An Introduction to Nonautonomous Dynamical Systems and their Attractors (Interdisciplinary Mathematical Sciences) 9811228655, 9789811228650

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An Introduction to Nonautonomous Dynamical Systems and their Attractors by Peter E Kloeden & Meihua Yang Stochastic PDEs and Modelling of Multiscale Complex System eds. Xiaopeng Chen, Yan Lv & Wei Wang Kernel-based Approximation Methods using MATLAB by Gregory Fasshauer & Michael McCourt Global Attractors of Non-Autonomous Dynamical and Control Systems (2nd Edition) by David N Cheban Festschrift Masatoshi Fukushima: In Honor of Masatoshi Fukushima’s Sanju eds. Zhen-Qing Chen, Niels Jacob, Masayoshi Takeda & Toshihiro Uemura Hilbert–Huang Transform and Its Applications (2nd Edition) eds. Norden E Huang & Samuel S P Shen Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis: Fractional Dynamics, Network Dynamics, Classical Dynamics and Fractal Dynamics with Their Numerical Simulations eds. Changpin Li, Yujiang Wu & Ruisong Ye Recent Developments in Computational Finance: Foundations, Algorithms and Applications eds. Thomas Gerstner & Peter Kloeden Stochastic Analysis and Applications to Finance: Essays in Honour of Jia-an Yan eds. Tusheng Zhang & Xunyu Zhou New Trends in Stochastic Analysis and Related Topics: A Volume in Honour of Professor K D Elworthy eds. Huaizhong Zhao & Aubrey Truman

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An Introduction to Nonautonomous Dynamical Systems and their Attractors

Dedicated to the memory of Karin Wahl-Kloeden (PEK) Dedicated to my parents and my daughter (MHY)

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An Introduction to Nonautonomous Dynamical Systems and their Attractors

Preface

The nature of time in nonautonomous dynamical systems is very different from that in autonomous systems, which depends only on the time that has elapsed since starting rather than on the actual time itself. Consequently, limiting objects may not exist in actual time as in autonomous systems. New concepts of attractors in nonautonomous dynamical systems are thus required. In addition, the definition of a dynamical system itself needs to be generalised to the nonautonomous context. Here two possibilities are considered: two-parameter semigroups or processes and the skew product flows. Their attractors are defined in terms of families of sets that are mapped onto each other under the dynamics rather than a single set as in autonomous systems. Two types of attraction are now possible: pullback attraction, which depends on the behaviour from the system in the distant past, and forward attraction, which depends on the behaviour of the system in the distant future. Pullback and forward attractors are invariant families of sets which depend on time and exist in actual time for all time. They are generally independent of each other. The theory of pullback attractors is now very well developed. In contrast the theory of forward attractors is not as satisfactory, since forward attractors often do not exist and when they do, they are often not unique. Moreover, they require the dynamical system to be defined for all time, including the distant past, even though they are concerned with the distant future. The asymptotic behaviour in the future limit seems to be better characterised by omega-limit sets, in terms of which form what are called forward attracting sets. They are generally not invariant in the conventional sense, but are asymptotically positively invariant in general and, if the future dynamics is appropriately uniform, also asymptotically negatively invariant. Much of this book is based on lectures given by the authors in Frankfurt and Wuhan. It was written mainly when the first author held a “Thousand Expert” Professorship at the Huazhong University of Science and Technology in Wuhan. The financial support from this program and the NSFC grants 11571125 and 11971184 is gratefully acknowledged. vii

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An Introduction to Nonautonomous Dynamical Systems and their Attractors

An Introduction to Nonautonomous Dynamical Systems and their Attractors

The material in the first nine chapters is now quite well known in the literature, see for example the monograph Nonautonomous Dynamical Systems [Kloeden and Rasmussen (2011)]. The material here is restricted to attractors. The emphasis is on systems with compact absorbing sets, thus essentially finite dimensional, and the presentation is also more didactical. This makes the book suitable for mathematics students at upper bachelors or master level and, in particular, for readers from other disciplines, who have a basic background in mathematics. Chapters 10–12 and Chapter 14 are based on more recent research results of the authors and their coworkers. Apart from some key references, most of the relevant literature is cited and discussed in the Notes at the end of the book. The authors thank Hongyong Cui, Christian P¨otzsche and Larissa Serdukova for many useful suggestions and assistance. They also thank the referees for their critical reading of the manuscript and insightful comments.

