s + 0.0113)(s + 10.294)(s + 9404.9)

1.8213(s + 0.7432)(s + 62.387)(s + 5237.2) . (s + 0.0113)(s + 0.9482)(s + 79.6002)(s + 6682.1)

(8.1) (8.2)

According to the number of poles and zeros, the tree equivalent circuit of Eq. (8.1) is designed as shown in Fig. 8.2a: The transfer function between A and B in Fig. 8.2a is expressed as     1 1 1 H (s) = R1 + (R2// ) // + (R3// ) sC2 sC1 sC3 =

1 R1+R2 C0 s 3 + ( R1C2R2 +

1 C3R3

+

C0 R1+R2 1 ( C0 C1 + C3 )(s + R1C2R2 )(s + C1R3+C3R3 ) C1+C3 R1+R2 1 C1+C3 2 +( 2 )s + C1R1C3 R1C2R2C3R3 C1R1C3R3 + C1R1C2R2C3 )s

+

1 C1R1C2R2C3R3

(8.3)

Among them, “//” represents the parallel relationship, and C0 is the unit parameter, and let C0 = 1 nF. Comparing Eq. (8.3) with Eq. (8.1), the parameter values of each component in the equivalent circuit can be obtained: R1 = 774.32 , R2 = 84.87 M, R3 = 36.18 k, C1 = 0.8681 nF, C2 = 0.1743 nF, C3 = 2.5406

8.1 Research on Analog Circuit of Fractional Chaos System

145 R3

R2 84.87M¡ R1 A

774.32¡ C1

(a)

0.8681nF

R2

C2 0.1743nF R3 36.18k¡ C3 2.5406nF

R1

B

1.2456k¡ A

(b)

77.52M¡ C2

83.136k¡ 0.1901nF C3

B

0.1556nF R4 C1 0.6892nF

396.98k¡ C4 2.700nF

Fig. 8.2 1/sq equivalent cell circuit. a q = 0.965, b q = 0.945

nF. The equivalent circuit of Eq. (8.2) and component parameters are obtained in the same way as shown in Fig. 8.2b. Based on the differential equations of the fractional-order simplified Lorenz system, the analog circuit is designed using a modular design method. According to the Laplace transform of the fractional-order system, the integral circuit of the integer-order system is replaced with 0.965 order and 0.945 order equivalent circuits respectively, and the circuit of fractional-order simplified Lorenz system is shown in Fig. 8.3. When c = 5, the corresponding attractor phase diagram obtained by Multisim simulation software is shown in Fig. 8.4. Given y = 1 dB, wmax = 100 rad/s, PT = 0.01, the fractional calculus operators with q = 0.929 and q = 0.930 are approximated in the frequency domain by the integer-order system function as shown in Eqs. (8.4) and (8.5). The phase diagrams of the attractor of the fractional-order simplified Lorenz system with q = 0.929 and q = 0.930 obtained by circuit simulation are shown in Fig. 8.5. It shows that in the simulation experiment of the analog circuit, when the fractional-order simplified Lorenz system is solved by employing the time–frequency domain approximation algorithm, the step size of q is 0.001, and the minimum order where the system is chaotic is q = 0.930. While in [4], under the same conditions, the fractionalorder simplified Lorenz system with q = 0.95 is realized, which is not the minimum order. Previously, Adomian decomposition algorithm is used to solve and analyze the fractional-order simplified Lorenz system. When c = 5, the minimum order where the system is chaotic is q = 0.595. The results obtained by the two algorithms are different.

H (s) =

1 1.7169(s + 0.2899)(s + 9.5140)(s + 312.2185) ≈ , (8.4) s 0.929 (s + 0.0113)(s + 0.3715)(s + 12.1901)(s + 400.0381) H (s) =

1 1.3453(s + 10.43)(s + 0.3036) ≈ . s 0.930 (s + 13.36)(s + 0.3889)(s + 0.01132)

(8.5)

146

8 Circuit Design and Realization of Fractional-Order …

Fig. 8.3 The analog circuit of fractional-order simplified Lorenz system

In practice, it is difficult to find the resistances and capacitances that meet the values in Fig. 8.2. Even if multiple resistors or capacitors are used in combination, it is difficult to completely match the values. When making circuits, custom or approximate resistors and capacitors can be used instead. Therefore, a deviation occurs

8.1 Research on Analog Circuit of Fractional Chaos System

147

Fig. 8.4 The attractor phase of the fractional-order simplified Lorenz system. a q = 0.965 b q = 0.945

Fig. 8.5 The attractor phase diagram of the fractional-order simplified Lorenz system from circuit simulation. a q = 0.929 b q = 0.930

at first when the analog circuit of the fractional-order simplified Lorenz system is realized according to Fig. 8.3. Figure 8.6 shows the fractional-order simplified Lorenz system with q = 0.965 realized by selecting resistors and capacitors approximate to the values in Fig. 8.2a. The corresponding attractor observed by the oscilloscope is shown in Fig. 8.7, which is similar to the simulation result showing a chaotic state. Through the above analysis, it can be found that some factors in the fractional-order chaotic system implemented by the analog circuit are difficult to control, and the realized results may deviate from the theoretical simulation results. It is inconvenient to adjust the parameters, and the implementation method is not flexible. These factors directly affect the performance and practical application of the fractional-order chaotic system.

148

8 Circuit Design and Realization of Fractional-Order …

Fig. 8.6 Analog circuit of fractional-order simplified Lorenz system

Fig. 8.7 The attractor of the fractional-order simplified Lorenz system from oscilloscope

8.2 Design and Implementation of Fractional-Order Chaotic System on DSP 8.2.1 Algorithm for Solving Fractional-Order Simplified Lorenz System DSP is widely used in engineering practice because of the characteristics of strong performance and low price. In order to save hardware resources, the complexity of the calculation process needs to be considered when using DSP to realize fractional-order chaotic system. Generally, the calculation shall be simplified as much as possible on the premise of ensuring the performance of chaotic system. The numerical solution of the fractional-order simplified Lorenz system is as follows according to the ADM decomposition algorithm [5]:

8.2 Design and Implementation of Fractional-Order …

149

⎧ hq ⎪ ⎪ x = x + 10(y − x ) + 10[(24 − 4c)xm ⎪ m+1 m m m ⎪ (q + 1) ⎪ ⎪ ⎪ ⎪ ⎪ h 2q ⎪ ⎪ ⎪ + cym − 10(ym − xm )] + ··· ⎪ ⎪ (2q + 1) ⎪ ⎪ ⎪ ⎪ hq ⎪ ⎪ ⎪ = y + [(24 − 4c)x + cy − x z ] + {10(24 − 4c)(ym − xm ) y m+1 m m m m m ⎨ (q + 1) ⎪ h 2q ⎪ ⎪ + c[(24 − 4c)x + cy − x z ] + · · · } + ··· ⎪ m m m m ⎪ (2q + 1) ⎪ ⎪ ⎪ ⎪ ⎪ 8 hq 8 8 ⎪ ⎪ = z + (x y − z ) + {(− )[(− )z m z m+1 m m m m ⎪ ⎪ 3 (q + 1) 3 3 ⎪ ⎪ ⎪ ⎪ 2q ⎪ h ⎪ ⎪ + xm ym ] + 10ym (ym − xm ) + · · · } + ··· ⎩ (2q + 1)

,

(8.6) where h is the iteration step. Equation (8.6) is a set of polynomials which are infinitely long. Because of the good convergence of Adomian decomposition algorithm [6–11], Eq. (8.6) will be appropriately intercepted when the system is solved and analyzed on computers. The different terms of polynomials are intercepted, and S is defined as the number of intercepted terms. When h = 0.01, and S equals 3, 4, and 5 respectively, the corresponding maximum Lyapunov exponent diagram are shown in the Fig. 8.8. In Fig. 8.8, the corresponding maximum Lyapunov exponent diagram when S = 3 has a significant change compared to when S = 4. The main reason is that inappropriate interception of the number of items affects the calculation accuracy. The corresponding maximum Lyapunov exponent diagrams for S = 4 and S = 5 are basically similar. Both the distribution of the chaotic region and the maximum Lyapunov exponent values at the corresponding position are very similar. Figure 8.8 shows that due to the fast convergence of the Adomian decomposition method, the dynamic characteristics of the fractional-order simplified Lorenz system can be reflected when the first four terms of the Adomian polynomial are intercepted. In practical applications, intercepting the first 4 terms of the Adomian polynomial can meet the application requirements, and the corresponding discrete iteration is ⎡

⎤ ⎡ ⎤ xm+1 C10 C11 C12 C13  ⎢ ⎥ ⎣ ⎣ ym+1 ⎦ = C20 C21 C22 C23 ⎦ 1 C30 C31 C32 C33 z m+1

hq h 2q h 3q (q+1) (2q+1) (3q+1)

T

,

(8.7)

where ⎧ ⎪ ⎨ C10 = xm C20 = ym , ⎪ ⎩ C30 = z m

(8.8)

150

8 Circuit Design and Realization of Fractional-Order …

Fig. 8.8 The largest Lyapunov exponent diagram of the fractional-order simplified Lorenz system with different numbers of terms. a S = 3 b S = 4 c S = 5

⎧ C11 = 10(C20 − C10 ) ⎪ ⎪ ⎨ C21 = (24 − 4c)C10 + cC20 − C10 C30 , ⎪ ⎪ ⎩ C = −8C + C C 31 30 10 20 3

⎧ C12 = 10(C21 − C11 ) ⎪ ⎪ ⎨ C22 = (24 − 4c)C11 + cC21 − C11 C30 − C10 C31 , ⎪ ⎪ ⎩ C = −8C + C C + C C 32 31 11 20 10 21 3

(8.9)

(8.10)

⎧ C13 = 10(C22 − C12 ) ⎪ ⎪ ⎪ ⎪ ⎪ (2q + 1) ⎨ C23 = (24 − 4c)C12 + cC22 − C12 C30 − C11 C31 2 − C10 C32 . (8.11)  (q + 1) ⎪ ⎪ ⎪ 8 (2q + 1) ⎪ ⎪ C33 = − C32 + C12 C20 + C11 C21 + C10 C22 ⎩ 3  2 (q + 1) In addition, in order to simplify the calculation, in Eqs. (8.7)–(8.11), define g1 = h q / (q + 1), g2 = h 2q / (2q + 1), g3 = h 3q / (3q + 1), and g4 = (2q + 1)/ 2(q + 1), respectively. In iterative calculations, g1 , g2 , g3 , and g4 are

8.2 Design and Implementation of Fractional-Order …

151

functions of h and q. After h and q are determined, g1 , g2 , g3 , and g4 are constants in each iteration, which greatly saves the computing resources of DSP.

8.2.2 Hardware Design of Fractional-Order Chaotic System In order to realize the fractional-order simplified Lorenz system on DSP, and to observe and test the experimental results, the modular design method is adopted. The block diagram of hardware circuit designed is shown in Fig. 8.9. TMS320F28335 in Fig. 8.9 is a representative 32-bit floating-point DSP processor of TI Company [12], and its highest clock frequency can reach 150 MHz. Due to highcost performance, powerful capability for digital signal processing, and excellent capability for event management, TMS320F28335 is suitable for applications in the field of digital signal processing. In the experiment, the DSP minimum system development board with an external 30 MHz crystal oscillator is employed. In order to observe the chaotic attractor on the oscilloscope, the chaotic sequences generated by the DSP are converted into analog signals through DAC8552. DAC8552 is a dual-channel digital to analog converter with 16-bit conversion accuracy, and its internal structure is shown in Fig. 8.10. A reasonable circuit design and operation control words can ensure that Vout A and Vout B complete the DA conversion at the same time, and output the analog signal converted synchronously, which is more conducive to observing the attractor of the chaotic system. The conversion speed of DAC8552 can reach 10 µs, and it is connected with DSP through SPI (Serial Peripheral Interface, SPI) interface, as shown in Fig. 8.11. During software design and programming, the operation control word for DAC8552 is shown in Fig. 8.12. DB15-DB0 is the 16-bit digital signal that needs to be converted. PD0 = 0 and PD1 = 0 mean normal operation. If Buffer Select is 0, it means that the operation data is input to channel A, and if it is 1, it is input to channel B. If LDA(B) = 0, A(B) channel does not output analog signals, and LDA(B) = 1, then A(B) channel outputs analog signals. LDA and LDB are both 1, which means the two channels output analog signals simultaneously. The interaction between the computer and the DSP is realized through the SCI (Serial Communication Interface, SCI) interface. After setting q, h, parameter c and

Fig. 8.9 Fractional-order simplified Lorenz system DSP implementation hardware block diagram

152

8 Circuit Design and Realization of Fractional-Order …

Fig. 8.10 DAC8552 structure diagram

Fig. 8.11 Circuit diagram of TMS320F28335 and DAC8552 connection

The first step is to write the 24 digits of the A data buffer Res

Res

LDB

LDA

DC

Buffer Select

PD1

PD0

DB15

—

DB1

DB0

0

0

0

0

×

0

0

0

D15

—

D1

D0

The second step is to write the 24 digits of the B data buffer Res

Res

LDB

LDA

DC

Buffer Select

PD1

PD0

DB15

—

DB1

DB0

0

0

1

1

×

1

0

0

D15

—

D1

D0

Fig. 8.12 Operation control word of DAC8552

8.2 Design and Implementation of Fractional-Order …

153

Fig. 8.13 TMS320F28335 and computer interface circuit

Fig. 8.14 DSP experimental platform of the fractional chaotic system

the initial values of the system, they are sent to DSP through the SCI interface. Similarly, the chaotic sequence obtained after DSP operation is sent to computer through SCI interface, so that the generated chaotic sequence can be collected and analyzed on the computer. The interface circuit between the computer and TMS320F28335 is shown in Fig. 8.13 where MAX3232 is the RS232 transceiver of MAXIM Company. The DSP experimental platform of the fractional chaotic system is shown in Fig. 8.14.

8.2.3 Software Design of Fractional Chaos System The software of fractional-order simplified Lorenz system based on DSP platform is designed by using module design method [13, 14], and the flowchart is shown in Fig. 8.15.

154

8 Circuit Design and Realization of Fractional-Order …

Fig. 8.15 The flowchart of the fractional-order simplified Lorenz system on DSP

It can be found from the simulation that there are often negative numbers in the chaotic sequence obtained by iterative calculations, but DAC8552 can only accept positive integers in [0, 216 − 1]. Therefore, in order to observe the chaotic attractor on the oscilloscope, the chaotic sequence needs to be processed through “Data Processing 1” in Fig. 8.15. It includes three steps. (1) An appropriate positive integer A is added to all the chaotic sequences obtained to ensure that the chaotic sequences all become positive values and do not change the relative values between the sequences. (2) Then enlarge each sequence by B times. (3) The integer part of the sequences is preserved and ensured the values in [0, 216 − 1] by rounding. It should be noted that due to the difference q and c, the size of the attractors will be different. Therefore, when implementing fractional simplified Lorenz systems with different q and c, the values of A and B may be different, and they will be adjusted depending on the specific attractor. Since data is transmitted in bytes in SCI communication, when the chaotic sequence is transmitted to the computer, “Data Processing 2” converts the chaotic sequence into ASCII code. In addition, push and pop operations are used to protect the chaotic sequence obtained in the iterative calculation on DSP, so that the iterative operation is not affected by data processing. Set h = 0.01, initial value (x 0 , y0 , z0 ) = (0.1, 0.2, 0.3), A = 15, B = 1400, the attractors of fractional-order simplified Lorenz system with different q and c implemented on DSP are observed on oscilloscope as shown in Fig. 8.16. The figure also shows the chaotic attractor phase diagrams obtained by MATLAB simulation under the same conditions. Figure 8.16a, c shows the corresponding chaotic state when q and c are different, and Fig. 8.16e shows the periodic state. By comparison, it

8.2 Design and Implementation of Fractional-Order …

155

18

z

z

12

6

(a)

0 -10

(b) 5

0 x

-5

x

10

30

z

z

20

10

(c)

0 -15

x

(d) -5

5

15

x

3

z

z

2

1

(e) x

0 -4

(f) 0 x

4

Fig. 8.16 The attractors of fractional-order simplified Lorenz system implemented on DSP and simulated on computer. a Implemented on DSP, q = 0.9, c = 5. b Simulated on computer, q = 0.9, c = 5. c Implemented on DSP, q = 0.7, c = 3. d Simulated on computer, q = 0.7, c = 3. e Implemented on DSP, q = 0.65, c = 7.5. f Simulated on computer, q = 0.65, c = 7.5

is found that the results of the fractional-order simplified Lorenz system implemented by DSP are consistent with the simulation results. By using the same DSP platform, based on the iterative equations to solve the fractional-order Rössler system and Lorenz–Stenflo system in Chap. 6, the two fractional-order chaotic systems are also realized. In Fig. 8.17, the attractors of the

156

8 Circuit Design and Realization of Fractional-Order … 4 (b)

(a)

2

y

y

0 -2 -4 -6 -8 x

-5

-3

-1

1 x

3

5

(c)

7

(d)

1

y

y

0

-1

-2

x

-3 -2

-1

0

x

1

2

3

Fig. 8.17 DSP implementation of fractional-order simplified Rössler system. a DSP implementation, q = 0.7, b computer simulation, q = 0.7, c DSP implementation, q = 0.3, d computer simulation, q = 0.3

fractional-order Rössler system with different q are shown when a = 0.55, b = 2, c = 4, h = 0.01. Figure 8.18 shows the attractors of the fractional-order Lorenz–Stenflo system with different σ, s, b, and r when q = 0.8, h = 0.001. The comparison between the implementation results and the corresponding simulation results shows that the two fractional-order chaotic systems have been successfully implemented on the DSP platform.

8.3 FPGA Design and Implementation of Fractional-Order Chaotic System 8.3.1 Design and Optimization of Circuit Structure At present, FPGA-based system design is getting more and more attention, and its application is becoming more and more extensive [15]. By using a top-down approach, fractional-order simplified Lorenz system is designed on FPGA. At the top level, sub modules are divided by function, and relationship and coordination between

8.3 FPGA Design and Implementation of Fractional-Order Chaotic System

157

Fig. 8.18 DSP implementation and computer simulation of fractional-order simplified Lorenz– Stenflo system. a DSP implementation, σ = 1.0, s = 1.5, b = 0.7, r = 26. b Computer simulation, σ = 1.0, s = 1.5, b = 0.7, r = 26. c DSP implementation, σ = 10, s = 30, b = 8/3, r = 340. d Computer simulation, σ = 10, s = 30, b = 8/3, r = 340

sub-modules are planned. At the bottom level, the functions of sub-modules are realized, respectively, by using Verilog hardware description language. The principle block diagram of the design is shown in Fig. 8.19. First, after defining g1 = h q / (q+1), g2 = h 2q / (2q+1), g3 = h 3q / (3q+1), and g4 = (2q + 1)/ 2(q + 1), and determining q and h, g1 , g2 , g3 , and g4 are calculated. Then they are sent to FPGA through UART together with the parameter c and the initial value (x 0 , y0 , z0 ). In FPGA, the chaotic sequence of the fractionalorder simplified Lorenz system generated by iteration can also be sent to the computer through UART for further analyzing and testing. In order to observe chaotic attractors through an oscilloscope, it is necessary to convert the generated chaotic sequence into

Fig. 8.19 The block diagram of the fractional-order simplified Lorenz system based on FPGA

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8 Circuit Design and Realization of Fractional-Order …

a fixed-point integer, and then are input into digital–analog converter unit through the SPI interface. The top-level design principle of fractional-order chaotic system on FPGA is shown in Fig. 8.20. The fractional chaotic signal generator calculates and outputs a set of fractional-order chaotic sequences according to Eqs. (8.7)–(8.11). The floatingpoint to fixed-point conversion unit converts the fractional chaotic sequence generated each time into a fixed-point number. UART is responsible for communication with the computer. SPI interface outputs the digital signals that need to be converted to the DA converter. The various modules coordinate their work through state control signals. The clk and r st signals ensure that clock and reset signals of each sub-module are synchronized. clko and f g are state control signals. Data[255...0] includes eight 32-bit floating-point numbers (initial values (x 0 , y0 , z0 ), parameters c and g1 , g2 , g3 , g4 ). When UART receives Data[255...0] from the computer, clkout sets cc = 0, and the fractional-order chaotic signal generator reads Data[255...0] and starts iteration calculation. After the fractional-order chaos generator completes one iteration, the chaotic sequence obtained is sent to the computer through UART. At the same time, through the clko of the fractional-order chaos generator, set f g = 0 and floatingpoint to fixed-point conversion unit start the conversion. By the conversion, the generated 32-bit floating-point number floatx[31...0], floaty[31...0], floatz[31...0] are converted into 16-bit fixed-point number flx[15...0], fly[15...0], flz[15...0] that are suitable for DAC8552. After the conversion is completed, the clko of the floatingpoint to fixed-point conversion unit sets f g of the SPI interface unit to 0. Then SPI outputs the data to DAC8552, and the chaotic attractor can be observed on oscilloscope. At the same time, clko = 0 of the SPI interface, the fractional-order chaos generator starts next iteration calculation.