T¨ ubingen Wuhan August 2020

Peter E. Kloeden Mathematisches Institut Universit¨ at T¨ ubingen D-72076 T¨ ubingen Germany Meihua Yang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China

Peter Kloeden Meihua Yang

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Contents

Preface

vii

Dynamical systems

1

1.

3

Autonomous dynamical systems 1.1 1.2 1.3 1.4 1.5

2.

. . . . . . .

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Nonautonomous dynamical systems: processes 2.1 2.2 2.3 2.4

3.

Autonomous difference equations . . . . . . . . . Discrete-time autonomous dynamical systems . . Continuous-time autonomous dynamical systems General case . . . . . . . . . . . . . . . . . . . . . Invariant sets, omega limit sets and attractors . . 1.5.1 Omega-limit sets . . . . . . . . . . . . . . 1.5.2 Attractors . . . . . . . . . . . . . . . . .

Nonautonomous difference equations . . . . Nonautonomous differential equations . . . Abstract nonautonomous dynamical systems A process as an autonomous semi-dynamical

13 . . . . . . . . . . . . . . . system

. . . .

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Skew product flows 3.1 3.2 3.3

3.4

3 4 6 7 8 8 9

13 15 16 17 19

Autonomous triangular systems . . . . . . . . . . . . . . . . Definition of a skew product flow . . . . . . . . . . . . . . . 3.2.1 Processes as skew product flows . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Autonomous triangular system . . . . . . . . . . . . 3.3.2 Driving system need not be an autonomous ODE . 3.3.3 Nonautonomous difference equations . . . . . . . . . Skew product flows as autonomous semi-dynamical systems ix

. . . . . . . .

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19 21 22 22 22 23 25 26

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An Introduction to Nonautonomous Dynamical Systems and their Attractors

Entire solutions and invariant sets

27

4.1 4.2

27 30

Autonomous case . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonautonomous case . . . . . . . . . . . . . . . . . . . . . . . . . .

Pullback attractors 5.

6.

35

5.1 5.2

36 38

6.4

43

The SIR model . . . . . . . . . . . . . . . . . . . . . . . The SI equations with variable population . . . . . . . . 6.2.1 Nontrivial limiting solutions . . . . . . . . . . . The SI equations with variable interaction . . . . . . . . 6.3.1 Nontrivial limiting solutions . . . . . . . . . . . 6.3.2 Critical case . . . . . . . . . . . . . . . . . . . . The SIR model with variable population . . . . . . . . . 6.4.1 Disease free nonautonomous equilibrium solution 6.4.2 Nontrivial dynamics . . . . . . . . . . . . . . . .

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43 45 47 49 50 52 53 53 54

Attractors for processes

55

7.1 7.2

57 61

Existence of a pullback attractor . . . . . . . . . . . . . . . . . . . Strictly contracting processes: an alternative existence proof . . . .

Examples of pullback attractors for processes 8.1 8.2 8.3 8.4 8.5

9.

Attractors of autonomous systems . . . . . . . . . . . . . . . . . . What is a nonautonomous attractor? . . . . . . . . . . . . . . . . .

Nonautonomous equilibrium solutions

6.3

8.

33

Attractors

6.1 6.2

7.

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A linear differential equation . . . . . . A linear difference equation . . . . . . . A Nonlinear differential equation . . . . A dissipative ODE . . . . . . . . . . . . One-sided dissipative Lipschitz condition

65 . . . . .

. . . . .

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Attractors of skew product flows 9.1 9.2

9.3 9.4 9.5

Invariant sets and attractors . . . . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Continuous-time example . . . . . . . . . . . . . . . . . . 9.2.2 Discrete-time example . . . . . . . . . . . . . . . . . . . . Basin of pullback attraction system . . . . . . . . . . . . . . . . . Upper semi-continuity of the set-valued mapping p 7→ Ap . . . . 9.4.1 The set-valued mapping p 7→ Ap need not be continuous A weak form of forward convergence . . . . . . . . . . . . . . . .

65 66 67 69 70 73

. . . . . . . .

73 75 75 77 78 79 81 82

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Contents

9.6

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Another type of attractor for skew product flows . . . . . . . . . .

84

Forward attractors and attracting sets

87

10. Limitations of pullback attractors of processes

89

10.1 10.2 10.3 10.4

An autonomous difference equation . . . . . A piecewise autonomous difference equation Fully nonautonomous system . . . . . . . . More information through skew products . .

. . . .

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. . . .

11. Forward attractors 11.1 11.2 11.3 11.4 11.5 11.6 11.7

95

Pullback attractors recalled . . . . . . . . . Nonautonomous forward attractors in Rd . Construction of possible forward attractors Forward asymptotic behaviour . . . . . . . Omega limit points of a process . . . . . . . Conditions ensuring forward convergence . . Asymptotically autonomous systems . . . .

. . . . . . .

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. 95 . 96 . 97 . 99 . 100 . 101 . 102

12. Omega-limit sets and forward attracting sets 12.1 Forward attracting sets . . . . . . . . . . . 12.1.1 Asymptotically positive invariance 12.2 Asymptotically negative invariance . . . . 12.3 Upper semi-continuity in a parameter . . 12.4 Concluding remark . . . . . . . . . . . . .

105 . . . . .

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Random attractors

13.0.1 A simple example . . . . . . . . . . . . . . . . . . . . . 13.1 Definition of a random dynamical system . . . . . . . . . . . . 13.2 Random attractors . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Examples of random attractors . . . . . . . . . . . . . . . . . . 13.3.1 Random attractors for RODES . . . . . . . . . . . . . . 13.3.2 Random attractors for stochastic differential equations

106 106 108 110 113

115

13. Random dynamical systems

14. Mean-square random dynamical systems

89 91 92 93

117 . . . . . .

. . . . . .

118 119 121 122 122 123 127

14.1 Mean-square random dynamical systems . . . . . . . . . . . . . . . 128 14.2 Mean-square random attractors . . . . . . . . . . . . . . . . . . . . 128 14.3 Example of a mean-square attractor . . . . . . . . . . . . . . . . . 129

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An Introduction to Nonautonomous Dynamical Systems and their Attractors

Notes

133

Bibliography

137

Index

141

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PART 1

Dynamical systems

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An Introduction to Nonautonomous Dynamical Systems and their Attractors

Chapter 1

Autonomous dynamical systems

Autonomous dynamical systems are briefly recalled here for use, comparison and contrast when we generalise them to the nonautonomous case. 1.1

Autonomous difference equations

Difference equations are a special kind of recursion relation. The adjective difference has historical origins from when such equations were expressed in terms of the difference operator ∆xn = xn+1 − xn in contrast to the differential operator in differential equations. Let f : Rd → Rd . Then xn+1 = f (xn ),

n = 0, 1, 2, . . . ,

(1.1)

d

is a first-order autonomous difference equation on R . It is called autonomous because f does not depend explicitly on n. The mapping f will be assume to be continuous here. A simple example is the scalar population growth equation xn+1 = axn (1 − xn ),

n = 0, 1, 2, . . . ,

where a is a positive constant and f (x) = ax(1 − x) with x ∈ R1 . The solution of the difference equation (1.1) is given through succesive iteration by xn = f n (x0 ) := f ◦ f ◦ · · · ◦ f (x0 ), | {z }

n = 0, 1, 2, . . . ,

n times

with f 0 defined to be the identity mapping, i.e., f 0 (x) ≡ x. Define the solution mapping π : Z+ × Rd → Rd by π(n, x0 ) = f n (x0 ) .

This mapping has the properties i) initial condition π(0, x0 ) = x0 for all x0 ∈ Rd , ii) semi-group under composition: π(n + m, x0 ) = π(n, π(m, x0 )) for all m, n ∈ Z+ and x0 ∈ Rd , 3

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iii) continuity: the mapping x0 7→ π(n, x0 ) is continuous for each n ∈ Z+ . The initial condition is obvious, while continuity follows by the fact that the composition of continuous mappings is continuous. To verify the semi-group property note that π(n + m, x0 ) = f n+m (x0 ) := f ◦ f ◦ · · · ◦ f (x0 ) | {z } n+m times

= f ◦ f ◦ · · · ◦ f ◦ f ◦ f ◦ · · · ◦ f (x0 ) | {z } | {z } n times

m times

= f n ◦ f m (x0 ) = π(n, π(m, x0 )) for all m, n ∈ Z+ and x0 ∈ Rd , by the associativity of the composition operation. Remark 1.1. The mapping f need not be invertible, e.g., f (x) = x(2 − x) on R. Suppose that f is a homeomorphism, i.e., f is continuous and its inverse f −1 exists and is continuous. Then we can define π(n, x0 ) for n = −1, −2, −3, . . . too, i.e., we can extend π from Z+ to Z: for n < 0 define
−n (x0 ) , π(n, x0 ) = f −1 ◦ f −1 ◦ · · · ◦ f −1 (x0 ) = f −1 {z } | −n times