Fig. 8.20 The schematic diagram of the top-level design

8.3 FPGA Design and Implementation of Fractional-Order Chaotic System

159

8.3.2 Design of Fractional Simplified Lorenz System Chaotic Signal Generator In Fig. 8.20, the fractional-order chaotic signal generator is the core of the system. Its internal structure is shown in Fig. 8.21, and it is composed of three parts: floatingpoint multiplier, floating-point adder, and operation controller. According to the calculation process in the iterative equations, the workflow of the fractional chaos operation controller is shown in Fig. 8.22. Clk j, clkc, clko, and f g are all control signals. clk j = 0 is used to select the floating-point number adder to perform floatingpoint addition operations for data A[31...0] and B[31...0]. clkc = 0 is to select the floating-point number multiplier to perform floating-point number multiplication for data A[31...0] and B[31...0]. After the arithmetic units complete the calculation, f gs = 0 or f gs1 = 0 through the respective clko, so as to notify the controller to read the calculation result Result[31...0] from the arithmetic units. According to the iterative equations, until the end of the iteration, the fractional-order chaotic sequence x[31...0], y[31...0], z[31...0] are obtained, and they are also the initial values for next iteration.

Fig. 8.21 Schematic diagram of the internal design of the fractional chaos generator

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8 Circuit Design and Realization of Fractional-Order …

Fig. 8.22 The working principle of the fractional chaos operation controller

8.3.3 Design of Floating-Point Arithmetic Units Because there is no integrated floating-point number arithmetic unit in FPGA, which is different from DSP, it is necessary to define floating-point number and design the floating-point number arithmetic unit by using IEEE785 standard. Table 8.1 shows the floating-point number representation mode specified by IEEE754 [16]. In Table 8.1, the design principles of floating-point numbers of three different widths are the same. For simplicity, a 32-bit single-precision floating-point number format is employed for experiments. The relationship between the floating-point number and its corresponding value is

8.3 FPGA Design and Implementation of Fractional-Order Chaotic System

161

Table 8.1 Representation of floating-point numbers specified by IEEE754 Floating point

Width

Sign (S)

Exponent (E)

Mantissa (M)

Single precision

32 bits

[31]

[30..23]

[22..0]

Exponent bias (B)

Double precision

64 bits

[63]

[62..52]

[51..0]

1023

Extended double precision

80 bits

[79]

[68..64]

[63..0]

16,383

127

A = (−1) S × (1.0 + M) × 2 E−127 ,

(8.12)

for S = 0, A is a positive number, and S = 1, A is a negative number. The order code E is an 8-bit binary number, which is in (0−255). Therefore, the value range of A is 2−127 to 2128 . The M in Eq. (8.12) is the mantissa which occupies 23 bits, and its value is M = M0 × 2−1 + M1 × 2−2 + M2 × 2−3 + · · · + M22 × 2−23 .

(8.13)

Given the floating-point numbers x and y x = (−1) Sx × (1.0 + Mx ) × 2 E x −127 ,

(8.14)

y = (−1) Sy × (1.0 + M y ) × 2 E y −127 ,

(8.15)

floating-point number addition is expressed as x + y = [(−1) Sx × (1.0 + Mx ) + (−1) Sy × (1.0 + M y ) × 2 E y −E x ] × 2 E x −127 , (8.16) and floating-point number multiplication is expressed as x y = (−1) Sx +Sy × [(1.0 + Mx ) × (1.0 + M y )] × 2 E x +E y −127 .

(8.17)

Figures 8.23 and 8.24 show the flowchart of floating-point number arithmetic. For floating-point number addition, according to Eq. (8.16), the steps are as follows: (1) Compare the order codes of two addends. (2) If they are not equal, the order codes need to be aligned. The smaller order code needs to be added with N to make the adjusted order code equal to the larger order code. (3) The mantissa of floating-point numbers with smaller order code also needs to be adjusted. That is, the mantissa (including hidden bits) is shifted to the right by N bits. (4) Calculate the sum of the mantissas (including hidden bits) of the two numbers after adjustment. If the hidden bits of the result are 10, the mantissa (together with the hidden bits) need to be moved to the right by one bit, and add 1 to the order code at the same time. (5) Determine the sign bit Sz, the result. If Sx = Sy, then Sz = Sx = Sy. Otherwise Sz is equal to the sign bit with the larger order code. And if the two order codes are also

162

8 Circuit Design and Realization of Fractional-Order …

Fig. 8.23 Flowchart of floating-point number addition

equal, Sz is equal to the sign bit with the larger mantissa. (6) The overflow check for the result is to judge that, if 00000000 < E z < 11111111, the result does not overflow, otherwise the result overflows. (7) Finally, remove the hidden bit to get the result of adding two floating-point numbers. For floating-point number multiplication, according to Eq. (8.17), the steps are as follows: (1) Determine the sign bit of the result for multiplying two floating numbers as Sz = Sx + Sy. (2) The temporary order code is E z = E x + E y − 127. (3) Multiply the two mantissas (including hidden bits) to get the temporary mantissa M z = M x × M y. (4) If the hidden bit of the temporary mantissa is not 01, shift Mz (together with the hidden bit) to the right by bits to make the hidden bit to be 01. Then take the high 23 bits of Mz shifted as the mantissa of the final result. (5) Format the order code through E z + 1. (6) Finally, the overflow check is also performed on

8.3 FPGA Design and Implementation of Fractional-Order Chaotic System

163

Fig. 8.24 Flowchart of floating-point number multiplication

the result. If there is no overflow, the hidden bit is removed to obtain the result of multiplying two floating-point numbers.

8.3.4 Experimental Results and Analysis In order to make the data format of the chaotic sequence calculated to be suitable for DAC8552, the chaotic sequence value is mapped to a positive integer in [0, 216 − 1]. The design idea and method are the same as the method in DSP. The design of interfaces (UART, SPI) of the system belongs to the category of FPGA design of SoC (System on Chip, SoC), and the relevant technical information [17, 18] can be referred. The FPGA implementation of the fractional-order simplified Lorenz

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8 Circuit Design and Realization of Fractional-Order …

system is shown in Fig. 8.25. In the experiment, FPGA chip EP1C6Q240 of the Altera Cyclone series is employed. The attractors with h = 0.01 and (x 0 , y0 , z0 ) = (0.1, 0.2, 0.3) observed on oscilloscope and the phase diagram of the attractors drawn after transferring the chaotic sequence to the computer are shown in Figs. 8.26 and 8.27, respectively. The system is tested comprehensively by using Quartus II. The relevant parameters of FPGA resource are occupied, and the maximum clock frequency are achieved as shown in Table 8.2.

Fig. 8.25 Fractional simplified Lorenz system platform implemented on FPGA

Fig. 8.26 The attractor of the fractional-order simplified Lorenz system implemented on FPGA (oscilloscope display). a q = 0.9, c = 5 b q = 0.8, c = 7.5

8.3 FPGA Design and Implementation of Fractional-Order Chaotic System

165

Fig. 8.27 The attractor of the fractional-order simplified Lorenz system implemented by FPGA (computer simulation). a q = 0.9, c = 5 b q = 0.8, c = 7.5

Table 8.2 Comparison of the implementation of the fractional-order system and the integer-order system on the FPGA platform Chaos generator

Total number of registers

Total number of logic devices

Total LABs

Maximum clock frequency (MHz)

The number of clock cycles required for each iteration

ADM decomposition method for solving fractional order system

1519 (12%)

6009 (50%)

653 (54%)

51.12

900

RK4 solves integer order system

1782 (14%)

6057 (50%)

849 (70%)

51.06

1006

The integer-order simplified Lorenz system is usually solved by using fourthorder Runge–Kutta (RK4). Under the same conditions and structure, the integer-order simplified Lorenz system based on RK4 is implemented on FPGA. The parameters after the comprehensive test are also listed in Table 8.2. Through comparison, it is found that the fractional-order simplified Lorenz system based on the Adomian decomposition method takes less FPGA resources and is faster than the integer-order simplified Lorenz system based on RK4.

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8 Circuit Design and Realization of Fractional-Order …

References 1. Ahmad WM, Sprott JC (2003) Chaos in fractional-order autonomous nonlinear systems. Chaos Solitons Fractals 16(2):339–351 2. Sun H, Abdelwahab A, Onaral B (1984) Linear approximation of transfer function with a pole of fractional power. IEEE Trans Autom Control 29(5):441–444 3. Liu C (2011) Theory and application of fractional order chaotic circuit (in Chinese). Xi’an Jiaotong University Press, Xi’an 4. Sun K, Yang J, Ding J, Sheng L (2010) Circuit design and implementation of Lorenz chaotic system with one parameter (in Chinese). Acta Physica Sinica 59(12):8385–8392 5. Adomian G (1984) A new approach to nonlinear partial differential equations. J Math Anal Appl 102(2):420–434 6. Adomian G (1984) On the convergence region for decomposition solutions. J Comput Appl Math 11(3):379–380 7. Cherruault Y (1990) Convergence of Adomian’s method. Math Comput Model 14(18):31–38 8. Cherruault Y, Saccomandi G, Some B (1992) New results for convergence of Adomian’s method applied to integral equations. Math Comput Model 16(2):85–93 9. Cherruault Y, Adomian G (1993) Decomposition methods: a new proof of convergence. Math Comput Model 18(12):103–106 10. Abbaoui K, Cherruault Y (1994) Convergence of Adomian’s method applied to nonlinear equations. Math Comput Model 20(9):69–73 11. Cherruault Y, Adomian G, Abbaoui K et al (1995) Further remarks on convergence of decomposition method. Int J Biomed Comput 38(1):89–93 12. Liu L, Gao Y, Zhang S et al (2011) Principle and development programming of TMS320F28335 DSP (in Chinese). Beijing University of Aeronautics and Astronautics Press, Beijing 13. Wang HH, Sun KH, He SB (2015) Characteristic analysis and DSP realization of fractionalorder simplified Lorenz system based on Adomian decomposition method. Int J Bifurc Chaos 25(06):1550085 14. Wang HH, Sun KH, He SB (2015) Dynamic analysis and implementation of a digital signal processor of a fractional-order Lorenz-Stenflo system based on the Adomian decomposition method. Phys Scr 90(1):015206 15. Ren W, Shen D, He L (2018) Based on the engineering application and practice of the FPGA technology (in Chinese). Science Press, Beijing 16. Kumutha A, Shobha P (2014) Implementation of IEEE754 floating point multiplier. Int J Eng Res Technol (ESRSA Publications) 17. Chu Z, Wen M, Gao K et al (2012) Design and application of FPGA (in Chinese). Xidian University Press, Xi’an 18. Ma L, Peng M (2015) Design and application of CPLD/FPGA (in Chinese). Huazhong University of science and Technology Press, Wuhan

Chapter 9

Applications of Fractional-Order Chaotic Systems in Secure Communications

The rich dynamics and the successful implementation of digital circuits of the fractional-order chaotic system lay the theoretical and hardware foundation for its application. This chapter focuses on the synchronization control of fractional-order chaotic systems, pseudo-random sequence generator based on DSP digital circuit implementation, and image encryption. In addition, the fractional-order chaotic system is applied to a spread spectrum communication system as a spread spectrum code, and the bit error rate (BER) of the system at different signal-to-noise ratios (SNR) is tested by simulation and compared with other spread spectrum codes.

9.1 Synchronous Control of the Fractional-Order Chaotic Systems The synchronization of the fractional-order chaotic systems is defined as follows. q

Definition 9.1 Consider the following two nonlinear dynamical systems Dto x = q F1 (t, x), Dto y = F2 (t, y) + U (t, x, y) where x, y ∈ Rn are the state variables of the systems, F 1 , F 2 : [R+ × Rn ] → Rn are nonlinear functions, U: [R+ × R n × Rn ] → Rn is the synchronization controller of the chaotic system, and R+ is the set of nonnegative real numbers. Two nonlinear dynamical systems reach synchronization if there exists D(t0 ) ⊆ R n , ∀x0 , y0 ∈ D(t0 ) such that x(t; t0 , x0 ) − y(t; t0 , x0 ) → 0 when t → ∞. Fractional-order chaotic system synchronization is the basis of its application to chaotic secure communication. Adomian decomposition algorithm is beneficial to the practical application of fractional-order chaotic systems. This chapter focuses on the fractional-order chaotic system synchronization algorithm (including network synchronization) based on the Adomian decomposition algorithm and its synchronization performance. © Science Press 2022 K. Sun et al., Solution and Characteristic Analysis of Fractional-order Chaotic Systems, https://doi.org/10.1007/978-981-19-3273-1_9

167

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9 Applications of fractional-order Chaotic Systems in Secure Communications

9.1.1 Coupled Synchronization of Fractional-Order Chaotic Systems Consider the following fractional-order continuous chaotic system: q

Dt0 x(t) = f (x(t)) + Cx(t).

(9.1)

Its coupled chaotic system is q

Dt0 y(t) = f (y(t)) + Cy(t) + ρ(x(t) − y(t)).

(9.2)

A perturbation term ρ(x(t) − y(t)) is added to the system (9.2) and coupled to the drive system (9.1), where ρ is the coupling coefficient. If lim y(t) − x(t) = 0,

(9.3)

t→∞

then the two systems reach coupled synchronization. Next, the fractional-order hyperchaotic Lorenz system is used as an example to study its synchronization problem. The equation of the fractional-order Lorenz hyperchaotic system is ⎧ q Dt0 x1 ⎪ ⎪ ⎪ q ⎨ Dt0 x2 q ⎪ ⎪ Dt0 x3 ⎪ ⎩ q Dt0 x4

= 10(x2 − x1 ) = 28x1 − x1 x3 + x2 − x4

(9.4)

= x1 x2 − 8x3 /3 = Rx2 x3

According to Eq. (9.2), the coupling system is ⎧ q Dt0 y1 ⎪ ⎪ ⎪ ⎨ Dq y t0 2 ⎪ Dtq0 y3 ⎪ ⎪ ⎩ q Dt0 y4

= 10(y2 − y1 ) + ρ(x1 − y1 ) = 28y1 − y1 y3 + y2 − y4 + ρ(x2 − y2 ) = y1 y2 − 8y3 /3 + ρ(x3 − y3 ) = Ry2 y3 + ρ(x4 − y4 )

.

(9.5)

The matrix form of the parameters and nonlinear terms of the system (9.4) is ⎡

−10 ⎢ 28 C=⎢ ⎣ 0 0

⎡ ⎤ ⎤ 0 10 0 0 ⎢ −x x ⎥ 1 0 −1 ⎥ 1 3⎥ ⎥, f (x(t)) = ⎢ ⎢ ⎥, ⎦ 0 −8/3 0 ⎣ x1 x2 ⎦ 0 0 0 Rx2 x3

Denote the error variable as

(9.6)

9.1 Synchronous Control of the Fractional-Order Chaotic Systems



e = y − x = [e1 , e2 , e3 , e4 ]T ei = yi − xi , i = 1, 2, 3, 4

169

,

(9.7)

then the error system of system (9.4) and system (9.5) is presented as q

q

q

Dt0 e(t) = Dt0 y(t) − Dt0 x(t) = (C − ρ)e(t) + f (y(t)) − f (x(t)),

(9.8)

where ⎡

0 x1 x3 − y1 y3

⎤

⎢ ⎥ ⎢ ⎥ f (x(t)) − f (y(t)) = ⎢ ⎥ = Be, ⎣ −x1 x2 + y1 y2 ⎦ −Rx2 x3 + Ry2 y3 ⎤ ⎡ 0 0 0 0 ⎢ −y3 0 −x1 0 ⎥ ⎥ B=⎢ ⎣ x2 y1 0 0 ⎦.

(9.9)

(9.10)

0 Ry3 Rx2 0 According to Eq. (9.3), the system error ei → 0 as t → ∞ (i = 1, 2, 3, 4) and system (9.4) and system (9.5) are synchronized. q

q

Lemma 9.1 When q ∈ (0, 1], Dt0 |x(t)| = sgn(x(t))Dt0 x(t). q

Proof If x(t) = 0, then Dt0 |x(t)| = 0. If x(t) > 0, then we have q Dt0 |x(t)|

1 = (1 − q)

t t0

y˙ (s) 1 ds = (t − s)q (1 − q)

t t0

x(s) ˙ q ds = Dt0 x(t). (t − s)q (9.11)

If x(t) < 0, then we have. q Dt0 |x(t)|

1 = (1 − q)

t t0

y˙ (s) 1 ds = − (t − s)q (1 − q)

t t0

x(s) ˙ q ds = −Dt0 x(t). (t − s)q (9.12)

Therefore, Lemma 9.1 is proved.

    ˜ + ψλmax B˜ , then system Theorem 9.1 When the coupling strength ρ > λmax C   ˜ = 0.5 C + CT , ψ = (9.4) and system (9.5) are synchronized globally, where C

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9 Applications of fractional-order Chaotic Systems in Secure Communications

max{|xi |, i = 1, 2, 3, 4}, B˜ = 0.5(B1 + B1T ), B1 = {[0, 0, 0, 0], [1, 0, 1, 0], [1, 1, 0, 0], [0, 1, 1, 0]}, and λmax (·) is the maximum eigenvalue. Proof Construct a Lyapunov exponent function as V = e.

(9.13)

According to Lemma 9.1, we have q

q

Dt0 V (t) = Dt0

4 

|ei | = sgn(eT )Dtq0 e.

(9.14)

i=1

Since R ∈ [0, 1], it follows that   ˜ ≤ ψλmax B˜ sgn(eT )e. sgn(eT )Be ≤ ψsgn(eT )Be

(9.15)

That is q

Dt0 V (t) = sgn(eT )(C + B − ρ)e C + CT B + BT e + ψsgn(eT ) e − ρsgn(eT )e 2  2  ˜ e + sgn(eT )ψλmax B˜ e − sgn(eT )ρe, ≤ sgn(eT )λmax C       ˜ I + ψλmax B˜ I − ρ e = sgn(eT ) λmax C

≤ sgn(eT )

= sgn(eT )Pe

(9.16)

where ⎧ ⎨ ρ = diag(ρ, ρ, ρ, ρ)       ˜ + ψλmax B˜ − ρ I ⎩ P = λmax C

(9.17)

    ˜ + ψλmax B˜ , Dtq V (t) < 0, the fractional-order derivaSo when ρ > λmax C 0 tive of the error system is negative at this point, i.e., the two systems are globally synchronized. It is proved. Remark 9.1 Theorem 9.1 is a sufficient condition for the coupling synchronization of fractional-order Lorenz hyperchaotic system. In other words, the system can definitely achieve synchronization when the above condition is satisfied, and the system can also achieve synchronization when the coupling strength ρ takes a smaller value, which is verified by numerical simulation results.