−n . i.e., for n < 0 we interpret f n as f −1 Then π satisfies the group property under composition. The main new thing to check is the case n < 0 < m.
−n π(n, π(m, x0 )) = f −1 ◦ f m (x0 ) = f −1 ◦ · · · ◦ f −1 ◦ f ◦ f ◦ · · · ◦ f (x0 ), | {z } | {z } −n times

( =

f −1 f


−n−m

n+m

m times

(x0 ) if n + m < 0,

(x0 )

if n + m > 0

= π(n + m, x0 )). Remark 1.2. The mapping f given by f (x) = x(2 − x) maps the unit interval [0, 1] into itself. It is invertible on this interval and generates a group under composition on it. 1.2

Discrete-time autonomous dynamical systems

More generally, instead of Rd , we could consider a metric space (X, dX ) as the state space.

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Autonomous dynamical systems

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Definition 1.1. A discrete-time autonomous dynamical system on a state space X is given by mapping π : Z × X → X, which satisfies the properties: i) initial condition: π(0, x0 ) = x0 for all x0 ∈ X, ii) group under composition: π(n + m, x0 ) = π(n, π(m, x0 )) for all m, n ∈ Z and x0 ∈ X, iii) continuity: the mapping x0 7→ π(n, x0 ) is continuous for each n ∈ Z. We have a semi-dynamical system if Z is replaced by Z+ ∪ {0} and ii) is replaced by the semi-group property, i.e., it holds just for n and m ∈ Z+ ∪ {0}. Note that the family of mappings {π(n, ·)}n∈Z is a group of mappings from X into itself under composition in the first case, while {π(n, ·)}n∈Z+ ∪{0} is a semi-group of mappings under composition. The following example of a discrete-time dynamical system will be important later in formulating nonautonomous difference equations generated by a finite number of mappings as a skew product flow, see Subsection 3.3.3. Example 1.1. Let X := {1, N }Z , where N ∈ N, be the space of bi-infinite integervalued sequences x = (. . . , x−1 , x0 , x1 , . . .} with xj ∈ {1, . . . , N }. Then X forms a compact metric space with the metric (the proof is left as an exercise) X 2−|n| |xn − yn | . dX (x, y) := n∈Z

Let θ : X → X be the left shift operator, which is defined as e := θ(x), x

en := (θ(x))n = xn+1 , x

n ∈ Z+ ∪ {0},

x = (. . . , x−2 , x−1 , x0 , x1 , x2 , . . .} ↓ e = (. . . , x−1 , x0 , x1 , x2 , x3 , . . .} . x | {z } ←shifted to the left

Note that θ is continuous and has a continuous inverse θ−1 given by the right shift operator
θ−1 (x) n := (θ(x))n−1 , n ∈ Z+ ∪ {0}. Then π : Z × X → X defined by ( π(n, x) =

θ−1 n


−n

θ (x)

(x) if n < 0, if n ≥ 0

is a discrete-time autonomous dynamical system on X. In particular, the family {θn : n ∈ Z} is a group under composition. This example can be represented as a first-order difference equation on the sequence space X with the mapping f : X → X defined by f (x) = π(1, x).

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1.3

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An Introduction to Nonautonomous Dynamical Systems and their Attractors

Continuous-time autonomous dynamical systems

Dynamical systems defined on the time set R instead of Z are called continuous-time dynamical system. Consider the autonomous ordinary differential equation (ODE) on Rd dx = f (x), x ∈ Rd , (1.2) dt and assume that the solution x(t) = x(t, t0 , x0 ) starting at x(t0 ) = x0 exists for all x0 ∈ Rd and t, t0 ∈ R, is unique, and continuous in the initial value (t0 , x0 ). (This is ensured if f satisfies a global Lipschitz condition or, more generally, satisfies a local Lipschitz condition and some kind of growth condition to prevent solutions from exploding in finite time.) Note that x(t; t0 , x0 ) ≡ x(t − t0 ; 0, x0 ), i.e., the solutions of an autonomous ODE depend only on the elapsed time t − t0 and not on the actual times t and t0 separately. This is a characteristic property of autonomous systems.