9.1 Synchronous Control of the Fractional-Order Chaotic Systems

171

Remark 9.2 According to Ref. [1], the lower limit of the coupling strength of the current coupled synchronous system cannot be found theoretically. It is worth pointing out that, compared to other synchronous controllers, the coupled synchronous controller is simple and easy to implement, so it has practical applications. The solution of the fractional-order Lorenz hyperchaotic system (9.4) is shown in Chap. 6. Using the Adomian decomposition algorithm, the solution of the response system (9.5) can be expressed as ⎧ 6 ⎪   ⎪ j ⎪ ⎪ y1,n+1 = ξ1 h jq ( jq + 1) ⎪ ⎪ ⎪ ⎪ j=0 ⎪ ⎪ ⎪ ⎪ 6 ⎪   ⎪ j ⎪ ⎪ y2,n+1 = ξ2 h jq ( jq + 1) ⎪ ⎪ ⎨ j=0 6 ⎪  ⎪  ⎪ j ⎪ = ξ3 h jq ( jq + 1) y ⎪ 3,n+1 ⎪ ⎪ ⎪ j=0 ⎪ ⎪ ⎪ ⎪ 6 ⎪  ⎪  ⎪ j ⎪ ⎪ = ξ4 h jq ( jq + 1) y 4,n+1 ⎪ ⎩

,

(9.18)

j=0

j

where the intermediate variables ξi (i = 1, 2, 3, 4; j = 1, 2, …, 6) are calculated by ξ10 = y1,n , ξ20 = y2,n , ξ30 = y3,n , ξ40 = y4,n , ⎧ 1 ξ1 ⎪ ⎪ ⎪ ⎪ ⎨ ξ1 2 1 ⎪ ξ ⎪ 3 ⎪ ⎪ ⎩ 1 ξ4

⎧ 2 ξ1 ⎪ ⎪ ⎪ ⎪ ⎨ ξ2 2 ⎪ ξ32 ⎪ ⎪ ⎪ ⎩ 2 ξ4

(9.19)

= 10(ξ20 − ξ10 ) + ρ(c10 − ξ10 ) = 28ξ10 − ξ10 ξ30 + ξ20 − ξ40 + ρ(c20 − ξ20 ) = ξ10 ξ20 − 8ξ30 /3 + ρ(c30 − ξ30 )

,

(9.20)

= Rξ20 ξ30 + ρ(c40 − ξ40 )

= 10(ξ21 − ξ11 ) + ρ(c11 − ξ11 ) = 28ξ11 − ξ10 ξ31 − ξ11 ξ30 + ξ21 − ξ41 + ρ(c21 − ξ21 ) = ξ11 ξ20 + ξ10 ξ21 − 8ξ31 /3 + ρ(c31 − ξ21 )   = R ξ21 ξ30 + ξ20 ξ31 + ρ(c41 − ξ21 )

,

(9.21)

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9 Applications of fractional-order Chaotic Systems in Secure Communications

⎧ 3 ξ1 = 10(ξ22 − ξ12 ) + ρ(c12 − ξ12 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 2 0 2 1 1 (2q + 1) ⎪ − ξ12 ξ30 + ξ22 ⎪ ξ2 = 28ξ1 − ξ1 ξ3 − ξ1 ξ3 2 ⎪  (q + 1) ⎪ ⎪ ⎪ ⎨ − ξ42 + ρ(c22 − ξ22 ) , ⎪ ⎪ 3 0 2 1 1 (2q + 1) 2 0 2 2 2 ⎪ ⎪ ξ3 = ξ1 ξ2 + ξ1 ξ2 2 + ξ1 ξ2 − 8ξ3 /3 + ρ(c3 − ξ3 ) ⎪ ⎪  (q + 1) ⎪ ⎪   ⎪ ⎪ (2q + 1) ⎪ 3 0 2 1 1 2 0 ⎪ ⎩ ξ4 = R ξ2 ξ3 + ξ2 ξ3 + ξ2 ξ3 + ρ(c42 − ξ42 )  2 (q + 1)

(9.22)

⎧ 4 ξ1 = 10(ξ23 − ξ13 ) + ρ(c13 − ξ13 ) ⎪ ⎪ ⎪ ⎪ ⎪ (3q + 1) ⎪ ⎪ ξ24 = 28ξ13 − ξ10 ξ33 − (ξ12 ξ31 + ξ11 ξ32 ) − ξ13 ξ30 + ⎪ ⎪ (q + 1)(2q + 1) ⎪ ⎪ ⎪ ⎨ ξ23 − ξ43 + ρ(c23 − ξ23 ) , ⎪ (3q + 1) ⎪ 4 0 3 2 1 1 2 3 0 3 3 3 ⎪ ⎪ ξ3 = ξ1 ξ2 + (ξ1 ξ2 + ξ1 ξ2 ) + ξ1 ξ2 + 8ξ3 /3 + ρ(c3 − ξ3 ) ⎪ ⎪ (q + 1)(2q + 1) ⎪ ⎪   ⎪ ⎪ (3q + 1) ⎪ ⎪ ⎩ ξ44 = R ξ20 ξ33 + (ξ22 ξ31 + ξ21 ξ32 ) + ξ23 ξ30 + ρ(c43 − ξ43 ) (q + 1)(2q + 1) (9.23) ⎧ 5 ξ1 = 10(ξ24 − ξ14 ) + ρ(c14 − ξ14 ) ⎪ ⎪ ⎪ ⎪ ⎪ (4q + 1) ⎪ ⎪ ⎪ ξ25 = 28ξ14 − ξ10 ξ34 − (ξ13 ξ31 + ξ11 ξ33 ) ⎪ ⎪ (q + 1)(3q + 1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − ξ 2 ξ 2 (4q + 1) − ξ 4 ξ 0 + ξ 4 − ξ 4 + ρ(c4 − ξ 4 ) ⎪ 1 3 2 1 3 2 4 2 2 ⎪ ⎪  (2q + 1) ⎪ ⎪ ⎪ ⎪ (4q + 1) ⎨ 5 − 8ξ34 /3 ξ3 = ξ10 ξ24 + (ξ13 ξ21 + ξ11 ξ23 ) , (9.24) (q + 1)(3q + 1) ⎪ ⎪ ⎪ (4q + 1) ⎪ ⎪ ⎪ + ξ14 ξ20 + ρ(c34 − ξ34 ) + ξ12 ξ22 2 ⎪ ⎪  (2q + 1) ⎪ ⎪ ⎪ ⎪ (4q + 1) ⎪ ⎪ ⎪ ξ45 = R(ξ20 ξ34 + (ξ23 ξ31 + ξ21 ξ33 ) ⎪ ⎪ (q + 1)(3q + 1) ⎪ ⎪ ⎪ ⎪ (4q + 1) ⎪ ⎪ ⎩ + ξ22 ξ32 2 + ξ24 ξ30 ) + ρ(c44 − ξ44 )  (2q + 1)

9.1 Synchronous Control of the Fractional-Order Chaotic Systems

⎧ 6 ξ1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ξ26 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 6 ξ3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ξ46 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

173

= 10(ξ25 − ξ15 ) + ρ(c15 − ξ15 ) (5q + 1) − ξ15 ξ30 + ξ25 (q + 1)(4q + 1) (5q + 1) − (ξ12 ξ33 + ξ13 ξ32 ) − ξ45 + ρ(c25 − ξ25 ) (2q + 1)(3q + 1) (5q + 1) = ξ10 ξ25 + (ξ11 ξ24 + ξ14 ξ21 ) − 8ξ35 /3+ (q + 1)(4q + 1) (5q + 1) + ξ15 ξ20 + ρ(c35 − ξ35 ) (ξ12 ξ23 + ξ13 ξ22 ) (2q + 1)(3q + 1) (5q + 1) = R(ξ20 ξ35 + (ξ21 ξ34 + ξ24 ξ31 ) + ξ25 ξ30 )+ (q + 1)(4q + 1) (5q + 1) + ρ(c45 − ξ45 ) (ξ22 ξ33 + ξ23 ξ32 ) (2q + 1)(3q + 1) = 28ξ15 − ξ10 ξ35 − (ξ11 ξ34 + ξ14 ξ31 )

(9.25)

Using the above numerical solution, the coupled system is solved. Take the parameter R = 0.2, the fractional order q = 0.96, the step size for the simulation is h = 0.01, the initial values of the drive system are x 1 (0) = 1, x 2 (0) = 2, x 3 (0) = 3, and x 4 (0) = 4, and the initial values of the coupled system are y1 (0) = 5, y2 (0) = 6, y3 (0) = 7, and y4 (0) = 8. Let ei = x i -yi , then the synchronous error curve is shown in Fig. 9.1. It can be seen that the drive system (9.4) and the coupled system (9.5) are synchronized at t ≈ 1.2 s. Next, we study the synchronization setup time of the fractional-order Lorenz hyperchaotic system when the order q and the coupling strength ρ are taken to different values. In Fig. 9.2a, when the fractional-order q is taken as 0.7, 0.8, 0.9, and 1.0, respectively, the coupling synchronization setup time decreases gradually

Fig. 9.1 Coupling synchronization error curves for the fractional-order Lorenz hyperchaotic systems a e1, b e2, c e3 , and d e4

174

9 Applications of fractional-order Chaotic Systems in Secure Communications

Fig. 9.2 Coupling synchronization error curves for the fractional-order Lorenz hyperchaotic systems with different parameters a Fractional order q variation and b Coupling strength ρ variation

with order q increasing. In Fig. 9.2b, when the coupling strength ρ is taken as 2, 4, 6, and 8, respectively, the synchronization setup time gradually decreases, which shows that a relatively large coupling strength should be selected to obtain a shorter synchronization setup time in practical applications. According to Theorem 9.1,     ˜ = 24.284, ψ = 81.555, and λmax B˜ = 1.618, the theoretical coupling λmax C strength ρ to achieve synchronization is 156.24. In fact, the simulations are using coupling strength values that are smaller than this theoretical value, i.e., Remark 9.1 is verified.

9.1.2 Generalized Function Projection Synchronization for Fractional-Order Chaotic Systems When the projective synchronization is achieved between two fractional-order chaotic systems, the state variables of the two systems evolve according to a certain proportional or functional relationship, which is defined as a generalized synchronization phenomenon. 1.

Generalized function projection synchronization algorithm for fractional-order chaotic systems

Consider the following fractional-order chaotic system: q

Dt0 x = f (x),

(9.26)

where x ∈ Rn is the system state variable and f : Rn → Rn is a nonlinear function. The response system equation is defined as q

Dt0 y = f (y) + U,

(9.27)

9.1 Synchronous Control of the Fractional-Order Chaotic Systems

175

where U is the nonlinear controller. Define the error system as e(t) = y − x,

(9.28)

where Φ is the n × n constant matrix. Definition 9.2 For the drive system (9.26) and the response system (9.27), the generalized function projection synchronization is achieved if there exists a scale factor matrix Φ such that lim e(t) = 0 holds. t→∞

Theorem 9.2 The fractional-order chaotic systems (9.26) and (9.27) achieve functional projection synchronization when the controller is designed with U(t) = u(t) + Ψ (t), where u(t) = Φf (x) - f (y), Ψ (t) = -κe, κ = [κ 1 , κ 2 , …, κ n ]T , κ i > 0 (i = 1, 2, …, n), and n ≥ 3. Proof The fractional-order error system equation corresponding to system (9.26) and system (9.27) is q

q

q

Dt0 e = Dt0 y − Dt0 x = f (y) −  f (x) + U.

(9.29)

Here, the nonlinear controller is defined as U(t) = u(t) + (t),

(9.30)

u(t) = f (x) − f (y) (t) = −κe

(9.31)

where

Substituting the above equation into Eq. (9.29), the fractional-order error system becomes q

Dt0 e = −κe.

(9.32)

It converges when κ i > 0 (i = 1, 2, …, n), implying that there is functional projection synchronization between the two chaotic system variables. Next, the relationship matrix between the variables of the two fractional-order chaotic systems is given, and from Eq. (9.28), the relationship matrix is a constant matrix, and it is defined as ⎡

α11 ⎢ α21 ⎢ =⎢ . ⎣ ..

α12 α22 .. .

··· ··· .. .

⎤ α1n α2n ⎥ ⎥ .. ⎥, . ⎦

αn1 αn2 · · · αnn

(9.33)

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9 Applications of fractional-order Chaotic Systems in Secure Communications

so ⎧ y1 = α11 x1 + α12 x2 + · · · + α1n xn ⎪ ⎪ ⎪ ⎪ ⎨ y2 = α21 x1 + α22 x2 + · · · + α2n xn .. ⎪ ⎪ . ⎪ ⎪ ⎩ yn = αn1 x1 + αn2 x2 + · · · + αnn xn

(9.34)

where y = [y1 , y2 , …, yn ] and x = [x 1 , x 2 , …, x n ]. It is found that the response system variables are linearly related to the weighted sum of all the driving system variables. The scale factor matrix Φ of the projection synchronization can be chosen flexibly with unpredictability, increasing the key space. On the other hand, the system variables of the response system are a linear superposition of the drive system variables, which makes the drive system attractor not similar to the response system attractor in shape and increases the complexity of the sequence. It can be seen that the designed generalized function projection synchronization scheme is beneficial to enhance the security of confidential communication. 2.

Function projection synchronization simulation of the fractional-order simplified Lorenz hyperchaotic system

Taking the fractional-order simplified Lorenz hyperchaotic system as an example, the function projection synchronization is studied, and the equations of the fractionalorder simplified Lorenz hyperchaotic drive system are ⎧ q Dt0 x1 ⎪ ⎪ ⎪ q ⎨ Dt0 x2 q ⎪ ⎪ Dt0 x3 ⎪ ⎩ q Dt0 x4

= 10(x2 − x1 ) = (24 − 4c)x1 − x1 x3 + cx2 + x4 = x1 x2 − 8x3 /3 = −kx1

(9.35)

The response system equation is ⎧ q Dt0 y1 ⎪ ⎪ ⎪ ⎨ Dq y t0 2 ⎪ Dtq0 y3 ⎪ ⎪ ⎩ q Dt0 y4

= 10(y2 − y1 ) + u 1 = (24 − 4c)y1 − y1 y3 + cy2 + y4 + u 2 = y1 y2 − 8y3 /3 + u 3 = −ky1 + u 4

(9.36)

According to Theorem 9.2, the design controllers (u1 , u2 , u3 , u4 ) are u 1 = −κ(y1 − α11 x1 − α12 x2 − α13 x3 − α14 x4 ) − 10(y2 − y1 )+ α11 (10(x2 − x1 )) + α12 ((24 − 4c)x1 − x1 x3 + cx2 + x4 )+, α13 (x1 x2 − 8x3 /3) − α14 kx1

(9.37)

9.1 Synchronous Control of the Fractional-Order Chaotic Systems

177

u 2 = − κ(y2 − α21 x1 − α22 x2 − α23 x3 − α24 x4 )− (24 − 4c)y1 + y1 y3 − cy2 − y4 + α21 10(x2 − x1 )+, α22 ((24 − 4c)x1 − x1 x3 + cx2 + x4 )+ α23 (x1 x2 − 8x3 /3) − α24 kx1

(9.38)

u 3 = − κ(y3 − α31 x1 − α32 x2 − α33 x3 − α34 x4 ) − y1 y2 + 8y3 /3 + α31 (10(x2 − x1 )) + α32 ((24 − 4c)x1 −, x1 x3 + cx2 + x4 ) + α33 (x1 x2 − 8x3 /3) − α34 kx1

(9.39)

u 4 = − κ(y1 − α41 x1 − α42 x2 − α43 x3 − α44 x4 ) + ky1 + α41 (10(x2 − x1 )) + α42 ((24 − 4c)x1 − x1 x3 + . cx2 + x4 ) + α43 (x1 x2 − 8x3 /3) − α44 kx1

(9.40)

Substituting Eqs. (9.37–9.40) into Eq. (9.36), the response system is obtained as ⎧ q Dt0 y1 = − κ(y1 − α11 x1 − α12 x2 − α13 x3 − α14 x4 ) + α11 (10(x2 − x1 ))+ ⎪ ⎪ ⎪ ⎪ ⎪ α12 ((24 − 4c)x1 − x1 x3 + cx2 + x4 ) + α13 (x1 x2 − 8x3 /3) − α14 kx1 ⎪ ⎪ q ⎪ ⎪ ⎪ Dt0 y2 =κ(y2 − α21 x1 − α22 x2 − α23 x3 − α24 x4 ) + α21 (10(x2 − x1 ))+ ⎪ ⎪ ⎪ ⎨ α22 ((24 − 4c)x1 − x1 x3 + cx2 + x4 ) + α23 (x1 x2 − 8x3 /3) − α24 kx1 q ⎪ Dt0 y3 = − κ(y3 − α31 x1 − α32 x2 − α33 x3 − α34 x4 ) + α31 (10(x2 − x1 )) + . ⎪ ⎪ ⎪ ⎪ ⎪ α32 ((24 − 4c)x1 − x1 x3 + cx2 + x4 ) + α33 (x1 x2 − 8x3 /3) − α34 kx1 ⎪ ⎪ ⎪ q ⎪ ⎪ ⎪ Dt0 y4 = − κ(y1 − α41 x1 − α42 x2 − α43 x3 − α44 x4 ) + α41 (10(x2 − x1 ))+ ⎪ ⎩ + α42 ((24 − 4c)x1 − x1 x3 + cx2 + x4 ) + α43 (x1 x2 − 8x3 /3) − α44 kx1 (9.41) Using the Adomian algorithm to solve the response system yields ⎧ 5 ⎪  ⎪ j ⎪ ⎪ y (m + 1) = ζ1 h jq / ( jq + 1) 1 ⎪ ⎪ ⎪ ⎪ j=0 ⎪ ⎪ ⎪ ⎪ 5 ⎪  ⎪ j ⎪ ⎪ y (m + 1) = ζ2 h jq / ( jq + 1) ⎪ 2 ⎪ ⎨ j=0 5 ⎪  ⎪ ⎪ j ⎪ y3 (m + 1) = ζ3 h jq / ( jq + 1) ⎪ ⎪ ⎪ ⎪ j=0 ⎪ ⎪ ⎪ ⎪ 5 ⎪ ⎪  ⎪ j ⎪ ⎪ y4 (m + 1) = ζ4 h jq / ( jq + 1) ⎪ ⎩ j=0

,

(9.42)

178

9 Applications of fractional-order Chaotic Systems in Secure Communications

where ζ10 = y1 (m), ζ20 = y2 (m), ζ30 = y3 (m), ζ40 = y4 (m),   ⎧ j j−1 j−1 j−1 j−1 j−1 ⎪ ζ1 = − κ ζ1 − α11 c1 − α12 c2 − α13 c3 − α14 c4 + ⎪ ⎪ ⎪ ⎪ ⎪ j j j j ⎪ ⎪ α11 c1 + α12 c2 + α13 c3 + α14 c4 ⎪ ⎪   ⎪ ⎪ j j−1 j−1 j−1 j−1 j−1 ⎪ ⎪ + ζ = − κ ζ − α c − α c − α c − α c 21 22 23 24 ⎪ 2 2 1 2 3 4 ⎪ ⎪ ⎪ ⎪ j j j j ⎨ α21 c1 + α22 c2 + α23 c3 + α24 c4  ,  j j−1 j−1 j−1 j−1 j−1 ⎪ ⎪ + ζ = − κ ζ − α c − α c − α c − α c ⎪ 31 32 33 34 3 3 1 2 3 4 ⎪ ⎪ ⎪ ⎪ j j j j ⎪ ⎪ α31 c1 + α32 c2 + α33 c3 + α34 c4 ⎪ ⎪   ⎪ ⎪ ⎪ ⎪ ζ4j = − κ ζ4j−1 − α41 c1j−1 − α42 c2j−1 − α43 c3j−1 − α44 c4j−1 + ⎪ ⎪ ⎪ ⎪ ⎩ j j j j α41 c1 + α42 c2 + α43 c3 + α44 c4

(9.43)

(9.44)

j

where ci ( i = 1, 2, 3, 4 and j = 1, 2, 3, 4, 5) are from the drive system (9.35), which is computed as described in Sect. 4.1. The following function projection simulation experiments are based on MATLAB for the fractional-order simplified Lorenz hyperchaotic system. The parameters of the fractional-order simplified Lorenz hyperchaotic system are q = 0.98, c = −2, k = 5, and the simulation time step size h = 0.01. The initial values of the drive system are x 1 (0) = 0.1, x 2 (0) = 0.2, x 3 (0) = 0.3, and x 4 (0) = 0.4, and the initial values of the response system are y1 (0) = 5, y2 (0) = 6, y3 (0) = 7, and y4 (0) = 8. Let the scale factor matrix Φ = α ij δ ij , where α ij is the scale factor and δ ij takes the value 0 or 1, i.e., Φ is a non-singular matrix, which has and only has one non-zero element value in each row and column of the matrix. When Φ is a diagonal matrix, the response system variables correspond one-to-one with the driving system variables. This is a generalized function projection synchronization. Since the dimension of the fractional-order simplified Lorenz hyperchaotic system is 4, i.e., there exist 4! − 1 = 23 possible generalized misalignment function projection synchronization schemes. Complete synchronization, projection synchronization, and generalized projection synchronization are all special cases of generalized misfit function projection synchronization. As an example, the following scale factor matrix is used to study the generalized misfit function projection synchronization of fractional-order simplified Lorenz hyperchaotic systems. ⎡

0 ⎢ −0.5 =⎢ ⎣ 0 0

0.5 0 0 0 0 0 0 −1.5

⎤ 0 0 ⎥ ⎥. 1.5 ⎦ 0

(9.45)

9.1 Synchronous Control of the Fractional-Order Chaotic Systems

179

Therefore, the generalized function projection synchronization error is ⎧ e1 ⎪ ⎪ ⎪ ⎨e 2 ⎪ e3 ⎪ ⎪ ⎩ e4

= y1 − 0.5x2 = y2 + 0.5x1 . = y3 − 1.5x4

(9.46)

= y4 + 1.5x3

The results of the simulation of the generalized dislocation function projection synchronization for the fractional-order simplified Lorenz hyperchaotic system are shown in Fig. 9.3. From Eq. (9.46), the variables y1 , y2 , y3 , and y4 are synchronized with 0.5x 2 , − 0.5x 1 , 1.5x 4 , and 1.5x 3 , respectively. The time series pairs (y1 , x 2 ), (y2 , x 1 ), (y3 , x 4 ), and (y4 , x 3 ) curves are shown in Fig. 9.3a, c, e, and g, and the corresponding synchronization graphs are shown in Fig. 9.3b, d, e, and f. It can be seen that the two systems achieve synchronization of the dislocation function projections. Let the scale factor matrix Φ = α ij , where α ij is a real constant, then the generalized dislocation function projection is synchronized as a special case of this situation. In particular, let the scale factor matrix take the values as ⎡

0.1 ⎢ 0.25 =⎢ ⎣ 0.1 0.3

0.2 0.25 0.4 0.2

0.3 0.25 0.4 0.2

⎤ 0.4 0.25 ⎥ ⎥. 0.1 ⎦ 0.3

(9.47)

Thus, according to Eq. (9.28), the error system can be expressed as ⎧ e ⎪ ⎪ 1 ⎪ ⎨e 2 ⎪ e3 ⎪ ⎪ ⎩ e4

= y1 − 0.1x1 − 0.2x2 − 0.3x3 − 0.4x4 = y2 − 0.25x1 − 0.25x2 − 0.25x3 − 0.25x4 = y3 − 0.1x1 − 0.4x2 − 0.4x3 − 0.1x4 = y4 − 0.3x1 − 0.2x2 − 0.2x3 − 0.3x4

(9.48)

Using the Adomian solution algorithm, the simulation results are shown in Fig. 9.4. The time series x 1 and y1 are shown in Fig. 9.4a. It can be seen that there is no obvious relationship between the two variables. In fact, according to Eq. 9.48, when t → ∞, y1 = 0.1x 1 + 0.2x 2 + 0.3x 3 + 0.4x 4 , the response system y1 − y2 attractor phase diagram is shown in Fig. 9.4b, compared with the fractional-order simplified Lorenz hyperchaotic attractor, the response system attractor phase diagram in shape does not have any similarity with it. Therefore, the type of drive system attractor cannot be inferred from the phase diagram of the response system attractor. We define ⎧ ϕ1 =0.1x1 + 0.2x2 + 0.3x3 + 0.4x4 ⎪ ⎪ ⎪ ⎨ ϕ =0.25x + 0.25x + 0.25x + 0.25x 2 1 2 3 4 (9.49) ⎪ ϕ =0.1x + 0.4x + 0.4x + 0.1x 1 2 3 4 ⎪ 3 ⎪ ⎩ ϕ4 =0.3x1 + 0.2x2 + 0.2x3 + 0.3x4

180

9 Applications of fractional-order Chaotic Systems in Secure Communications

Fig. 9.3 Dislocation function projection synchronization of the fractional-order simplified Lorenz hyperchaotic system a Time series y1 and x 2 , b y1 –x 2 , c Time series y2 and x 1 , d y2 –x 1 , e Time series y3 and x 4 , f y3 –x 4 , g Time series y4 and x 3 , and h y4 –x 3

From Fig. 9.4d–g, it can be seen that the values of y1 –ϕ 1 , y2 –ϕ 2 , y3 –ϕ 3, and y4 –ϕ 4 are consistent between them. It follows that the drive system (9.26) and the response system (9.27) achieve generalized linear scale function projection synchronization. 3.