Example 1.2. The scalar ODE dx =x dt with initial value x(t0 ) = x0 has the explicit solution x(t, t0 , x0 ) = x0 et−t0 . This means that the solution curves are invariant under time translation. x

4

x(t, x(s, x0))

3.5

x(t+s, x0)

3 2.5 2 1.5 1 x(s, x0)

0.5

x0

0

0

s

t

s+t

1.5

2

−0.5 −0.5

Fig. 1.1

0

0.5

1

Invariance of solution curves under time translation.

Henceforth we will write just x(t − t0 , x0 ) for x(t, t0 , x0 ) and consider the solution mapping x as a mapping from R × Rd into Rd . This mapping has the following important properties: 1)

initial condition x(0, x0 ) = x0 .

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Autonomous dynamical systems

2)

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group under composition x(t + s, x0 ) = x(t, x(s, x0 )) for all t, s ∈ R. These correspond to the uniqueness of solutions to initial values problems.

3)

continuity (t, x0 ) 7→ x(t, x0 ) is continuous.

Remark 1.3. The solution of the ODE is in fact differentiable in t, hence continuous. From the theory of differential equations we know that it is also continuous in the initial condition x0 . 1.4

General case

Again, instead of Rd , we consider a general complete metric space (X, dX ) as the state space. In addition, we consider the time set T, which is Z in the discrete-time case and R in the continuous-time case. Similarly, T+ denotes Z+ and R+ in these cases. Definition 1.2. An autonomous dynamical system on a state space X is given by mapping π : T × X → X, which satisfies the properties: i) initial condition: π(0, x0 ) = x0 for all x0 ∈ X, ii) group under composition: π(s + t, x0 ) = π(s, π(t, x0 )) for all s, t ∈ T and x0 ∈ X, iii) continuity: the mapping (t, x) 7→ π(t, x) is continuous at all points (t0 , x0 ) ∈ T × X. Remark 1.4. If T = Z, then continuity in t in iii) is with respect to the discrete topology, so we essentially just have continuity in x0 for each t. For many differential equations the solutions may exist forward in time, but not backwards in time. Example 1.3. Consider the scalar differential equation
dx = x 1 − x2 dt with the explicit solution (see Fig. 1.2) x(t, x0 ) = p

x0 et 1 + x20 (e2t − 1)

.

The solution mapping x : R+ × R1 → R1 satisfies the semi-group rather than group property since the solutions do not exist for all negative time when |x0 | > 1. A similar situation arises if the mapping f in the difference equation (1.1) is not invertible.

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These examples motivate the definition of an abstract semi-dynamical system π on the state space X, in which the time set T is replaced by T+ ∪ {0} and the group property (ii) is replaced by the semi-group property over T+ ∪ {0}. Definition 1.3. An autonomous semi-dynamical dynamical system on a state space X is given by mapping π : T+ ∪ {0} × X → X, which satisfies the properties: i) initial condition: π(0, x0 ) = x0 for all x0 ∈ X, ii) semi-group under composition: π(s + t, x0 ) = π(s, π(t, x0 )) for all s, t ∈ T+ ∪ {0} and x0 ∈ X, iii) continuity: the mapping (t, x) 7→ π(t, x) is continuous at all points (t0 , x0 ) ∈ T+ ∪ {0} × X. 1.5

Invariant sets, omega limit sets and attractors

We are interested in the long term, i.e., asymptotic, behaviour of the system, so we will consider a semi-dynamical system π on a complete metric state space (X, dX ). The invariant sets in X w.r.t. π provide us with a lot of useful information about the dynamical behaviour of π. Definition 1.4. A nonempty subset A of X is called an invariant set of a semidynamical system π if for all t ∈ T+ ,

π(t, A) = A where π(t, A) :=

[

{π(t, a)}.

a∈A

Example 1.4. Consider the scalar ODE
dx = x 1 − x2 , (1.3) dt which generates semi-dynamical system on X = R. It has steady state (i.e., constant) solutions x ¯ = 0, ±1. These steady states are invariant sets as are combinations of them: {0}, {−1}, {1}, {−1, 1}, {0, 1}, {0, −1}, {0, −1, 1}. Other closed invariant sets are the intervals [−1, 0], [0, 1], [−1, 1]. One could also consider unions of intervals and disjoint steady states such as {−1} ∪ [0, 1]. In addition, one has invariant sets such as (−∞, −1] and [1, ∞). The set [−1, 1] is the largest compact invariant set for this example. It has a special significance. 1.5.1

Omega-limit sets

The ω-limit sets of a semi-dynamical system characterise its asymptotic behaviour as t → ∞.

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x

2 1.5 1

1

0.5 t

0

0 −0.5

−1

−1 −1.5 −2 −0.5

Fig. 1.2

0

0.5

1

1.5

2

Steady states and typical solutions of the ODE (1.3).

Definition 1.5. (Omega-limit sets) The ω-limit set of a bounded set B ⊂ X is defined by ω(B) = {x ∈ X : ∃tk → ∞, yk ∈ B with π(tk , yk ) → x} . When B = {y}, i.e., just one point, one usually writes ω(y) instead of ω({y}). In general, [ ω(B) ) ω(y). y∈B

The ω-limit sets have the following properties. Theorem 1.1. For a nonempty bounded subset B of X, \ [ ω(B) = π(s, B) . t≥0 s≥t

The next theorem is stated for the state space X = Rd . A similar result holds in more general state spaces provide π satisfies an appropriate compactness property, see Remark 1.5. S Theorem 1.2. Let π be a semi-dynamical system on Rd . If t≥0 π(t, B) is bounded, then the ω-limit set ω(B) for every nonempty bounded set B ⊂ X is nonempty, compact and invariant. Note in Theorem 1.2 that ω(B) is connected for continuous time systems when B is connected. It need not, however, be connected for discrete time systems, e.g., consider the autonomous difference equation with f (x) = −x, so f n (−1) = (−1)n+1 . 1.5.2

Attractors

An attractor is an invariant set of special interest since contains all the long term dynamics of a dissipative dynamical system, i.e., it is where every thing ends up. In particular, it contains the omega limit set ω(D) of every nonempty bounded

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subset D of the state space X. In addition, an attractor is the omega limit set of a neighbourhood of itself, i.e., it attracts a neighbourhood of itself. This additional stability property distinguishes an attractor from omega limit sets in general. The distance of a point x from a subset A of X, where (X, dX ) is a complete metric space, is defined by distX (x, A) := inf dX (x, a), a∈A

where the infimum is attained and can be replaced by the minimum when A is a compact set. The distance of a compact set B from a compact set A is then defined by distX (B, A) := max distX (b, A) = max inf dX (b, a). b∈B a∈A

b∈B

In general, distX (A, B) 6= distX (B, A), for example,

0 = distR1 ({0}, [0, 1]) 6= distR1 ([0, 1], {0}) = 1

for the subsets A = {0} and B = [0, 1] of R1 .

Definition 1.6. A (global) attractor of a semi-dynamical system π is nonempty compact invariant set A of X which attracts all nonempty bounded subsets D of X, i.e., distX (π(t, D), A) → 0

as t → ∞.

The global attractor in Example 1.4 is A = [−1, 1]. It is the largest compact invariant set that attracts all other nonempty bounded subsets. The set {1} is a local attractor in this example since it attracts only bounded subsets of the positive half line. We will mainly talk about global attractors and often omit the adjective global. An attractor may have a very complicated geometrical shape, e.g., the fractal dimensional set in the Lorenz ODE system. It is often easier to determine an absorbing set with a simpler geometrical shape such as a ball. Definition 1.7. A nonempty subset B of X is called an absorbing set of π if for every nonempty bounded subset D of X there exists a TD ≥ 0 such that for all t ≥ TD .

π(t, D) ⊂ B

All of the future dynamics is in B, which need not be invariant, but often it is positively invariant, i.e., π(t, B) ⊂ B for all t ∈ T+ . Theorem 1.3. If a semi-dynamical system π on a complete metric space (X, dX ) has a compact absorbing set B ⊂ X, then it has an attractor A, which is contained in B and is given by \[ A= π(s, B). t≥0 s≥t

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If B is positively invariant then A=

\

π(t, B).