Analysis of the function projection properties of the fractional-order simplified Lorenz hyperchaotic systems

First, taking the above generalized linear scale function projection synchronization as an example, we analyze the relationship between the synchronization setup time

9.1 Synchronous Control of the Fractional-Order Chaotic Systems

181

Fig. 9.4 Generalized linear scale function projection synchronization for the fractional-order simplified Lorenz hyperchaotic system a Time series y1 and x 1 , b x 1 –y1 , c y1 –y2 , d y1 –ϕ 1 , e y2 –ϕ 2 , f y3 –ϕ 3 , and g y4 –ϕ 4

and the variation of the system order q and the control parameters κ = [κ 1 , κ 2 , κ 3 , κ 4 ]T . From Fig. 9.5, it can be seen that the system synchronization establishment time increases with the order q, which means that synchronization is easier to establish when the system is in the fractional-order case. The synchronization establishment time increases with the control parameter κ as shown in Fig. 9.5, where the synchronization error is defined as

182

9 Applications of fractional-order Chaotic Systems in Secure Communications

Fig. 9.5 Synchronization setup time with parameters variation a the order q variation and b the control parameter κ variation

Err or =

4 

|yi − αi1 x1 − αi2 x2 − αi3 x3 − αi4 x4 |.

(9.50)

i=1

Obviously, the synchronization establishment time gradually decreases as the control parameter κ increases. For practical applications, a relatively large control parameter should be selected to achieve synchronization more easily. Based on the characteristics of the Adomian algorithm for solving the synchronous system, a secure communication scheme is designed as shown in Fig. 9.6. The signal transmitted in the channel is the state variable of the chaotic system, and at the receiving end, the response system needs to calculate the intermediate variables of the driving system once again for use in the response system solution. The characteristic is that the synchronous secure communication scheme is consistent with the conventional scheme. The information transmitted by the channel is small, but the computation amount in the driving system is relatively large. Fig. 9.6 Synchronous and secure communication scheme based on chaotic variables

9.1 Synchronous Control of the Fractional-Order Chaotic Systems

183

9.1.3 Network Synchronization of Fractional-Order Chaotic Systems With the rapid development of complexity science as well as Internet technology, various complex networks appeared in the life of human society, such as the Internet, World Wide Web (WWW), science citation network, metabolic network, biological network, and social network. The application of complex networks composed of chaotic systems to secure communication has important research significance, and their synchronization has attracted the researcher’s attention. 1.

Synchronization algorithm for ring networks of fractional-order time-delay chaotic systems

A ring bidirectional coupling complex network composed of N fractional-order timedelay chaotic system is defined by ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

q

Dt0 x1 (t) =L(x1 , x1 (t − τ )) + N (x1 (t), x1 (t − τ )) + ρ(x N + x2 − 2x1 ) q

Dt0 x2 (t) =L(x2 , x2 (t − τ )) + N (x2 (t), x2 (t − τ )) + ρ(x1 + x3 − 2x2 ) .. , . ⎪ ⎪ ⎪ q ⎪ ⎪ ⎪ Dt0 x N −1 (t) =L(x N −1 , x N −1 (t − τ )) + N (x N −1 (t), x N −1 (t − τ )) ⎪ ⎪ ⎪ ⎪ + ρ(x N −2 + x N − 2x N −1 ) ⎪ ⎪ ⎪ q ⎪ ⎪ D x (t) =L(x N , x N (t − τ )) + N (x N (t), x N (t − τ )) t0 N ⎪ ⎪ ⎩ + ρ(x N −1 + x1 − 2x N )

(9.51)

where x1 , x2 , …, xN ∈ Rn are the state variables of each fractional-order time-delay chaotic system and ρ is the coupling coefficient. L(xi , x(t – τ ) is the linear part of the ith time-delayed chaotic system, and N(xi , x(t – τ ) is the nonlinear part of the ith time-delayed chaotic system. The ring network structure is a representative, yet relatively simple network structure. On the other hand, the coupled synchronization algorithm is easier to implement physically in practice compared to other algorithms, so it is of practical importance to study this structure. The network structure is shown in Fig. 9.7, where each node in the channel is a fractional-order chaotic system and interacts with only two adjacent systems for information interaction. Definition 9.3 Define the error ei = xi – xi+1 (i = 1, 2, …, N – 1). When i = N, eN = xN -x1 , the network achieves as t → ∞ for each ||ei ||→0, where     synchronization  ei  = pj=1 ei j  = pj=1 xi j − x(i+1) j , and p is the dimension of the system.

184

9 Applications of fractional-order Chaotic Systems in Secure Communications

Fig. 9.7 Structure diagram of the two-way coupled ring coupling network

Definition 9.3 shows that network synchronization of the fractional-order chaotic systems is achieved when the dynamical behavior of each node in the chaotic network (fractional-order chaotic system) is consistent. Theorem 9.3 For the two-way ring-coupled network system described by Eq. (9.51), synchronization is achieved when the coupling coefficient ρ > 0. Proof From Definition 9.3, we construct the Lyapunov function as V (t) =

N 

ei ,

(9.52)

i=1

where V (t) is the synchronization error of this network. Introducing the fractionalorder calculus operator, we obtain q

Dt0 V (t) =

N 

q

Dt0 ei  =

i=1

p 

q

Dt0 |e1 j | +

j=1

p 

q

Dt0 |e2 j | · · · +

j=1

p 

q

Dt0 |e N j |. (9.53)

j=1

By Lemma 9.1, we have q Dt0 V (t)

=

p 

q sgn(e1 j )Dt0 e1 j

j=1

=

+

p 

q sgn(e1 j )Dt0 e2 j

··· +

j=1

p 

q

q

j=1

p 

p

j=1

q

q

q

q

sgn(e2 j )[Dt0 x2 j − Dt0 x3 j ]+

j=1



q

sgn(e1 j )Dt0 e N j

j=1

sgn(e1 j )[Dt0 x1 j − Dt0 x2 j ] +

··· +

p 

sgn(e N j )[Dt0 x N j − Dt0 x1 j ].

9.1 Synchronous Control of the Fractional-Order Chaotic Systems

≤

p 

q

q

[Dt0 x1 j − Dt0 x2 j ] +

j=1

p 

q

185

q

[Dt0 x2 j − Dt0 x3 j ]+

j=1

 p

··· +

q

q

[Dt0 x N j − Dt0 x1 j ]

j=1

=0

(9.54)

q

When Dt0 V (t) = 0, it means that ei j ≥ 0, for i = 1, 2, …, N and j = 1, 2, …, p, q q q q q q q q i.e.,Dt0 x1i ≥ Dt0 x2i , Dt0 x2i ≥ Dt0 x2i , …, Dt0 x(N −1)i ≥ Dt0 x N i ,Dt0 x N i ≥ Dt0 x1i . So q q q q Dt0 V (t) = 0 holds when and only when Dt0 x1i = Dt0 x2i = · · · = Dt0 x N i (i = 1, 2, q …, N), otherwise Dt0 V (t) < 0. It implies that V (t) → 0 as t → ∞. So, the two-way coupled ring network system achieves synchronization. Remark 9.3 For any N > 3, the bidirectional coupled ring network is synchronized. Remark 9.4 According to experiments, the value of the coupling strength ρ cannot be taken too small, otherwise, synchronization is difficult to establish. In fact, generally ρ ≥ 1. Remark 9.5 Bidirectional coupling synchronization is a type of coupling synchronization and is easier to achieve between different systems compared to unidirectional coupling. 2.

Network synchronization solution for the fractional-order time-delayed Lorenz chaotic systems

The fractional-order Lorenz system equation is ⎧ q ⎪ ⎨ Dt0 x1 = a(x2 − x1 ) q Dt0 x2 = cx1 − x1 x3 + d x2 , ⎪ ⎩ q Dt0 x3 = x1 x2 − bx3

(9.55)

where x 1 , x 2 , and x 3 are the system state variables and a, b, c, and d are the system parameters. The controller u = (u1 , u2 , u3 ) is introduced to the fractional-order Lorenz system, and we obtained ⎧ q ⎪ ⎨ Dt0 x1 = a(x2 − x1 ) + u 1 q Dt0 x2 = cx1 − x1 x3 + d x2 + u 2 ⎪ ⎩ q Dt0 x3 = x1 x2 − bx3 + u 3

(9.56)

When the controllers are u1 = κx 1 (t – τ ), u2 = 0, and u3 = 0, respectively, the fractional-order time-delay Lorenz chaotic system is obtained as

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9 Applications of fractional-order Chaotic Systems in Secure Communications

⎧ q ⎪ ⎨ Dt0 x1 = a(x2 − x1 ) + κ x1 (t − τ ) q Dt0 x2 = cx1 − x1 x3 + d x2 , ⎪ ⎩ q Dt0 x3 = x1 x2 − bx3

(9.57)

where κ is the control parameter and τ is the time delay. Applying the system to a two-way coupled ring network, and then the network is defined as ⎧ q ⎪ ⎨ Dt0 xi1 =a(xi2 − xi1 ) + κ xi1 (t − τ ) + ρ(x(i−1)1 + x(i+1)1 − 2xi1 ) q Dt0 xi2 =cxi1 − xi1 xi3 + d xi2 + +ρ(x(i−1)2 + x(i+1)2 − 2xi2 ) ⎪ ⎩ q Dt0 xi3 =xi1 xi2 − bxi3 + ρ(x(i−1)3 + x(i+1)3 − 2xi3 )

(9.58)

Next, the numerical solution of the fractional-order time-delay network system is studied based on the Adomian algorithm. Firstly, the time is gridded {t n = nh, n = −m, −(m − 1), …, −1, 0, 1, 2, …, N}, where m = τ /h and N is the length of the time series. The system is solved by using the time-delay Adomian decomposition algorithm [2], and the solution can be expressed as ⎧ 5  ⎪ j ⎪ ⎪ x (n + 1) = K i1 h jq / ( jq + 1) 1i ⎪ ⎪ ⎪ ⎪ j=0 ⎪ ⎪ ⎪ ⎪ 5 ⎨  j x2i (n + 1) = K i2 h jq / ( jq + 1) ⎪ ⎪ j=0 ⎪ ⎪ ⎪ ⎪ 5 ⎪  ⎪ ⎪ j ⎪ ⎪ x (n + 1) = K i3 h jq / ( jq + 1) ⎩ 3i

(9.59)

j=0 j

The intermediate variables K i = 0 (i = 1, 2, 3, j = 0, 1, 2, 3, 4, 5) are expressed as 0 0 0 K i1 = x1i (n), K i2 = x2i (n), K i3 = x3i (n),

⎧ 1 0 0 0 0 0 0 K =a(K i2 − K i1 ) + κ K i1τ + ρ(K (i−1)1 + K (i+1)1 − 2K i1 ) ⎪ ⎨ i1 1 0 0 0 0 0 0 0 K i2 =cK i1 − K i1 K i3 + d K i2 + ρ(K (i−1)2 + K (i+1)2 − 2K i2 ) , ⎪ ⎩ 1 0 0 0 0 0 0 K i3 =K i1 K i2 − bK i3 + ρ(K (i−1)3 + K (i+1)3 − 2K i3 )

(9.60)

(9.61)

⎧ 2 1 1 1 1 1 1 K =a(K i2 − K i1 ) + κ K i1τ + ρ(K (i−1)1 + K (i+1)1 − 2K i1 ) ⎪ ⎨ i1 2 1 0 1 1 0 1 1 1 1 K i2 =cK i1 − K i1 K i3 − K i1 K i3 + d K i2 + ρ(K (i−1)2 + K (i+1)2 − 2K i2 ), ⎪ ⎩ 2 1 0 0 1 1 1 1 1 K i3 =K i1 K i2 + K i1 K i2 − bK i3 + ρ(K (i−1)3 + K (i+1)3 − 2K i3 ) (9.62)

9.1 Synchronous Control of the Fractional-Order Chaotic Systems

⎧ 3 0 0 2 2 2 2 K i1 =a(K i2 − K i1 ) + κ K i1τ + ρ(K (i−1)1 + K (i+1)1 − 2K i1 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 2 0 2 1 1 (2q + 1) 2 0 ⎪ − K i1 K i3 + ⎪ K i2 =cK i1 − K i1 K i3 − K i1 K i3 2 ⎪  (q + 1) ⎪ ⎨ 2 2 2 2 + ρ(K (i−1)2 + K (i+1)2 − 2K i2 ) d K i2 ⎪ ⎪ ⎪ (2q + 1) ⎪ 0 2 1 1 2 0 2 ⎪ ⎪ K 3 =K i1 K i2 + K i1 K i2 + K i1 K i2 − bK i3 + ⎪ 2 (q + 1) ⎪ i3  ⎪ ⎪ ⎩ 2 2 2 + K (i+1)3 − 2K i3 ) ρ(K (i−1)3

⎧ 4 3 3 3 3 3 3 K i1 =a(K i2 − K i1 ) + κ K i1τ + ρ(K (i−1)1 + K (i+1)1 − 2K i1 ) ⎪ ⎪ ⎪ ⎪ ⎪ (3q + 1) ⎪ 4 3 0 3 2 1 1 2 ⎪ − ⎪ K i2 =cK i1 − K i1 K i3 − (K i1 K i3 + K i1 K i3 ) ⎪ (q + 1)(2q + 1) ⎪ ⎨ 3 0 3 3 3 3 K i3 + d K i2 + ρ(K (i−1)2 + K (i+1)2 − 2K i2 ) K i1 , ⎪ ⎪ ⎪ (3q + 1) ⎪ 0 3 2 1 1 2 3 0 ⎪ ⎪ K 4 =K i1 K i2 + (K i1 K i2 + K i1 K i2 ) + K i1 K i2 − ⎪ ⎪ i3 (q + 1)(2q + 1) ⎪ ⎪ ⎩ 3 3 3 3 + ρ(K (i−1)3 + K (i+1)3 − 2K i3 ) bK i3

187

(9.63)

(9.64)

⎧ 5 4 4 4 4 4 4 ⎪ ⎪ K i1 =a(K i2 − K i1 ) + κ K i1τ + ρ(K (i−1)1 + K (i+1)1 − 2K i1 ) ⎪ ⎪ ⎪ (4q + 1) ⎪ 5 4 0 4 3 1 1 3 ⎪ ⎪ K i2 =cK i1 − K i1 K i3 − (K i1 K i3 + K i1 K i3 ) ⎪ ⎪ (q + 1)(3q + 1) ⎪ ⎪ ⎪ ⎪ (4q + 1) ⎨ 2 2 4 0 4 4 4 4 − K i1 K i3 − K i1 K i3 + d K i2 + ρ(K (i−1)2 + K (i+1)2 − 2K i2 ) 2 (2q + 1) ,  ⎪ ⎪ ⎪ (4q + 1) ⎪ 5 0 4 3 1 1 3 ⎪ ⎪ K i3 =K i1 K i2 + (K i1 K i2 + K i1 K i2 ) ⎪ ⎪ (q + 1)(3q + 1) ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 (4q + 1) 4 0 4 4 4 4 ⎪ ⎩ + K i1 K i2 + K i1 K i2 − bK i3 + ρ(K (i−1)3 + K (i+1)3 − 2K i3 )  2 (2q + 1) (9.65) j

j

where K i1τ = K i1 (t − τ ). When t n ≤ mh, t n – τ lies between (n – 1 – m)h and (n – m)h. When t n > mh, t n −τ lies between (n – m)h and (n – m + 1)h. Based on j the linear fitting technique, the intermediate variable K τ 1 value of the time-delay variable is calculated by ⎧  τ j τ j ⎪ K 1 (tn−1−m ) + m − K 1 (tn−m ), for, tn ≤ mh ⎨ 1−m+ j h h   K τ 1 (tn ) =  . τ τ j j ⎪ ⎩ 1−m+ K 1 (tn−m ) + m − K 1 (tn−m+1 ), for, tn > mh h h (9.66) Therefore, the numerical solution of the fractional-order time-delay network can be obtained according to the above equation. From the above numerical solution expression, it is clear that the current value is determined by the previous moment

188

9 Applications of fractional-order Chaotic Systems in Secure Communications

value as well as the intermediate variable values and variable values before the moment τ when solving using the time-delay Adomian algorithm. Also in the actual solution, the historical values of the state variables and intermediate variables at [−τ, 0] need to be set for subsequent calculations. 3.

Network synchronization simulation of the fractional-order time-delay Lorenz chaotic systems

The following numerical simulations are performed for the network synchronization of the fractional-order time-delay Lorenz chaotic systems. Let the parameters of the fractional-order time-delay Lorenz system be a = 40, b = 3, c = 10, and d = 25; the parameters of the time-delay controller be κ = 1 and τ = 0.5; the value of the coupling strength be ρ = 5; the fractional-order order be q = 0.98; and the value of the simulation time step be h = 0.01. The synchronization network contains five fractional-order time-delay chaotic systems, i.e., the network state matrix is ⎡

−2 ⎢ 1 ⎢ ⎢ G=⎢ 0 ⎢ ⎣ 0 1

1 −2 1 0 0

0 1 −2 1 0

0 0 0 −2 1

⎤ 1 0 ⎥ ⎥ ⎥ 0 ⎥. ⎥ 1 ⎦ −2

(9.67)

In the time period [−τ, 0], the initial values of the state variables of each system from system 1 to system 5 are (1, 2, 3), (4, 5, 6), (7, 8, 9,), (10, 11, 12), and (13, 14, 15), respectively, and the initial values of the intermediate variable values are set to j K τ i = i × 10 j−1 . The synchronization error curves between the systems are shown in Fig. 9.8, where eij = x i –x j. As seen in Fig. 9.8, the two systems are synchronized

Fig. 9.8 Synchronization error curve between systems a e12 , b e23 , c e34 , and d e45

9.1 Synchronous Control of the Fractional-Order Chaotic Systems

189

0 (x ) − Fig. 9.9 Intermediate variable attractor for System 1 (SYS1 is shown in Fig. 6.14) a K 11 11 0 0 1 1 1 2 2 2 3 3 3 4 4 −K4 , K 13 (x12 )− K 12 (x13 ), b K 11 − K 13 − K 12 , c K 11 − K 13 − K 12 , d K 11 − K 13 − K 12 , e K 11 − K 13 12 5 − K5 − K5 and f K 11 13 12

with each other in the ring network. It can be seen that the whole network is also synchronized between the systems. According to the network synchronization scheme as shown in Fig. 9.7, S i (i = 1, 2, …, N) is sent to the target system S i − 1 and system S i + 1. From the numerical solution of the time-delayed system, S i consists of K ilk (k = 0, 1, 2, 3, 4, 5, l = 1, 2, 3, 4). Taking System 1 (SYS1 in Fig. 9.7) as an example, the attractor phase diagram consisting of its intermediate variables is shown in Fig. 9.9, which shows that the intermediate variables in the channel are also chaotic. Define the fractional-order time-delay network synchronization error as Ek =

4  3   k  K − K k  ij (i+1) j ,

(9.68)

i=1 j=1

where k = 0, 1, …, 5. Obviously, when k = 0, E 0 denotes the sum of the synchronization errors between the system state variables in the network. The synchronization errors between different intermediate variables in the fractional-order time-delay Lorenz chaotic system synchronization network are shown in Fig. 9.10, and it shows that in the actual simulation, synchronization also exists between different system intermediate variables, and in fact, the K ilk (k = 0, 1, 2, 3, 4, 5, l = 1, 2, 3, 4) in the transmitted signals S i in the channel are correspondingly synchronized after the ring network reaches synchronization. The variation of the synchronization network with order q, coupling strength ρ, and time delay τ for the fractional-order time-delay Lorenz chaotic system is shown in Fig. 9.11. In Fig. 9.11a, the synchronization establishment time decreases as the fractional order q decreases. Similarly, the synchronization establishment

190

9 Applications of fractional-order Chaotic Systems in Secure Communications

Fig. 9.10 Synchronization error curves for the fractional-order Lorenz hyperchaotic systems with intermediate variables a E 0 , b E 1 , c E 2 , d E 3 , e E 4 , and f E 5

Fig. 9.11 Synchronization establishment time with parameters variation a Order q variation, b Coupling strength ρ variation, and c Time delay τ variation

time decreases as the coupling strength ρ increases as shown in Fig. 9.11b, while the synchronization establishment time increases with the time delay τ as shown in Fig. 9.11c. When smaller order of the fractional-order system is chosen, the looped bidirectional coupled fractional-order time-delay Lorenz system network is easier to establish the synchronization, and in practical applications, a relatively strong coupling strength should be chosen. In addition, the larger the value of the time delay, the initial value part has a greater impact on the synchronization establish time at the beginning. The synchronization of the network is more difficult to establish. In practical secure communication, the time delay of the chaotic system should be minimized.

9.2 Design of the Fractional-Order Chaotic Pseudo-Random Sequence Generator

191

9.2 Design of the Fractional-Order Chaotic Pseudo-Random Sequence Generator 9.2.1 Design of Chaotic Pseudo-Random Sequence Generator Based on the Fractional-Order Simplified Lorenz System When we design a pseudo-random sequence generator based on the chaotic system, the chaotic sequence is generally converted into binary pseudo-random sequences by quantization algorithms, which play a critical role in this process and directly determine the performance of the generated pseudo-random sequence. People use integerorder chaotic systems or discrete chaotic systems to design pseudo-random sequence generators, and many kinds of quantization algorithms have been proposed [3–5], including the simple direct interception and complex shift-and-crossover heterodyne operations. The purpose of the quantization algorithm is mainly to get the high-performance pseudo-random sequence from the chaotic sequence to meet the requirements. When the chaotic sequence complexity is not enough, we can meet the requirements through design the complex quantization algorithm. If the highperformance pseudo-random sequence can be obtained by using the simple quantization algorithm, it indicates that it has a high complexity itself. In addition, the quantization algorithm needs extra resources for the system, and the complex quantization algorithm occupies more system resources and complicates the program, so the application system tries to use the simple quantization algorithm to meet the requirements. Based on the Adomian decomposition algorithm to solve the fractional-order simplified Lorenz system, the obtained chaotic sequence is converted into a binary pseudo-random sequence by the quantization algorithm as shown in Fig. 9.12. Firstly, the chaotic sequence x n+1 , yn+1 , zn+1 is obtained from each iteration multiplied by 1011 , and its integer part is taken to obtain three 64-bit binary integers (DB63-DB0), which are defined as x I , yI , zI . The last 8 bits of x I (DB7-DB0) are chosen as the 8 bits of the pseudo-random binary sequence. As the iteration proceeds, a sufficiently long set of binary pseudo-random sequences BS1 is obtained. Meanwhile, x I ⊕ yI ⊕ zI is calculated and its last 8 bits are taken as the 8 bits of another pseudo-random binary sequence (BS2). In this way, two sets of pseudo-random binary sequences are

Fig. 9.12 Binary quantization algorithm for pseudo-random sequence generator

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9 Applications of fractional-order Chaotic Systems in Secure Communications

generated simultaneously. Obviously, the above quantization algorithms for generating two sets of pseudo-random sequences are simple. If BS1 and BS2 can satisfy the requirements of pseudo-random sequences, the complexity of the original chaotic sequence is high. Most of the current pseudo-random sequence generators based on chaotic systems are implemented by computer simulation, while microprocessors are generally used instead of PCs in engineering applications. DSPs are often used in engineering practice due to their superior performance in digital signal processing. Considering the different hardware structures, different data types, and different compilation systems of PC and DSP, etc., even if both use the same algorithm and initial conditions, there are still differences in the obtained chaotic sequence results. So it is more meaningful to implement a pseudo-random sequence generator based on fractional-order chaotic system on the DSP platform. Let the parameters of the fractional-order simplified Lorenz system c = 5, the differential order q = 0.65, and the initial values (x 0 , y0 , z0 ) = (0.1, 0.2, 0.3), then the system is chaotic, and the maximum Lyapunov exponent is 3.0679, and the corresponding attractor phase diagram is shown in Fig. 9.13a. The DSP platform is used to implement the fractional-order simplified Lorenz system, and the attractor displayed on the oscilloscope is shown in Fig. 9.13b. Finally, the quantization algorithm as shown in Fig. 9.12 is then used to quantize the chaotic sequences and obtain two sets of pseudo-random binary sequences BS1 and BS2. The software flow of the pseudo-random sequence generator is shown in Fig. 9.14, where the generated pseudo-random binary sequences are converted to ASCII codes before being transmitted to the computer for analysis and testing and are displayed directly on the computer side binary numbers or using MATLAB for analysis.

Fig. 9.13 Attractor of the fractional-order simplified Lorenz system a Computer simulation and b DSP implementation

9.2 Design of the Fractional-Order Chaotic Pseudo-Random Sequence Generator

193

Fig. 9.14 Software flow of the pseudo-random sequence generator

9.2.2 Performance Tests for Chaotic Pseudo-Random Sequences 1.

Statistical performance testing

To evaluate the performance of the generated pseudo-random sequences, statistical tests need to be performed on them. Among the many standard testing tools for pseudo-random sequences, NIST’s (National Institute of Science and Technology, NIST) STS (Statistical Test Suite, STS) is the more authoritative testing tool [6]. NIST released the statistical test package STS 2. 1. 2, and it was designed to test the random performance of obtained binary sequences. It includes 15 tests for a total of 188 tests. Almost all common tests are included, such as frequency, cumulative sum, and complexity. In the report of statistical test results from STS, there are two indicators of whether the binary pseudo-random sequence being tested is a pass or fail. A test is considered passed only if the p-values for each of the 188 tests are greater than 0.0001 and the percentage of sequences that pass the test are within the confidence interval. For a known threshold value α (typically set to 0.01), the confidence interval is defined as 

  (1 − α)α (1 − α)α , (1 − α) + 3 (1 − α) − 3 , m m 

where m is the length of the sequence for the test.

(9.69)

194

9 Applications of fractional-order Chaotic Systems in Secure Communications

For the fractional-order simplified Lorenz system with c = 5 and q = 0.65, two sets of pseudo-random binary sequences BS1 and BS2 are obtained. 100 pseudo-random binary sequences are selected for each set, each of which is 106 bits long. Defining α = 0.01, the confidence interval for most of the test items derived from Eq. (9.69) is [96.015%, 1]. It does not include the 8-item Random Excursions test and the 18-item Random Excursions Variant test. When we do these two types of tests, the test system automatically selects 57 sequences for testing in BS1 and 69 sequences for testing in BS2. So for BS1 and BS2, the confidence intervals in these 26 tests become [95.050%, 1] and [95.410%, 1], respectively. Figures 9.15 and 9.16 show the results for BS1 and BS2, respectively, and for the 188 tests for BS1. Figure 9.15a shows that the p-values are all greater than 0.0001, and Fig. 9.15b shows that the pass proportions of the sequences are all within the confidence intervals. For the 188 tests of BS2, the same test results are obtained in Fig. 9.16a, b. According to Figs. 9.15 and 9.16, the final results of the STS test for BS1 and NIST test for BS2 are obtained as shown in Tables 9.1 and 9.2, where five test items need to be tested multiple times, then the corresponding p-values listed and the percentage

Fig. 9.15 BS1 p-values and pass-through ratios a p-values and b Proportion of sequences that pass the test

Fig. 9.16 BS2 p-values and pass-through ratios, a p-values and b Proportion of sequences that pass the test

9.2 Design of the Fractional-Order Chaotic Pseudo-Random Sequence Generator

195

Table 9.1 NIST test results for BS1 No

Test index

P-values

Proportion

Results

1

Frequency

Number of tests 1

0.401199

100/100

Pass

2

Block frequency

1

0.494392

97/100

Pass

3

Cumulative sums

2

0.366918

100/100

Pass

4

Runs

1

0.289667

99/100

Pass

5

Longest run

1

0.494392

97/100

Pass

6

Rank

1

0.514124

100/100

Pass

7

FFT

1

0.816537

99/100

Pass

8

Non overlapping template

148

0.001509

97/100

Pass

9

Overlapping template

1

0.911413

98/100

Pass

10

Universal

1

0.383827

98/100

Pass

11

Approximate entropy

1

0.834308

97/100

Pass

12

Random excursions

8

0.102526

55/57

Pass

13

Random excursions variant

18

0.019188

56/57

Pass

14

Serial

2

0.115387

98/100

Pass

15

Linear complexity

1

0.924076

100/100

Pass

Table 9.2 NIST test results for BS2 No

Test index

P-values

Proportion

Results

1

Frequency

Number of tests 1

0.080519

99/100

Pass

2

Block frequency

1

0.699313

99/100

Pass

3

Cumulative sums

2

0.383827

98/100

Pass

4

Runs

1

0.779188

99/100

Pass

5

Longest run

1

0.798139

99/100

Pass

6

Rank

1

0.383827

97/100

Pass

7

FFT

8

Non overlapping template

9

1

0.017912

100/100

Pass

148

0.012650

96/100

Pass

Overlapping template

1

0.012650

100/100

Pass

10

Universal

1

0.025193

100/100

Pass

11

Approximate entropy

1

0.224821

99/100

Pass

12

Random excursions

8

0.063482

67/69

Pass

13

Random excursions variant

18

0.009422

66/69

Pass

14

Serial

2

0.137282

100/100

Pass

15

Linear complexity

1

0.040108

100/100

Pass

196

9 Applications of fractional-order Chaotic Systems in Secure Communications

Fig. 9.17 Correlation of BS1 and BS2

0.15 0.1 0.05 0 -0.05 -0.1 -1000

-600

-200 0 200 Related interval

600

1000

of passing the test are the minimum values among multiple tests. Tables 9.1 and 9.2 illustrate that both sets of pseudo-random binary sequences obtained from the fractional-order simplified Lorenz system pass the NIST STS test, and it has good random performance. 2.

Inter-correlation test for two sets of sequences

Both sets of pseudo-random sequences generated above passed the NIST test, and it indicates that each sequence exhibits good balance, autocorrelation, spectral properties, etc., and has good randomness. The following test is the intercorrelation performance of the two. Binary sequences of equal length are randomly selected from BS1 and BS2, respectively, and “1” and “0” in the sequences are mapped to “1” and “−1”, and calculate the periodic correlation of the two sequences. The results are shown in Fig. 9.17, and the correlation coefficients are all between ±0.1, indicating that BS1 and BS2 are mutually uncorrelated. In addition, the Hamming distance is a useful tool specifically for testing the mutual correlation between two binary sequences [7]. The Hamming distance represents the number of positions in two sequences that correspond to the same position but have different values and assuming that both binary sequences under test are of length M. The Hamming distance of two binary sequences S 1 and S 2 is defined as d(S1 , S2 ) =

M 

(x j ⊕ y j ),

(9.70)

j=1

where x j and yj are the elements in S 1 and S 2, respectively. Theoretically, for two mutually uncorrelated pseudo-random binary sequences, the Hamming distance is about M/2, which is usually expressed in the form of a percentage, i.e., d(S1, S2)/M is about 50%. The Hamming distance of the two pseudo-random binary sequences BS1 and BS2 obtained above is calculated according to Eq. (9.70) to determine their mutual correlation, and the results are shown in Table 9.3.

9.2 Design of the Fractional-Order Chaotic Pseudo-Random Sequence Generator Table 9.3 Hamming distances of BS1 and BS2 for different lengths

Length

d(BS1, BS2)/M (%)

M=

104

50.07000

M = 105

49.97500

M=

106

49.95480

M = 107

49.99899

197

The Hamming distance for both BS1 and BS2 is about 50% for different values of M chosen, further indicating that the two pseudo-random binary sequences generated above are uncorrelated with each other. 3.

Key space analysis for generating pseudo-random sequences

For a pseudo-random sequence generator based on a chaotic system, the sensitivity of the system to the initial values and system parameters determines the number of different pseudo-random sequences that can be generated. The range of initial values and system parameters corresponding to the generation of different pseudorandom sequences is usually called the “key space” of the pseudo-random sequence generator. According to the previous analysis, in a fractional-order chaotic system, the differential order q of the system is also a bifurcation parameter, and q is one of the initial conditions that affects the characteristics of the system in addition to the initial value and the system parameters. That is, the value of q for a pseudo-random sequence generator based on a fractional-order chaotic system also determines its key space. For the above-designed pseudo-random binary sequence generator based on the fractional-order simplified Lorenz system, the focus here is on the effect of q. A binary sequence KS1 of length 107 bit is obtained from BS1. Then q = 0.9 + 10–7 is changed, and a binary sequence KS2 of the same length is obtained from the same position in the new BS1. According to Eq. (9.70), the Hamming distances of KS1 and KS2 are calculated for different M, and the results are shown in Table 9.4. It shows that the key space of the pseudo-random sequence generator based on the fractionalorder simplified Lorenz system is at least 107 times larger after considering the order q. 4.

Speed analysis for generating pseudo-random sequences

Evaluating the performance of a pseudo-random sequence generator often requires considering the speed of generating the sequence. In the above experiments, the TMS320F28335 is clocked at 30 MHz and the compiled environment is CCS4.1. Table 9.4 Hamming distances for different lengths of KS1 and KS2

Length

d(KS1, KS2)/(%)

M=

104

49.47000

M = 105

49.89200

M = 106

49.97560

M=

50.03064

107

198

9 Applications of fractional-order Chaotic Systems in Secure Communications

From the software design, it can be found that most of the time spent on generating pseudo-random sequences is spent on iterative operations for fractional-order chaotic system. According to the iterative equation of the fractional-order simplified Lorenz system solved by the Adomian solution algorithm, one iteration consists of 30 additions and 35 times multiplication, so the measured rate of generating a pseudo-random binary sequence is 1.84 × 106 (bits/sec). For comparison, the integer-order simplified Lorenz system was solved using the commonly used fourth-order Runge–Kutta method, and its iterative equation was obtained by ⎧ ⎪ ⎨ xn+1 =xn + h(K 11 + 2K 12 + 2K 13 + K 14 )/6 yn+1 =yn + h(K 21 + 2K 22 + 2K 23 + K 24 )/6 , ⎪ ⎩ z n+1 =z n + h(K 31 + 2K 32 + 2K 33 + K 34 )/6

(9.71)

⎧ ⎪ ⎨ K 11 =10[(yn − xn )] K 21 =[(24 − 4c)xn − xn z n + cyn ] , ⎪ ⎩ K 31 =[xn yn − 8z n /3]

(9.72)

where

⎧ K 12 =10[(yn + 0.5h K 21 ) − (xn + 0.5h K 11 )] ⎪ ⎪ ⎪ ⎨ K 22 =[(24 − 4c)(xn + 0.5h K 11 ) − (xn + 0.5h K 11 )(z n + 0.5h K 31 ) , ⎪ + c(yn + 0.5h K 21 )] ⎪ ⎪ ⎩ K 32 =[(xn + 0.5h K 11 )(yn + 0.5h K 21 ) − 8(z n + 0.5h K 31 )/3] ⎧ K 13 =10[(yn + 0.5h K 22 ) − (xn + 0.5h K 12 )] ⎪ ⎪ ⎪ ⎨ K =[(24 − 4c)(x + 0.5h K ) − (x + 0.5h K )(z + 0.5h K ) 23 n 12 n 12 n 32 , ⎪ + c(y + 0.5h K )] n 22 ⎪ ⎪ ⎩ K 33 =[(xn + 0.5h K 12 )(yn + 0.5h K 22 ) − 8(z n + 0.5h K 32 )/3] ⎧ K 14 =10[(yn + h K 23 ) − (xn + h K 13 )] ⎪ ⎪ ⎪ ⎨ K =[(24 − 4c)(x + h K ) − (x + h K )(z + h K ) 24 n 13 n 13 n 33 ⎪ + c(y + h K )] n 23 ⎪ ⎪ ⎩ K 34 =[(xn + h K 13 )(yn + h K 23 ) − 8(z n + h K 33 )/3]

(9.73)

(9.74)

(9.75)

Equations (9.71–9.75) include 59 additions and 57 multiplications, and the rate of generating a pseudo-random binary sequence is 1.28 × 106 (bits/sec) under the same conditions. It can be seen that the former generates pseudo-random binary sequences about 1.5 times faster based on the fractional-order simplified Lorenz system compared to the integer-order simplified Lorenz-based system.

9.2 Design of the Fractional-Order Chaotic Pseudo-Random Sequence Generator

199

9.2.3 Design of the Fractional-Order Lorenz Hyperchaotic Pseudo-Random Sequence Generator Compared with general chaotic systems, hyperchaotic systems have two or more positive Lyapunov exponent spectrums and have higher complexity, which is beneficial for practical applications. Based on the fractional-order Lorenz hyperchaotic system, the pseudo-random sequence generator is designed on the DSP development board to lay the foundation for the practical application. Firstly, the fractional-order Lorenz hyperchaotic system is implemented on the DSP development board, and the results of its implementation are shown in Fig. 9.18. The design of the DSP chaotic pseudo-random sequence generator based on the fractional-order Lorenz hyperchaotic system is shown in the following steps. Step 1: Let n = 1, R = 0.26, q = 0.7, the initial values of the system x 0 are (1, 2, 3, 4), and M = 0.125 × 108 + 100 and iterate the fractional-order Lorenz hyperchaotic system solution 1000 times. Let n = 1001 and x 0 (n) = (x(n), y(n), z(n), u(n)). Step 2: Iterate the fractional-order Lorenz hyperchaotic system solution one time to obtain the numerical solution of the system such that data = x(n + 1) and perform the following operations:   data = r ound |data| × 1011 .

(9.76)

That is, data can be represented by a 64-bit binary number DB63 -DB0 . Step 3: Make data1 = DB7 -DB0 . Then enter the number data1 into the PC via the serial port MAX3232 (RS-232 transceiver) and save it in a txt. document for further testing. Step 4: Let n = n + 1 and x0 (n) = [x(n), y(n), z(n), u(n)]. Step 5: Repeat steps 2 to 4 until n > M.