t≥0

The proof is a special case on the corresponding proof for nonautonomous dynamical systems that will be given later. Note that the sets [ π(s, B) At := s≥t

are compact as closed subsets of the compact set B (provided t is large enough) and nested, i.e., As ⊂ At if s ≥ t. This means that A exists and is a nonempty compact subset of B. We will say more about attractors of autonomous dynamical systems in Chapter 5 Remark 1.5. Theorem 1.3 is useful in locally compact state spaces such as Rd where the closed and bounded subsets are the compact subsets. In infinite dimensional dynamical systems the absorbing sets are usually assumed to be just closed and bounded, which are more common and easily determined than compact sets in such spaces. Some additional compactness property of the semi-group π such as its asymptotic compactness is then needed to ensure the nonemptiness of the attractor in Theorem 1.3. Definition 1.8. A semi-dynamical system π on a complete metric space (X, dX ) is said to be asymptotically compact if, for every sequence {sk }k∈N in R+ with sk → ∞ as k → ∞ and every bounded sequence {xk }k∈N in X, the sequence {π(sk , xk )}k∈N has a convergent subsequence.

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Chapter 2

Nonautonomous dynamical systems: processes

The properties of the solution mappings of nonautonomous difference and differential equations are considered here to motivate the formulation of a nonautonomous dynamical system as a process of 2-parameter semi-group. 2.1

Nonautonomous difference equations

A nonautonomous difference equation on Rd has the form xn+1 = fn (xn ), where the fn : Rd → Rd are continuous mappings. The mappings fn may be completely different from each other, but in concrete applications they are often similar but with different parameters. For example, let d = 1 and fn (x) = an x(1 − x), where an ∈ R+ . Here xn+1 = an xn (1 − xn ), is a population growth model with a (possibly) different growth rate an each year. As another example, the Euler scheme with different step sizes ∆n > 0 for an autonomous ODE dx = f (x), dt

x ∈ Rd ,

is given by xn+1 = xn + f (xn )∆n . Here fn (x) = x + f (x)∆n with the parameter ∆n . In general, the mappings fn need not be invertible. 13

(2.1)

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The solution mapping Consider a nonautonomous difference equation xn+1 = fn (xn ) on Rd with the initial value xn0 = x0 . Then xn0 +1 = fn0 (xn0 ) and xn0 +2 = fn0 +1 (xn0 +1 ) = fn0 +1 (fn0 (xn0 )) = fn0 +1 ◦ fn0 (xn0 ) . | {z } composition

Similarly, xn = fn−1 ◦ · · · ◦ fn0 (xn0 ) | {z } composition

for all n ≥ n0 + 1. This defines a solution mapping φ(n, n0 , x0 ) := fn−1 ◦ · · · ◦ fn0 (xn0 ) for all n ≥ n0 + 1 in Z and x0 ∈ Rd with φ(n0 , n0 , x0 ) := x0 . The solution mappping φ(n, n0 , x0 ) has the following properties: 1)

initial condition φ(n0 , n0 , x0 ) = x0 . (This is part of the definition above.)

2)

2-parameter semi-group under composition φ(n2 , n0 , x0 ) = φ(n2 , n1 , φ(n1 , n0 , x0 ))

for all n0 ≤ n1 ≤ n2 in Z and x0 ∈ Rd . This follows from the associativity of composition φ(n2 , n0 , x0 ) = fn2 −1 ◦ · · · ◦ fn0 (xn0 ) = fn2 −1 ◦ · · · ◦ fn1 (fn1 −1 ◦ · · · ◦ fn0 (xn0 )) = φ(n2 , n1 , fn1 −1 ◦ · · · ◦ fn0 (xn0 )). | {z } =xn1

3)

continuity x0 7→ φ(n1 , n0 , x0 ) is continuous for all n1 ≥ n0 . This follows from the continuity of the composition of continuous functions x 7→ fn1 −1 ◦ · · · ◦ fn0 (x),

n1 > n0 .

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Nonautonomous dynamical systems: processes

2.2

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Nonautonomous differential equations

Consider a nonautonomous ODE on Rd dx = f (t, x), t ∈ R, x ∈ Rd , dt with initial value x(t0 ) = x0 . Assume the existence and uniqueness of the solutions and their continuity in initial conditions. In particular, the solution mapping x(t) = x(t, t0 , x0 ) is then defined for all x0 ∈ Rd and t ≥ t0 in R. (This is guaranteed by appropriate assumptions on the vector field f , e.g., if it is continuous in both variables and satisfies a global Lipschitz condition in the second variable uniformly in the first, though weaker conditions are possible.) Example 2.1. The nonautonomous ODE on R, dx = 2tx dt has the solution x(t, t0 , x0 ) = x0 et

2

−t20

.