Fig. 9.18 Attractors of the fractional-order Lorenz hyperchaotic system based on DSP a q = 0.65, R = 0.26 and b q = 0.72, R = 0.26

200

9 Applications of fractional-order Chaotic Systems in Secure Communications

Table 9.5 NIST test results

Test index (number of tests)

P-value

Proportion (%)

Frequency (1)

0.366918

100

B.Frequency (1)

0.002559

99

C.Sums (2)

0.275709

99

Runs (1)

0.048716

98

Longest Run (1)

0.935716

99

Rank (1)

0.883171

99

FFT (1)

0.574903

100

N.O.Temp (148)

0.002203

97

O.Temp (1)

0.834308

100

Universal (1)

0.759756

99

App.Entropy (1)

0.758354

98

R.Excur. (8)

0.005166

96.7

R. Excur.V. (18)

0.105618

96.7

Serial (2)

0.075719

99

L.Complexity (1)

0.554420

99

Results √ √ √ √ √ √ √ √ √ √ √ √ √ √ √

At the end of the algorithm, the txt. file of the pseudo-random sequence with 0 and 1 is obtained on the computer. Here, the length of each test sequence is 106 bits and 100 sets of data are used for testing. If α = 0.01, the confidence space for the pass rate is [0.96015, 1] for most tests. The test results are shown in Table 9.5, where only the lowest results are shown for the items that are tested multiple times. From Table 9.5, it is clear that all p-value values are greater than 0.0001 and the pass rates lie within the confidence interval. So the fractional-order chaotic pseudo-random sequence designed in this chapter has good pseudo-random. At present, based on DSP/FPGA technology, chaotic systems are widely used in engineering fields, especially in image encryption [8] and secure communication [9, 10]. In these applications, chaotic pseudo-random sequences usually play an important role. In addition, the research results show that fractionalorder chaotic systems can also generate pseudo-random sequences with practical applications, laying the foundation for the practical applications of fractional-order chaotic systems.

9.3 Study of Fractional-Order Chaotic Image Encryption Algorithm

201

9.3 Study of Fractional-Order Chaotic Image Encryption Algorithm 9.3.1 Image Encryption Algorithm Based on Fractional-Order Hyperchaotic System The previous study of the dynamics of fractional-order chaotic systems and the results of the complexity analysis of fractional-order chaotic systems provide a basis for the selection of system parameters and show the value of fractional-order chaotic systems applications. Image encryption is an important application aspect of fractional-order chaotic systems. Taking the system parameters as R = 0.26 and order q = 0.7, the Lyapunov exponent spectrum of the fractional-order Lorenz hyperchaotic system is Lyi (i = 1, 2, 3, 4) = (0.8768, 0.4697, 0, −52.6598), so the system is hyperchaotic, and also its complexity value is relatively large, which is beneficial for practical applications. In the previous section, a chaotic pseudo-random sequence generator based on a fractional-order Lorenz hyperchaotic system was designed and passed the NIST test. Next, a fast image encryption algorithm is designed based on the pseudorandom sequence generated by this chaotic pseudo-random sequence generator, and the algorithm security is analyzed. Consider a grayscale image of size N × N and denote the pixel matrix by P, i.e., the pixel value at position (i, j) is Pij . The encryption algorithm is designed as follows. Step 1: Let n = 1, R = 0.26, q = 0.7, and the initial values of the system x 0 be [1–4] and iterate the fractional-order Lorenz hyperchaotic system solution 1000 times. Let n = 1001 and x 0 (n) = [x(n), y(n), z(n), u(n)]. Step 2: Iterate the solution of the fractional-order Lorenz hyperchaotic system N times to obtain the numerical solution of the system and perform the following operations: ⎧   xk = r ound |xk | × 101 ⎪ ⎪ ⎪ ⎪ ⎨ yk = r ound |yk | × 101   ,  1 ⎪ | |z × 10 z = r ound ⎪ k k ⎪ ⎪   ⎩ wk = r ound |wk | × 101

(9.77)

where k = 1,2, …, N. That is, each sequence can be represented by a 64-bit binary number DB63 -DB0 . Let {X k }, {Y k }, {Z k }, and {U k } be the first 8 bits (DB7 -DB0 ) of x k , yk , zk , and uk respectively, i.e., four integer chaotic sequences of length N and with sequence values ranging from 0 to 255 are obtained. Step 3: Perform the following operation on the obtained fractional-order chaotic time series

202

9 Applications of fractional-order Chaotic Systems in Secure Communications

⎧ X n = (X n + ϕ) mod 256 ⎪ ⎪ ⎪ ⎨ Y = (Y + ϕ) mod 256 n n , ⎪ Z = (Z n n + ϕ) mod 256 ⎪ ⎪ ⎩ Un = (Un + ϕ) mod 256

(9.78)

where ϕ=

N N  

Pi, j

(9.79)

i=1 j=1

be the sum of all pixel values of the plaintext image. Let i = 1 and j = 1. Step 4: Let ⎥ ⎛⎢ ⎞ ⎢ ⎥ ⎢ N ⎥ η = ⎝⎣ Pi, j ⎦ + X i ⎠ mod N .

(9.80)

j=1

This step starts with a permutation of the pixel position of the ith row of the image. Consider the ith row of the image to be a ring and move that ring η steps to the right, and the operation is ⎧ ⎪ ⎨ β = Pi,1:N −η Pi,1:η = Pi,N −η+1:N ⎪ ⎩ Pi,η+1:N = β

(9.81)

Next, the pixel values in row i of the image are scrambled. If i = 1, then Eq. (9.82) is executed. If i = 1, then Eq. (9.83) is executed. P1, j = P1, j ⊕ Pi, j = Pi, j ⊕

  Z 1 + PN , j mod 256 ,

(9.82)

  Z i + Pi−1, j mod 256 ,

(9.83)





where j = 1, 2, …, N. Step 5: Let  η=

N 

! Pi, j

" + Yj

mod 256.

(9.84)

i=1

This step starts by dislocating the pixel position of the jth column of the image. Consider the jth column of the image as a ring and move that ring η steps upwards. So the operation is

9.3 Study of Fractional-Order Chaotic Image Encryption Algorithm

⎧ ⎪ ⎨ β = P1:N −η, j P1:η, j = PN −η+1:N , j ⎪ ⎩ Pη+1:N , j = β

203

(9.85)

The following permutation of the pixel values in column j of the image is performed. If j = 1, then Eq. (9.86) is executed. If j = 1, then Eq. (9.87) is executed    U1 + Pi,N mod 256 ,

(9.86)

   U j + Pi, j−1 mod 256 ,

(9.87)

Pi,1 = Pi,1 ⊕ Pi, j = Pi, j ⊕

where i = 1, 2, …, N. Step 6: Let i = i + 1 and j = j + 1. Step 7: Repeat steps 4–6 until i > N. The decryption algorithm is similar to the encryption algorithm, and it is the inverse process of the encryption algorithm. The exact execution process is described below. Step 1: Let n = 1, R = 0.26, q = 0.7, and the initial values of the system x 0 coincide with those of the encryption process as [1–4] and iterate the fractional-order Lorenz hyperchaotic system solution 1000 times and let n = 1001 with x 0 (n) = [x(n), y(n), z(n), u(n)]. Step 2: Iterate the solution of the fractional-order Lorenz hyperchaotic system N times to obtain the numerical solution of the system and perform the following operations:   ⎧ xk = r ound |xk | × 101 ⎪ ⎪ ⎪ ⎪ ⎨ yk = r ound |yk | × 101   ,  ⎪ z k = r ound |z k | × 101 ⎪ ⎪ ⎪   ⎩ wk = r ound |wk | × 101

(9.88)

where k = 1,2, …, N. That is, each sequence can be represented by a 64-bit binary number DB63 -DB0 . Let {X k }, {Y k }, {Z k }, and {U k } be the first 8 bits (DB7 -DB0 ) of x k , yk , zk , and uk respectively, i.e., we obtain four integer chaotic sequences of length N and with sequence values ranging from 0 to 255. Step 3: Perform the following operation on the obtained fractional-order chaotic time series: ⎧ X n = (X n + ϕ) mod 256 ⎪ ⎪ ⎪ ⎨ Yn = (Yn + ϕ) mod 256 , (9.89) ⎪ Z n = (Z n + ϕ) mod 256 ⎪ ⎪ ⎩ Un = (Un + ϕ) mod 256

204

9 Applications of fractional-order Chaotic Systems in Secure Communications

where ϕ is the sum of all pixel values of the plaintext image and is calculated as shown in Eq. (9.79). Let i = N and j = N. Step 4: Let  η=

N 

!

" + Yj

Pi, j

mod 256.

(9.90)

i=1

The pixel values in column j are first recovered by performing Eq. (9.91) if j = 1 and Eq. (9.92) if j = 1    U1 + Pi,N mod 256 ,

(9.91)

   U j + Pi, j−1 mod 256 ,

(9.92)

Pi,1 = Pi,1 ⊕ Pi, j = Pi, j ⊕

where i = 1, 2, …, N. Next, the pixel position of the jth column of the image is recovered. Consider the jth column of the image as a ring and move the ring η steps down, which is shown below ⎧ ⎪ ⎨ β = P1:N −η, j P1:η, j = PN −η+1:N , j (9.93) ⎪ ⎩ Pη+1:N , j = β Step 5: Let  η=

N 

! Pi, j

" + Yj

mod 256.

(9.94)

i=1

Firstly, we recover the pixel value of row i of the image. If i = 1, then Eq. (9.95) is executed. If i = 1, then Eq. (9.96) is executed. P1, j = P1, j ⊕ Pi, j = Pi, j ⊕

  Z 1 + PN , j mod 256 ,

(9.95)

  Z i + Pi−1, j mod 256 ,

(9.96)





where j = 1, 2, …, N. Next, the pixel position of the ith row of the image is recovered by considering the ith row of the image as a ring and moving the ring η steps to the left, which is done as ⎧ ⎪ ⎨ β = Pi,1:N −η Pi,1:η = Pi,N −η+1:N . (9.97) ⎪ ⎩ Pi,η+1:N = β

9.3 Study of Fractional-Order Chaotic Image Encryption Algorithm

205

Fig. 9.19 Encryption Simulation Results a Original image, b Encrypted image, and c Correctly decrypted image

Step 6: Let i = i-1 and j = j - 1. Step 7: Repeat steps 4 to 6 until i < 1. Here, a 256 × 256 Lena image as in Fig. 9.19a is used for encryption and decryption operations, and the security of the encryption algorithm is analyzed. Using the above algorithm, the Lena image encrypted image is shown in Fig. 9.19b, which shows that the encrypted image has completely hidden the original image information. The corresponding decrypted image with the correct key is shown in Fig. 9.19c, and it is seen that the correct plaintext image is obtained. Next, the security of the algorithm will be analyzed.

9.3.2 Security Analysis of Image Encryption Algorithm 1.

Statistical analysis

Statistical analysis is an effective method to analyze the security of image encryption algorithms. The result of the statistical analysis shows the ability of the image to resist statistical attacks. A cipher image obtained by a good image encryption algorithm should be resistant to any statistical attack. Currently, statistical analysis mainly uses grayscale histogram distribution and correlation methods between adjacent pixels. A comparison of the grayscale histogram distribution of the image before and after encryption is shown in Fig. 9.20. It can be seen that before encryption, the image shows a non-uniform distribution, and after encryption, the ciphertext shows a uniform distribution, which can resist statistical attacks. The adjacent pixels in digital images are not independent of each other and have a high correlation. One of the goals of image encryption is to reduce the correlation between adjacent pixels of the image, mainly including the correlation between horizontal pixels, vertical pixels, and diagonal pixels. The smaller the correlation, the better the image encryption effect and the higher the security. Figure 9.21 shows the correlation between the adjacent pixels of the original image and the improved hybrid encrypted image in the horizontal and vertical directions. It can be seen that the correlation between pixels

206

9 Applications of fractional-order Chaotic Systems in Secure Communications

Fig. 9.20 Histogram of the original and encrypted images a Histogram of the original image and b Histogram of the encrypted image

Fig. 9.21 Adjacent pixel correlation in vertical, diagonal, and horizontal a Original image vertical orientation, b Original image diagonal orientation, c Original image horizontal orientation, d Encrypted image vertical orientation, e Encrypted image diagonal orientation, and f Encrypted image horizontal orientation

of the original image shows a clear linear relationship, while the correlation between pixels of the improved hybrid encrypted image shows a random correspondence. The correlation coefficient ρ x,y between adjacent pixels x and y is defined as cov(x, y) ρx,y = √ , D(x)D(y) where

(9.98)

9.3 Study of Fractional-Order Chaotic Image Encryption Algorithm

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

E(x) =

N N 1  1  xi , D(x) = [xi − E(x)]2 N i=1 N i=1

207

(9.99)

cov(x, y) = E[(x − E(x))(y − E(y))]

Here, N = 104 pairs of pixels selected randomly are used to calculate the correlation coefficients. The correlation coefficient values in diagonal, horizontal, and vertical directions are 0.972, 0.9318, and 0.9456, respectively. For Lena image, the large correlation coefficient values indicate that there is a large correlation between pixels, while the correlation coefficient values in diagonal, horizontal, and vertical directions, respectively, for encrypted images are −0.0203, 0.0017, and 0.0302, which are small coefficient values, indicating that there is almost no correlation between adjacent pixels of the encrypted image. So the effectiveness of the algorithm is verified. 2.

Resistant to differential attacks

According to cryptographic principles, a good image encryption algorithm should be sensitive to plaintext changes, and the stronger its sensitivity, the better the algorithm’s resistance to differential attacks. The algorithm resistance to differential attacks is generally characterized by the number of pixels change rate (NPCR) and the unified average changing intensity (UACI). Let C 1 be the encryption result of the original image and C 2 be the encryption result of increasing or decreasing one pixel value at a position in the original image. The NPCR is calculated as N N NPCR =

i=1

j=1

D(i, j)

N×N

,

(9.100)

where # D(i, j) =

1, i f C1 (i, j) = C2 (i, j) 0, i f C1 (i, j) = C2 (i, j)

(9.101)

UACI is defined by the equation. UACI =

 |C1 (i, j) − C2 (i, j)| × 100% 255 × M × N i, j

(9.102)

Here, the pixel value is randomly increased by 1 for a certain location of the Lena image, i.e., each time the two original image pixel values differ by only 1. The experiment is performed 1000 times and the corresponding NPCR as well as UACI values are calculated for each time. The results are shown in Table 9.6. The ideal value of NPCR is 99.6094%, while the ideal value of UACI is generally 33%. From Table 9.6, the obtained NPCR and UACI calculation results are close to the ideal

208

9 Applications of fractional-order Chaotic Systems in Secure Communications

Table 9.6 Results of NPCR and UACI calculations

Minimum

Maximum

Average

NPCR

0.9952

0.9968

0.9960

UACI

0.3188

0.3275

0.3241

values, indicating that the proposed image encryption algorithm has good resistance to differential attacks. 3.

Key sensitivity analysis

Sensitivity to the key is one of the characteristics of a good image encryption algorithm. A small difference in the key will cause the image to not be decrypted correctly. In Fig. 9.22, the decrypted image is shown when the key values (including the system initial values x 0 , y0 , z0 , and u0 , the system order q, and the system parameter R) are changed by 10–10 , respectively. It can be seen that the encrypted image can be decrypted correctly by the improved hybrid encryption algorithm, and if there is a very small change in the initial value, it will lead to decryption failure, which shows that the algorithm is highly sensitive to the key and has high security. The order of the number of calculations required to use the exhaustive attack method is at least O(1060 ) when counted with 10–10 accuracy, which shows that this algorithm is resistant to exhaustive attacks.

Fig. 9.22 Decryption diagram with error key a x 0 = x 0 + 10−10 , b y0 = y0 + 10−10 , c z0 = z0 + 10−10 , d u0 = u0 + 10−10 , e q0 = q0 + 10−10 , and f R0 = R0 + 10–10

9.4 Fractional-Order Chaotic Spread Spectrum Communication Systems

209

9.4 Fractional-Order Chaotic Spread Spectrum Communication Systems 9.4.1 Spread Spectrum Codes in Spread Spectrum Communication Systems The principle of a direct sequence spread spectrum communication system is shown in Fig. 9.23. In a spread spectrum communication system, the performance of the spread spectrum code is closely related to the system’s anti-interference, anti-noise, anti-interception, and information data concealment capabilities, as well as to the system’s multipath protection, anti-fading, multi-access capability, and difficulty in achieving synchronization and capture. Therefore, a binary pseudo-random sequence as a spread spectrum code should have the following basic properties. (1)

(2)

(3)

It has sharp autocorrelation, and the cross-correlation function should be close to 0. In spread spectrum communication, if there is mutual interference between two channels, their mutual correlation value must not be 0. To synchronize the signals quickly during communication and have good multiple access capability, the spread spectrum codes need to have sharp autocorrelation characteristics, which can reduce the false capture rate at the receiver side, and the mutual correlation between the spread spectrum codes function close to zero. It can reduce the interference between different users and improve the multiple access capability of the system. There is a long enough code period and complexity to ensure the requirement of interception and interference resistance. One of the main purposes of using pseudo-random sequences as spreading codes for spread spectrum communications is to improve the confidentiality and anti-interference performance of the system. The longer and more complex the spreading code, the better the interference and confidentiality performance of the spread spectrum signal. The number of sequences required to meet the requirement is sufficiently large to increase the multiple access capability of the system. The multiple access capability of a spread spectrum communication system is a very important indicator. That is, it directly reflects the number of users that can be accommodated in the same channel. The more the number of mutually uncorrelated spread spectrum codes, the stronger the multiple access capability of the system. On the contrary, the multi-access capability of the system is limited, and the number

Fig. 9.23 Block diagram of a direct sequence spread spectrum communication system

210

(4)

9 Applications of fractional-order Chaotic Systems in Secure Communications

of users accommodated will be less, thus the advantages of spread spectrum communication technology cannot be exploited. Sequences are easy to generate and process. Spreading codes that are easy to generate and process in a spread spectrum communication system will allow more system resources to be used to handle other important aspects of the communication system, resulting in more stable system operation and better and more efficient communication performance.

In practice, the most used spreading code sequences to meet the spreading requirements are m-sequence, Gold sequence, and chaotic pseudo-random sequence. However, the number of spreading codes for m-sequences and Gold sequences is limited. The limitations of the traditional use of integer order and discrete chaos to generate spreading sequences are gradually emerging [11–13]. The following is based on the fractional-order simplified Lorenz chaotic system to design the spreading codes in the spread spectrum communication system.

9.4.2 Design of a Single-User Fractional-Order Chaotic Spread Spectrum Communication System Based on Correlated Reception The design of a point-to-point spread spectrum communication system based on correlation reception is based on the principle as shown in Fig. 9.23. The main purpose of designing this spread spectrum communication system is to study the performance of the pseudo-random sequence obtained by the above method as a spread spectrum code. So, for the transmitter side, in order to reflect the application of the above DSPgenerated pseudo-random sequence in the spread spectrum communication system, a certain length of pseudo-random sequence is randomly intercepted in BS2 and used directly as a spread spectrum code. The same pseudo-random sequence is used as a spread spectrum demodulation at the receiver side and it is assumed that the transmitter and receiver have achieved synchronization. To test the performance of the communication system at different signal-to-noise ratios (SNR), the noise of the channel is adjustable. The Simulink simulation diagram of the designed single-user spread spectrum communication system based on correlated reception is shown in Fig. 9.24. The Random integer module in the figure generates the data to be sent, and the T Codes Generator and the Buffer unit are responsible for generating the spreading codes from the intercepted pseudo-random sequences in the format of frames. The function of polarity conversion modules (Unipolar to Bipolar Converter, “0” to “−1”; Bipolar to Unipolar Converter, “−1” to “0”), is unipolar binary code elements through the heterodyne operation can achieve spread spectrum and despreading, but the process of “0” is special. When the noise signal is added, the signal is not binary, so the heterodyne operation is invalid. It can be avoided if bipolar binary code elements are used, and the same way of phase multiplication can be used to achieve spread

9.4 Fractional-Order Chaotic Spread Spectrum Communication Systems

211

Fig. 9.24 Simulink simulation block diagram of a point-to-point spread spectrum communication system based on correlated reception

spectrum and despreading. The carrier modulation and demodulation method use BPSK (binary phase shift keying). The internal structure of the receiver INTEGER associated with the receiver side is shown in Fig. 9.25, which works as follows. In1 is the input, and each of its frames of input (N × 1) needs to be transformed into the form of (1 × N) by Unbuffer. It is integrated into the Integrate and Dump module and converted into frame format by the Sampling module after the Zero-Order Hold module. Since bipolar binary code element is used, the judgment module is done by the Sign module. If the result of integration is greater than 0, the result is 1, otherwise, it is −1. Finally, the received data and the original data sent (Transmit Data) are analyzed for BER in the BER Calculation unit (Error Rate Calculation).

Fig. 9.25 Related receiver simulation

212

9 Applications of fractional-order Chaotic Systems in Secure Communications

9.4.3 Performance Analysis of the Spread Spectrum Communication Systems The performance of BS2 as a spreading code is analyzed based on the simulation in Fig. 9.24. Pseudo-random sequences of 8 bits, 16 bits, and 32 bits are randomly intercepted from BS2 as spreading codes, and 106 -bit code elements are transmitted in the test to obtain the variation of bit error rate (BER) with a signal-to-noise ratio (SNR) as shown in Fig. 9.26. When 32-bit spreading code is chosen, the BER is close to 0 with SNR = −2 dB. While when 8-bit spreading code is chosen, the BER is close to 0 only with SNR = 4 dB. When the SNR of the spreading communication system is constant, the more the bits of the spreading code, the smaller the BER. It can be seen that the number of bits of the spreading code has a great influence on the performance of the spread spectrum communication system. The longer the number of bits, the better the performance of the system. The pseudo-random sequence generated based on the Hénon map is chosen as the spreading code under the same conditions. The equation of the Hénon map is [14]

xn+1 = −axn2 + yn + 1 . yn+1 = bxn

(9.103)

The system is chaotic when a = 1.4 and b = 0.3, and the chaotic sequence is obtained by randomly selecting the initial values [x 0 , y0 ] = [0.7, 0.5]. The chaotic sequence x n is transformed by |xn | = 0.b1 (xn )b2 (xn ) · · · bi (xn ) · · · , where Fig. 9.26 Performance of a single-user spread spectrum communication system based on a fractional-order simplified Lorenz system

(9.104)

9.4 Fractional-Order Chaotic Spread Spectrum Communication Systems

$ % ⎧ b (x ) = sng0.5 (2i−1 |xn | − 2i−1 |xn | ) ⎪ ⎨ i n , 0 (x < 0.5) ⎪ ⎩ sng0.5 (x) = 1 (x ≥ 0.5)

213

(9.105)

x denotes rounding down, and the new binary number is obtained as ⎧ |x1 | = 0.b1 (x1 )b2 (x1 ) · · · bi (x1 ) · · · ⎪ ⎪ ⎪ ⎨ |x | = 0.b (x )b (x ) · · · b (x ) · · · 2 1 2 2 2 i 2 . ⎪ ······ ⎪ ⎪ ⎩ |xn | = 0.b1 (xn )b2 (xn ) · · · bi (xn ) · · ·

(9.106)

The ith bit in |x n | selected from the above equation can form a binary pseudorandom sequence. i = 1, 2, 3, and the binary pseudo-random sequence BH based on the Hénon map is obtained, which is analyzed to have good statistical properties [14]. The binary sequences of 8 bits, 16 bits, and 32 bits are randomly intercepted from BH as spreading codes for simulation, and the obtained results are shown in Fig. 9.27. The pseudo-random sequence generated based on Chen’s system is then chosen as the spreading code. The equation of Chen’s system is [15] ⎧ ⎪ ⎨ x˙ = a(y − x) y˙ = (c − a)x − x z + cy . ⎪ ⎩ z˙ = x y − bz

(9.107)

When a = 35, b = 3, c = 28, the system is chaotic and let the initial values be x 0 = − 3, y0 = 2, z0 = 20. The resulting chaotic sequence x(i), y(i), z(i), i = 0, 1, 2, …, ∞, is transformed by Fig. 9.27 Performance of single-user spread spectrum communication system based on the Hénon map

214

9 Applications of fractional-order Chaotic Systems in Secure Communications

Fig. 9.28 Performance of a single-user spread spectrum communication system based on the Chen system

⎧ ⎪ ⎨ v(3i) = 3000(x(i) + 45) v(3i + 1) = 3000(y(i) + 35) , ⎪ ⎩ v(3i + 2) = 3000(z(i) + 45)

(9.108)

K j = v( j) mod256,

(9.109)

where j = 0, 1, 2, …, ∞, then K j is the obtained pseudo-random binary sequence, which has been tested and also passed the NIST test with good statistical properties and pseudo-randomness. Similarly, a random binary sequence is intercepted from K j as a spreading code. The spreading code lengths are taken as 8 bits, 16 bits, and 32 bits, respectively, and the results obtained from the simulation are shown in Fig. 9.28. All three cases above show that the longer the number of bits of the spreading code, the better the communication effect. In the following, all three cases are compared when 32-bit spreading codes are selected. The variation of BER with SNR is shown in Fig. 9.29. At SNR < −5 dB, the BERs are similar and larger for all three cases, indicating that all three cases perform poorly in very harsh communication environments. When SNR ≥ −4 dB, the BER of the fractional-order simplified Lorenz-based system is slightly lower than that of the other two cases. When SNR = −2 dB, the BER of the fractional-order simplified Lorenz-based system is close to 0, but the other two cases still have high BER. When SNR ≥ −1 dB, the BER of all three cases is close to 0. It shows that the BER of the fractional-order simplified Lorenzbased system of spread spectrum communication system performs better than the other two cases. Gold sequences and m-sequences are widely used as spreading codes and their performance is recognized. Equivalently, both sending 106-bit code elements, a comparison of the performance of the pseudo-random sequence based on the fractional-order simplified Lorenz system with the Glod and m-sequences for a spread spectrum communication system is shown in Fig. 9.30. When SNR = − 1 dB, the BER of the communication system based on the fractional-order simplified

9.4 Fractional-Order Chaotic Spread Spectrum Communication Systems

215

Fig. 9.29 Performance comparison of different spread spectrum codes, (Hénon map, Chen system)

Fig. 9.30 Performance comparison of different spread spectrum codes (Gold sequence, m-sequence)

Lorenz system is close to 0, which is significantly smaller than the BER of the Gold and m-sequences. It shows that the pseudo-random sequence designed based on the fractional-order chaotic system performs better than the Gold and m-sequences.

9.4.4 Design of a Multi-user Chaotic Spread Spectrum Communication System Based on Correlated Reception One advantage of spread spectrum communication is that multiple users can communicate on the same channel without interfering with each other. The number of users of the following communication system is 4. The Simulink simulation diagram of the system is shown in Fig. 9.31, where the transmitter and receiver of each user are the same as in Fig. 9.24. Four random 32-bit pseudo-random sequences are intercepted

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9 Applications of fractional-order Chaotic Systems in Secure Communications

Fig. 9.31 Four-user spread spectrum communication system based on correlated reception

from BS2 as the spreading codes for each of the four users, and the transmitted signals of the four users are mixed together and sent out, with each user sending 104 code elements. By detecting the variation of BER with SNR for each user separately, the results are obtained as shown in Fig. 9.32. It can be seen that the communication performance of each of the four users varies with the SNR in a similar pattern, maintaining a very low BER even in an environment with a very low SNR, indicating that the four users interfere with each other very little due to the excellent performance of the spread spectrum codes, although they are in the same channel and use the same carrier frequency. Fig. 9.32 Performance of a four-user correlated receive spread spectrum communication system

9.4 Fractional-Order Chaotic Spread Spectrum Communication Systems

217

Fig. 9.33 Performance of the multi-user spread spectrum communication system

Under the same conditions, the number of users in the multi-user system was varied to 2, 4, 6, 8, and 10. To compare the effect of the number of users on the performance of the communication system, the BER of all users was averaged for each case. The test results are shown in Fig. 9.33. From the figure, the communication performance in several cases is very close, and the BER is approximately 0 for SNR ≥ −1 dB. It indicates that the used spreading codes have good orthogonality and the system has good multiple access capability.

9.4.5 Design of a Multi-User Chaotic Spread Spectrum Communication System Based on the Rake Reception In direct sequence code division multiple access communication systems, multipath interference is generated due to multipath. In the case of using spread spectrum codes, multipath interference is also generated. Therefore, when multipath exists, the performance of conventional direct correlation receivers often fails to meet the requirements. Rake reception technology can overcome these shortcomings, and it is a multipath separation-based reception technology, which can use multipath to improve the output signal-to-noise ratio. This reception technology can play a good role in suppressing multipath interference caused by time-delay expansion and is also very effective for anti-fading. Rake receivers essentially use correlated receivers but use them in combination and by correlating the output of each correlated receiver with a delayed form of the received signal. The output of each correlated receiver needs to be weighted according to their strength, and the weighted outputs are all combined into one output. The Simulink simulation of the multi-user spread spectrum communication system based on the Rake receiver is shown in Fig. 9.34. In the figure, the transmitter side is the same as Fig. 9.24, and the spread spectrum signal is sent through

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9 Applications of fractional-order Chaotic Systems in Secure Communications

Fig. 9.34 Simulation model of a spread spectrum communication system based on the Rake receiver

the multipath transmission channel (Propagate through) and received at the receiver side using the Rake receiver. The number of users of the spread spectrum communication system based on the Rake receiver is set to four. Four 32-bit pseudo-random sequences are randomly selected as spread spectrum codes in BS2, respectively, to monitor the BER with signal-to-noise ratio, as shown in Fig. 9.35. 10-1

Fig. 9.35 Performance of the Rake receiver-based spread spectrum communication system

User1 User2 User3 User4 4 users

BER

10-2

10-3

10-4

10-5 -11 -10

-9

-8

-7

-6

-5

SNR(dB)

-4

-3

-2

-1

0

References

219

When SNR > −6 dB, the BER of all four users gradually converges to 0, and it shows very good performance. The average BER of the four-user communication system based on coherent reception is also shown in Fig. 9.35, and it is clear that the BER is higher than that of the Rake receiver at any SNR. It can be seen that the pseudo-random sequence obtained based on the fractional-order chaotic system is also applicable to the spread spectrum communication system with the Rake receiver. The performance is better than that of the communication system with coherent reception. Based on a fractional-order simplified Lorenz system, we design a pseudo-random sequence generator, which can generate two pseudo-random sequences simultaneously and is implemented in a DSP platform. The two sequences are uncorrelated with each other and both pass the NIST test with good randomness, large secret key space, and fast generation of pseudo-random sequences. The generated pseudorandom sequences are applied as spreading codes in a spread spectrum communication system. The performance of the communication system is better compared with the Chen system, Hénon map, typical Gold sequence, and m-sequence by simulation tests, with good multi-access capability. It has the same good performance in a multi-user system based on the Rake receiver [16], indicating that the fractional-order chaotic system has broad application prospects.

References 1. Jia Z, Deng G (2007) Linear and nonlinear coupled synchronization of hyperchaotic Lü systems. J Dyn Control 5(3):220–223 2. He S, Sun K, Wang H (2016) Synchronization of fractional-order time delay chaotic systems with ring connection. Eur Phys J Spec Top 225(1):97–106 3. Kanso A, Smaoui N (2009) logistic chaotic maps for binary numbers generation. Chaos, Solitons Fractals 40:2557–2568 4. Singla P, Sachdeva P, Ahmad M (2014) A chaotic neural network based cryptographic pseudo-random sequence design. In: IEEE 2014 fourth international conference on advanced computing & communication technologies (ACCT), pp 301–306 5. Liu NS (2011) Pseudo-randomness and complexity of binary sequences generated by the chaotic system. Commun Nonlinear Sci Numer Simul 16(2):761–768 6. Bassham Iii LE, Rukhin AL, Soto J et al (2010) SP 800–22 Rev. 1a. A statistical test suite for random and pseudorandom number generators for cryptographic applications. NIST Special Publication 7. François M, Defour D, Berthomé P (2014) A pseudo-random bit generator based on three chaotic logistic maps and IEEE 754–2008 floating-point arithmetic. Theory Appl Models Comput. Springer International Publishing 229–247 8. Rhouma R, Belghith S (2011) Cryptanalysis of a chaos-based cryptosystem on DSP. Commun Nonlinear Sci Numer Simul 16(2):876–884 9. Hidalgo RM, Fern Aacute JG, Rivera RR et al (2001) Versatile DSP-based chaotic communication system. Electron Lett 37(19):1204–1205 10. Tlelo-Cuautle E, Carbajal-Gomez VH, Obeso-Rodelo PJ et al (2015) FPGA realization of a chaotic communication system applied to image processing. Nonlinear Dyn 82(4):1879–1892 11. Wang XY, Liu LT (2013) Cryptanalysis and improvement of a digital image encryption method with chaotic map lattices. Chin Physica B 22(5):198–202

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12. Li C, Zhang L, Ou R et al (2012) Breaking a novel color image encryption algorithm based on chaos. Nonlinear Dyn 70(4):2383–2388 13. Zhu CX, Xu SY, Hu YP, Sun KH (2015) Breaking a novel image encryption scheme based on Brownian motion and PWLCM chaotic system. Nonlinear Dyn 79(2):1511–1518 14. Li JB, Zeng YC, Chen SB et al (2011) Improved Hénon mapping for generating chaotic pseudorandom sequences and performance analysis. J Phys 6:60508 15. Hu HP, Liu LF, Ding ND (2013) Pseudorandom sequence generator based on the Chen chaotic system. Comput Phys Commun 184(3):765–768 16. Wang H, Sun K, He S (2018) Design of fractional-order chaotic spread-spectrum communication system. J Cent South Univ (Natural Science Edition) 49(4):874–870

Chapter 10

Solution and Characteristic Analysis of Fractional-Order Discrete Chaotic System

10.1 Research Progress of Fractional-Order Discrete Chaotic Systems Due to the rich dynamic characteristics of fractional-order chaotic systems, in recent years, its related topics have become a new research hotspot in nonlinear chaos theory. The history of fractional-order calculus theory is almost the same as that of integerorder calculus, and it has been more than 300 years ago. Although fractional calculus has many very interesting special properties in the field of mathematics, due to the lack of practical application background, its theoretical research has been stagnant for a long time. Until the middle of the twentieth century, people discovered that many physical systems have fractional-order dynamics, such as viscous systems, colored noise, electromagnetic waves, and so on. Compared with integer-order differential equations, fractional-order differential equations can describe natural phenomena more accurately due to their special memory effects [1], such as the description of memory and mechanical properties of various materials, viscoelastic dampers, electronic circuits, Chemistry, and fractional capacitance theory and control of flexible construction objects. In the early studies of chaotic systems, the main research is based on integerorder chaotic systems, but when people introduce fractional differential operators into nonlinear dynamic systems, they find that the system can still exhibit chaotic behavior. So far, the research on fractional-order chaotic systems has been carried out for decades, such as fractional-order Chua’s circuit [2], the fractional-order Jerk system [3], the fractional-order Rössler system [4], the fractional-order Chen system [5], and many more. However, these studies are almost focused on fractionalorder continuous chaotic systems, and there are very few studies on fractional-order discrete chaotic systems. In fact, as early as 1989, Miller and Ross began the study of discrete fractionalorder difference and gave a preliminary definition [6]. Until recently, due to the important application of discrete dynamical systems in the engineering field, people began to pay more attention to the research on the definition of discrete fractional © Science Press 2022 K. Sun et al., Solution and Characteristic Analysis of Fractional-order Chaotic Systems, https://doi.org/10.1007/978-981-19-3273-1_10

221

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10 Solution and Characteristic Analysis of Fractional-Order …

order, and some groundbreaking discussions were carried out, such as the initial value problem in the definition of discrete fractional-order difference [7], discrete variation calculation problem [8], the Laplace transform problem in discrete fractional-order calculus [9], and discussion of the characteristics of the Caputo fractional-order difference definition, the Riemann–Liouville fractional-order difference definition [10], and so on. On these foundations, Edelman, a Professor at New York University in the United States, introduced the definition of fractional-order difference based on the Caputo operator into the discrete chaotic map [11, 12] and proposed the fractional-order standard map and the fractional-order Logistic map. This is the first time that the concept of discrete fractional-order has been introduced into discrete chaotic systems. Since then, the study of fractional-order discrete chaotic systems based on the definition of fractional-order difference has aroused widespread interest, and many reports on the properties of new fractional-order discrete chaotic systems have been reported [13–16], and other applications such as chaos synchronization and control [17, 18], image encryption [19], parameter recognition [20], and other practical engineering applications have made it a research hotspot in nonlinear dynamics in recent years. However, although there have been many research results on fractional-order discrete chaotic systems, the analysis of system dynamics is still not comprehensive, and there are many topics that need in-depth exploration. For example, in a multi-dimensional fractional discrete chaotic system, what is the dynamic behavior of the system in the case of non-identical orders (the fractional orders of each dimension are not completely the same), and whether the system can exhibit hyperchaotic behavior? How to quickly and efficiently calculate the Lyapunov exponent of a high-dimensional fractional-order discrete chaotic system? How does the complexity of a fractional-order discrete chaotic system change with the order? How to find a fractional-order discrete chaotic system with a practical background? The current research on fractional discrete chaotic systems is still in its infancy. Therefore, people need to continue to explore and research. Only by solving these obvious problems can we lay the foundation for the actual engineering application of the fractional-order discrete chaotic system.

10.2 Definition of Discrete Fractional-Order Difference At present, the definition of fractional-order discrete chaotic systems is mainly based on three definitions: the Riemann–Liouville definition [10], the Caputo definition [11–20], and the Grunwald–Letnokov definition [21, 22]. Among them, because the Riemann–Liouville difference equation needs to define the initial conditions, it is difficult to find the actual background in the application, while the Caputo definition does not have the problem of initial value setting. Therefore, people generally study fractional-order discrete chaotic systems based on Caputo’s definition. In addition, there are a few studies on the fractional-order discrete chaotic system defined by Grunwald–Letnokov, and many characteristics analysis are far from enough. Therefore, in this chapter, we will focus on introducing the fractional-order discrete chaotic

10.2 Definition of Discrete Fractional-Order Difference

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system and its characteristic analysis based on the Caputo fractional-order difference definition. First, define an independent time scalar Na ={a, a + 1, a + 2, ...},a ∈ R, and define a function x(n) whose antecedent difference operator is x(n) = x(n + 1) − x(n). Definition 10.1 There is a relationship f : Na → R, then the fractional-order sum of the order ν (v > 0 is the fractional-order sum, v < 0 is the fractional-order difference) is defined as 1  (t − σ (s))(ν−1) x(s). (ν) s=a t−ν

a−ν x(t) =

(10.1)

Among them, t ∈ Na+ν , a is the initial point, σ (s) = s + 1, () is the Gamma ∞ function, and (n) = 0 t n−1 e−t dt, t (v) is the decreasing factorial factor, which is defined as t (ν) =

(t + 1) . (t + 1 − ν)

(10.2)

Definition 10.2 When the fractional-order v > 0, the Caputo difference operator is defined as C

aν x(t) = a−(m−ν) m x(t) =

t−(m−ν)  1 (t − σ (s))(m−ν−1) m s x(s), (m − ν) s=a

(10.3) among them, t ∈ Na+m−ν , and m = ν. From the above definition, the following theorem can be obtained. Theorem 10.1 The Caputo fractional-order difference equation is defined as C

aν x(t) = f (t + ν − 1, x(t + ν − 1)),

(10.4)

where k x(a) = ck , k = 0, 1, …, m − 1. Equation (10.4) can continue to be equivalent to Equation x(t) = x0 (t) +

t−ν  1 (t − σ (s))(ν−1) f (s + ν − 1, x(s + ν − 1)), (10.5) (ν) s=a+m−ν

where t ∈ Na+m , the initial value x 0 (t) is defined as

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10 Solution and Characteristic Analysis of Fractional-Order …

x0 (t) =

m−1  k=0

(t − a)(k) k  x(a). k!

(10.6)

It can be observed from Eq. (10.5) that when the order v = 1, the system equation will degenerate to a general constant difference equation. In addition, because there is a cumulative term in the equation, the current iteration value of the difference equation in the fractional-order will be related to the value generated by each previous iteration, which reflects the special memory effect of the fractional-order difference equation.

10.3 Characteristic Analysis of Fractional-Order Discrete Chaotic System Through the definition of the fractional-order difference equation in the Sect. 10.2 of this chapter, in theory, any discrete chaotic system can be transformed into a fractional-order form in any dimension. Therefore, this section will take the onedimensional fractional-order Logistic map and the two-dimensional fractional-order Hénon map as examples to solve them and analyze the dynamic characteristics.

10.3.1 The Fractional-Order Logistic Map According to the definition of fractional-order difference and the equation of Logistic map, the equation of fractional-order Logistic map can be derived as C

aν x(t) = μx(t + ν − 1)(1 − x(t + ν − 1)),

(10.7)

where t ∈ Na+1−ν , μ is the system parameter. Convert it into the fractional-order difference equation defined by Caputo as x(t) = x(0) +

t−ν  1 (t − s − 1)(ν−1) × [μx(s + ν − 1)(1 − x(s + ν − 1) − x(s + ν − 1))]. (ν)

(10.8)

s=1−ν

Suppose s + ν = j, and then according to Eq. (10.2), the numerical formula of the fractional-order logistic map can be derived as 1  (t − j + ν) ×[μx( j − 1)(1 − x( j − 1) − x( j − 1))]. (ν) j=1 (t − j + 1) t

x(t) = x(0) +

(10.9)

10.3 Characteristic Analysis of Fractional-Order …

225

It can be seen from Eq. (10.9) that when ν = 1, the fractional-order Logistic map degenerates to the classic Logistic map, and the t-th iteration value x(t) will be related to the value x(0), x(1), …, x(t − 1) produced by each previous iteration. The bifurcation diagrams of the fractional-order logistic map under different orders are shown in Figs. 10.1 and 10.2, and the corresponding Lyapunov exponents are plotted. At present, there are only two methods for describing the Lyapunov exponents of fractional discrete chaotic systems: one is the Jacobian matrix method [23], and the other is the Wolf algorithm [24]. As the calculation speed of the Wolf algorithm is slow, the Jacobian matrix method is used here. The simulation experiment is implemented in MATLAB, where the initial value is set to x(0) = 0.24, the maximum number of iterations is t = 1700, and the previous 1500 states are discarded as the transition process. As shown in Figs. 10.1 and 10.2, the fractional-order Logistic map produces different dynamic behaviors as the order v changes. Moreover, the bifurcation diagram of the fractional-order Logistic map, relative to the integer-order Logistic

Fig. 10.1 Fractional-order Logistic map, bifurcation diagram and Lyapunov exponent when the order v = 0.8. a Bifurcation diagram and b Lyapunov exponent

Fig. 10.2 Fractional-order Logistic map, order v = 0.5. a Bifurcation diagram and b Lyapunov exponent

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10 Solution and Characteristic Analysis of Fractional-Order …

map, seems to gradually “move” to the left on the abscissa axis as the order v decreases. The smaller the order v, the smaller the value of the system parameter μ when the system enters the chaotic state. Figure 10.3 shows the sequence generated by the fractional-order Logistic map when the order v = 0.5. Figure 10.3a is the sequence generated when the system parameter μ = 3.1, which shows the four-cycle state of the system. Figure 10.3b shows that the system is in a chaotic state when the parameter μ = 3.3. This result is consistent with the dynamic behavior shown in the bifurcation diagram in Fig. 10.2. As shown in Fig. 10.4, using the Spectrum Entropy (SE) complexity algorithm, a chaos diagram in which system complexity varies with the fractional order v and the system parameter μ is calculated. It can be concluded from the figure that the change trend of the complexity of the Logistic map with the system parameter μ in the fractional-order is roughly the same as the change trend of the classic integer-order Logistic map. The complexity increases as the parameter μ increases.

Fig. 10.3 Sequence generated by fractional-order Logistic map under different system parameters when order v = 0.5. a μ = 3.1 and b μ = 3.3

Fig. 10.4 SE complexity chaos diagram of fractional-order Logistic map

10.3 Characteristic Analysis of Fractional-Order …

227

10.3.2 The Fractional-Order Hénon Map Similar to the process of deriving the fractional-order Logistic map equation, the derivation step is omitted here, and the numerical formula of the fractional-order Hénon map is directly written as ⎧ ⎪ ⎪ ⎨ x(t) = x(0) +

1 (ν1 )

⎪ ⎪ ⎩ y(t) = y(0) +

1 (ν2 )

t  j=1 t  j=1

(t− j+ν1 ) (t− j+1)



× 1 + y( j − 1) − ax( j − 1)2 − x( j − 1)

(t− j+ν2 ) (t− j+1)

×[bx( j − 1) − y( j − 1)]

, (10.10)

where a and b are system parameters, and v1 and v2 are the order of the fractional-order of the two dimensions in the system. The bifurcation diagrams of the fractional-order Hénon map under different orders with the parameter a (parameter b = 0.2) and the corresponding Lyapunov exponents are shown in Figs. 10.5, 10.6 and 10.7. The initial value is set to x(0) = −0.3, y(0) = 0.2, the maximum number of iterations is t = 1700, and the previous 1500 states are discarded as the transition process. Figures 10.5 and 10.6 show the bifurcation diagram and the Lyapunov exponent of the fractional-order Hénon map in the case of the same order (v1 = v2 ). Similar to the results obtained in the fractional-order Logistic map, the bifurcation graph of the fractional-order Hénon map also gradually “moves to the left” as the order v decreases. When the system enters the chaotic state, the value of parameter a also decreases with the decrease of v. Figure 10.7 shows the bifurcation diagram and the Lyapunov exponent of the fractional-order Hénon map in the case of non-identical orders (v1 = v2 ). It shows that the dynamic behavior of the system is different from that of the same order. It

Fig. 10.5 The bifurcation graph and Lyapunov exponent of the fractional-order Hénon map when the order v1 = v2 = 0.9. a Bifurcation diagram and b Lyapunov exponent

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Fig. 10.6 The bifurcation graph and Lyapunov exponent of the fractional-order Hénon map when the order v1 = v2 = 0.6. a Bifurcation diagram and b Lyapunov exponent

Fig. 10.7 The bifurcation graph and Lyapunov exponent of the fractional-order Hénon map when the order v1 = 0.9, v2 = 0.6. a Bifurcation diagram and b Lyapunov exponent

seems to be a transition process of the bifurcation diagram between v1 = v2 = 0.9 and v1 = v2 = 0.6. Figure 10.8 shows the x sequence generated by the fractional-order Hénon map under different system parameters. The behavior of the sequence is consistent with the dynamic behavior shown in the bifurcation diagram in Fig. 10.6. Similarly, the SE algorithm is used to draw a chaos diagram with the complexity of the fractional-order Hénon map corresponding to the order v and the system parameters a and b, as shown in Fig. 10.9. From the abscissa in the figure, as the order v decreases, the effective value range of the chaotic system gradually decreases. From the ordinate point of view, the complexity of the system will increase as the value of the system parameter increases.

10.3 Characteristic Analysis of Fractional-Order …

229

Fig. 10.8 Sequence generated by fractional-order Hénon map under different parameters, order v1 = v2 = 0.6. a a = 0.65, b = 0.2 and b a = 0.9, b = 0.2

Fig. 10.9 SE complexity chaos diagram of fractional-order Hénon map. a b = 0.2, the complexity when the parameters v and a change and b a = 1.4, the complexity when the parameters v and b change

10.3.3 The Fractional-Order High-Dimensional Chaotic Map Introduce the concept of fractional difference to a chaotic map with highly complex behavior, that is, a high-dimensional hyperchaotic map with a grid sinusoidal cavity (SI-CMCM, sine iterative chaotic map with infinite collapse modulation map) [25], and analyze its dynamic behavior. The system equation of fractional-order SI-CMCM is

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10 Solution and Characteristic Analysis of Fractional-Order …

⎧  

n 1  (n − j + v1 ) c ⎪ ⎪ x1 (n) = x1 (0) + a sin[ωxm ( j − 1)] sin − x1 ( j − 1) ⎪ ⎪ ⎪ (v1 ) (n − j + 1 ) x1 ( j − 1) ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪  

n ⎪ ⎪

1  (n − j + v2 ) c ⎪ ⎪ ⎪ a sin ωx1 ( j) sin − x2 ( j − 1) x (n) = x2 (0) + ⎪ ⎨ 2 (v2 ) (n − j + 1 ) x2 ( j − 1) j=1 , ⎪ ⎪ ⎪ . ⎪ ⎪ . ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪  

n ⎪  ⎪



1 (n − j + vm ) c ⎪ ⎪ a sin ωxm−1 ( j) sin + h x1 ( j − 1) − xm ( j − 1) ⎪ xm (n) = xm (0) + ⎩ (vm ) (n − j + 1 ) xm ( j − 1) j=1

(10.11) where a, c, and ω are system parameters. The step function h(x) is a control parameter that can generate multiple rows of sinusoidal cavities, defined as follows:     N   N N sgn h(x) = a x − (2k − 1) + sgn x + (2k − 1) , N = odd, a a k=1 (10.12)     N   N N sgn x − 2k + sgn x + 2k h(x) = a a a k=1   N x , N = even, + a × sgn a

(10.13)

where k is a non-negative constant, sgn(·) is a sign function, and N is the number of rows of the grid sine cavity chaotic map. When the number of rows N is an odd number, it corresponds to Eq. (10.12), and when it is an even number, it corresponds to Eq. (10.13). In order to facilitate the analysis of the dynamic characteristics of the fractionalorder SI-CMCM, the system with dimension m = 2 is taken as an example in this chapter, and the two-dimensional fractional-order SI-CMCM system equation is given here as ⎧  

n ⎪ 1  (n − j + v1 ) c ⎪ ⎪ x(n) = x(0) + a sin[ωy( j − 1)] sin − x( j − 1) ⎪ ⎪ ⎪ (v (n − j + 1 x( j − 1) ) ) 1 j=1 ⎨  

n ⎪ ⎪ 1  (n − j + v2 ) c ⎪ ⎪ a sin[ωx( j)] sin + h(x( j − 1)) − y( j − 1) ⎪ ⎪ y(n) = y(0) + (v ) ⎩ (n − j + 1 y( j − 1) ) 2 j=1

,

(10.14)

For a two-dimensional fractional SI-CMCM, the number of sinusoidal cavities is 2π/aω × N. When the system parameters a, ω, and N are of different values, grid sinusoidal cavities with different numbers of rows are generated. Set x (0) = 0.3, y (0) = 0.5, a = 2, c = 50, ω = π, n = 30,000, and v1 = v2 = 1, the attractor phase diagram of the fractional-order SI-CMCM at different number of rows N is shown in Fig. 10.10.

10.3 Characteristic Analysis of Fractional-Order …

231

Fig. 10.10 Attractor of fractional-order SI-CMCM at v = 1. a N = 1, b N = 2, c N = 3, and d N =4

Here, first analyze the characteristics of the fractional SI-CMCM system when N = 3 and change ν 1 and ν 2 (0 < v1 < 1, 0 < v2 < 1, and v = v1 = v2 ). The phase diagram of the attractor is shown in Fig. 10.11. As the fractional-order ν decreases, the boundary range of the chaotic attractor keeps increasing. In addition, it can be noticed that the shape of the chaotic attractor has changed from a regular sinusoidal cavity to an elliptical diverging chaotic attractor. What is interesting is that when the order ν drops to 0.01, the attractors become sparse and disordered. Secondly, analyze its bifurcation diagram with different parameters, as shown in Fig. 10.12. In the following simulation experiment, the ranges of a, c, and ω are set to [1, 5]. (1)

Change of amplitude a

When c = 50 and ω = π, Fig. 10.12 shows the bifurcation diagrams at different ν, respectively. When v changes from integer-order to fractional-order, the periodic window at c ∈ (4.550, 4.571) becomes a chaotic state, and there is no other periodic window. It can be seen from Fig. 10.12 that the smaller the order, the larger the chaotic range of the system. The results show that the fractional-order chaotic system has a

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10 Solution and Characteristic Analysis of Fractional-Order …

Fig. 10.11 The attractors of the fractional-order SI-CMCM when N = 3. a v = 0.95, b v = 0.9, c v = 0.7, d v = 0.5, e v = 0.1, and f v = 0.01

Fig. 10.12 The bifurcation diagrams of the fractional-order SI-CMCM when N = 3. a c = 50, ω = π, v = 1, b c = 50, ω = π, v = 0.9, c c = 50, ω = π , v = 0.7, d c = 50, ω = π, v = 0.5, e a = 2, c = 50, v = 1, f a = 2, c = 50, v = 0.9, g a = 2, c = 50, v = 0.7, h a = 2, c = 50, v = 0.5, i a = 2, ω = π, v = 1, j a = 2, ω = π, v = 0.9, k a = 2, ω = π, v = 0.7, and l a = 2, ω = π, v = 0.5

10.3 Characteristic Analysis of Fractional-Order …

233

larger key space and better ergodicity, and it has more advantages than integer-order SI-CMCM in terms of information encryption. (2)

Change of internal disturbance frequency c

When a = 2 and ω = π, Fig. 10.12 shows the bifurcation diagrams of different v, respectively. As shown in Fig. 10.12e, there is an obvious periodic window at c∈[1, 1.381]. It can be observed that as the order ν decreases, the periodic window disappears and the range of chaos gradually expands. (3)

Change of parameter ω

When a = 2 and c = 50, Fig. 10.12 depicts the bifurcation diagram for different ν. Like integer-order systems, fractional-order systems are chaotic in the entire range and have no periodic windows. Obviously, the fractional-order system has a wider chaotic space than the integer-order system. Consider the situation when N = 4. Change v1 and v2 (0 < v1 < 1, 0 < v2 < 1, and v = v1 = v2 ), the attractor is shown in Fig. 10.13. In Fig. 10.13, the sub-pictures a–e are similar to those in Fig. 10.11, except for Fig. 10.13f. It can be seen from the figure that the shape of the attractor is very different from that drawn in Fig. 10.11f. As the fractional order ν drops to 0.01, the attractor has a dense and ordered shape. It can be seen that with the change of ν, the change law of the attractor of the fractional SI-CMCM system with different N is different. (1)

Change of amplitude a

When c = 50 and ω = π, the bifurcation diagrams for different ν are shown in Fig. 10.14. Unlike the system with odd N in Fig. 10.12a, there is no period window

Fig. 10.13 The attractors of the fractional-order SI-CMCM when N = 4. a v = 0.95, b v = 0.9, c v = 0.7, d v = 0.5, e v = 0.1, and f v = 0.01

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10 Solution and Characteristic Analysis of Fractional-Order …

Fig. 10.14 The bifurcation diagrams of the fractional-order SI-CMCM when N = 4. a c = 50, ω = π, v = 1, b c = 50, ω = π, v = 0.9, c c = 50, ω = π , v = 0.7, d c = 50, ω = π, v = 0.5, e a = 2, c = 50, v = 1, f a = 2, c = 50, v = 0.9, g a = 2, c = 50, v = 0.7, h a = 2, c = 50, v = 0.5, i a = 2, ω = π, v = 1, j a = 2, ω = π, v = 0.9, k a = 2, ω = π, v = 0.7, and l a = 2, ω = π, v = 0.5

within the parameter range in Fig. 10.14a. Similar to the situation in Fig. 10.12, as ν decreases, the fractional-order chaotic system is in global chaos, and the chaotic range of the fractional-order system expands accordingly. (2)

Change of internal disturbance frequency c

When a = 2 and ω = π, the bifurcation diagrams for different ν are shown in Fig. 10.14. When the order of the system is an integer, the integer-order system has two obvious periodic windows at c ∈ (1.170,1.411) ∪ (1.622,1.962), and there are several narrower periodic windows. When the order of the system changes to a fractional-order, the periodic window disappears completely. (3)

Change of parameter ω

When a = 2 and c = 50, the bifurcation diagram of different ν is shown in Fig. 10.14. In Fig. 10.14i, there seems to be a very small period window at ω ∈ (1.551, 1.581). However, it can be seen from the enlarged view in the upper right corner of Fig. 10.14i that the system is still chaotic in a very small range. In addition, the range of the chaotic state of the system continues to expand as ν decreases. Figure 10.15 depicts the fractional-order SI-CMCM system when the parameters a = 2, c = 3, and ω = π change with the permutation entropy (PE) [26] of the order ν. The red line represents the system at N = 4, and the blue line refers to

10.3 Characteristic Analysis of Fractional-Order …

235

Fig. 10.15 The PE complexity varies with the order v of the system when a = 2, c = 3, and ω = π

the system at N = 3. It can be seen that when ν ∈ (0, 0.8), the complexity of PE gradually increases. When v is greater than 0.8, the curve tends to stabilize and maintain a higher value. This shows that when ν ∈ (0.8, 1), the fractional-order SICMCM has extremely high complexity. The higher the complexity, the more suitable for confidential communication, so in practical applications, we’d better choose an order within the range of ν ∈ (0.8, 1). Figure 10.16 shows the PE complexity chaos diagram of choosing different parameter planes (a, v), (c, v), (ω, v), and (N, v). The yellower the color, the higher the complexity. In Fig. 10.16a, b, there are many small dark-colored areas, which indicates that the PE complexity is low there. There are two obvious dark-colored areas in Fig. 10.16c, and PE has a lower complexity, as in Fig. 10.16d. Comparing Fig. 10.16e with Fig. 10.16f, the same conclusion was obtained. As the fractional-order and system parameters increase, the PE complexity of the system also increases. In addition, when N = 3, the system has a wider area of high complexity than the system when N = 4, indicating that the system has higher PE complexity when N = 3. In Fig. 10.16h, i, the PE complexity of the system changes with the fractional order ν and the number of rows N, which are completely similar. Higher order ν and system parameters mean higher complexity and randomness. Therefore, in the practical application of confidential communication, we should avoid the value of the dark blue area when selecting parameters. Figure 10.16 shows a reasonable range of parameter selection. In fact, according to the Caputo fractional-order difference equation described in this chapter, we can transform any discrete chaotic system into a fractional-order form. The emergence of fractional-order chaotic maps can not only serve as a good supplement to the theory of fractional calculus, but also promote more in-depth research in the field of discrete chaos and become a potential application in the further development of chaotic secure communication.

236

10 Solution and Characteristic Analysis of Fractional-Order …

Fig. 10.16 The PE complexity chaos diagram of fractional-order SI-CMCM on the parameter plane (a, v), (c, v), (ω, v), and (N, v). a c = 3, ω = π, N = 3, b c = 3, ω = π, N = 4, c a = 2, ω = π, N = 3, d a = 2, ω = π, N = 4, e a = 2, c = 3, N = 3, f a = 2, c = 3, N = 4, g a = 2, c = 3, ω = π, N = odd, and h a = 2, c = 3, ω = π, N is even

References

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Appendix

Program Code

Taking the fractional-order simplified Lorenz system as an example, the MATLAB code of the system’s prediction–correction algorithm and Adomian decomposition algorithm is given [1].

© Science Press 2022 K. Sun et al., Solution and Characteristic Analysis of Fractional-order Chaotic Systems, https://doi.org/10.1007/978-981-19-3273-1

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Appendix: Program Code

Prediction–Correction Algorithm

function [x y z]=jianhua(c,q) if nargin==0 c=5; q=1; q1=q;q2=q;q3=q; end %c=5; h=0.01;N=5000; q1=q;q2=q;q3=q; x0=-6.7781;y0=-8.3059;z0=9.2133; %x0=7.7053;y0=8.7546;z0=10.6567; M1=0;M2=0;M3=0; x(N+1)=[0];y(N+1)=[0];z(N+1)=[0]; x1(N+1)=[0];y1(N+1)=[0];z1(N+1)=[0]; x1(1)=x0+h^q1*10*(y0-x0)/(gamma(q1)*q1); y1(1)=y0+h^q2*((24-4*c)*x0-x0*z0+c*y0)/(gamma(q2)*q2); z1(1)=z0+h^q3*(x0*y0-8*z0/3)/(gamma(q3)*q3); x(1)=x0+10*h^q1*(y1(1)-x1(1)+q1*(y0-x0))/gamma(q1+2); y(1)=y0+h^q2*((24-4*c)*x1(1)-x1(1)*z1(1)+c*y1(1)+q2*((24-4*c)*x0-x0*z 0+c*y0))/gamma(q2+2); z(1)=z0+h^q3*(x1(1)*y1(1)-8*z1(1)/3+q3*(x0*y0-8*z0/3))/gamma(q3+2); for n=1:N M1=(n^(q1+1)-(n-q1)*(n+1)^q1)*10*(y0-x0); M2=(n^(q2+1)-(n-q2)*(n+1)^q2)*((24-4*c)*x0-x0*z0+c*y0); M3=(n^(q3+1)-(n-q3)*(n+1)^q3)*(x0*y0-8*z0/3); N1=((n+1)^q1-n^q1)*10*(y0-x0); N2=((n+1)^q2-n^q2)*((24-4*c)*x0-x0*z0+c*y0); N3=((n+1)^q3-n^q3)*(x0*y0-8*z0/3); for j=1:n M1=M1+((n-j+2)^(q1+1)+(n-j)^(q1+1)-2*(n-j+1)^(q1+1))*10*(y(j)-x(j)); M2=M2+((n-j+2)^(q2+1)+(n-j)^(q2+1)-2*(n-j+1)^(q2+1))*((24-4*c)*x(j)-x (j)*z(j)+c*y(j)); M3=M3+((n-j+2)^(q3+1)+(n-j)^(q3+1)-2*(n-j+1)^(q3+1))*(x(j)*y(j)-8*z(j )/3);

Appendix: Program Code

241

N1=N1+((n-j+1)^q1-(n-j)^q1)*10*(y(j)-x(j)); N2=N2+((n-j+1)^q2-(n-j)^q2)*((24-4*c)*x(j)-x(j)*z(j)+c*y(j)); N3=N3+((n-j+1)^q3-(n-j)^q3)*(x(j)*y(j)-8*z(j)/3); end x1(n+1)=x0+h^q1*N1/(gamma(q1)*q1); y1(n+1)=y0+h^q2*N2/(gamma(q2)*q2); z1(n+1)=z0+h^q3*N3/(gamma(q3)*q3); x(n+1)=x0+h^q1*(10*(y1(n+1)-x1(n+1))+M1)/gamma(q1+2); y(n+1)=y0+h^q2*((24-4*c)*x1(n+1)-x1(n+1)*z1(n+1)+c*y1(n+1)+M2)/gamma( q2+2); z(n+1)=z0+h^q3*(x1(n+1)*y1(n+1)-8*z1(n+1)/3+M3)/gamma(q3+2); end

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Appendix: Program Code

Adomian Decomposition Algorithm

function dy=SimpleFratral(x0,h,q,c) dy=zeros(1,3); b=8/3; a=10; if q0 c10=x0(1); c20=x0(2); c30=x0(3); end if q>1&& q