Unlike as an autonomous ODE, the solution here depends explicitly on both t and t0 , and not just on the elapsed time since starting t − t0 . In particular, here t2 − t20 = (t − t0 )(t + t0 ) cannot be written in terms of t − t0 only. The solution mappping x(t, t0 , x0 ) has the following properties: 1)

initial condition: x(t0 , t0 , x0 ) = x0 .

2)

2-parameter semi-group under composition: x(t2 , t0 , x0 ) = x(t2 ; t1 , x(t1 , t0 , x0 ))

for all t0 ≤ t1 ≤ t2 in R. This is a consequence of the existence and uniqueness of solutions. The solution starting at (t1 , x1 ), where x1 = x(t1 , t0 , x0 ) is unique, so we have x(t2 , t0 , x0 ) = x(t2 , t1 , x1 ) = x(t2 , t1 , x(t1 , t0 , x0 )), This is called the 2-parameter semi-group property , i.e., with parameters t and t0 instead of just the elapsed time t − t0 in an autonomous system. 3)

continuity: (t, t0 , x0 ) 7→ x(t, t0 , x0 ) is continuous.

The solution of the ODE is differentiable in t, hence continuous. From the theory of differential equations we know that it is also continuous in the initial condition (t0 , x0 ).

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Fig. 2.1 Under the 2-parameter semi-group property the dashed full solution and the concatenated solutions coincide.

2.3

Abstract nonautonomous dynamical systems

We will consider two abstract formulations of nonautonomous dynamical systems in this book. The first is a more direct generalisation of the definition of an abstract autonomous semi-dynamical system and is based on the properties of the solution mappings of nonautonomous differential and difference equations. It is called a process or 2-parameter semi-group. The other case, called a skew product flow, will be introduced in Chapter 9. Instead of Rd we consider a complete metric space (X, dX ) as the state space. In addition, we consider a time set T, which is Z in the discrete-time case and R in the continuous-time case. Similarly, T+ , denotes Z+ and R+ in these time cases. Define T2≥ := {(t, t0 ) ∈ T × T : t ≥ t0 } . Definition 2.1. (Process) A process is a mapping φ from T2≥ × X → X with the following properties: i) initial condition: φ(t0 , t0 , x0 ) = x0 for all x0 ∈ X and t0 ∈ T. ii) 2-parameter semi-group property: φ(t2 , t0 , x0 ) = φ(t2 , t1 , φ(t1 , t0 , x0 )) for all (t1 , t0 ), (t2 , t1 ) ∈ T2≥ and x0 ∈ X. iii) continuity: the mapping (t, t0 , x0 ) 7→ φ(t, t0 , x0 ) is continuous. The continuity (t, t0 , x0 ) 7→ φ(t, t0 , x0 ) in the discrete-time case T = Z reduces to continuity x0 7→ φ(t, t0 , x0 ) in x0 for all t ≥ t0 in T. Remark 2.1. We can consider a process φ as a 2-parameter family of mappings φt,t0 (·) on X that forms a 2-parameter semi-group under composition, i.e., for all t0 ≤ t1 ≤ t2 in T.

φt2 ,t0 (x) = φt2 ,t1 ◦ φt1 ,t0 (x)

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Remark 2.2. For an autonomous system, a process reduces to π(t − t0 , x0 ) = φ(t, t0 , x0 ),

since the solutions depend only on the elapsed time t − t0 , i.e., just one parameter instead of independently on the actual time t and the initial time t0 , i.e., two parameters. 2.4

A process as an autonomous semi-dynamical system

We can consider a process φ on a state space X and time set T as an autonomous semi-dynamical system on the extended state space X := T × X. Define Π : T+ × X → X by

Π(τ, (t0 , x0 )) := (τ + t0 , φ(τ + t0 , t0 , x0 ))

for all t ∈ T and (t0 , x0 ) ∈ X. i) initial condition:

For τ = 0 we have Π(0, (t0 , x0 )) := (t0 , φ(t0 , t0 , x0 )) = (t0 , x0 ),

which is just the initial condition property. ii) semi-group property: For σ, τ ∈ T+ , using the 2-parameter semi-group property in the second to third line, we have Π(σ + τ, (t0 , x0 )) = (σ + τ + t0 , φ(σ + τ + t0 , t0 , x0 )) = (σ + (τ + t0 ), φ(σ + (τ + t0 ), τ + t0 , φ(τ + t0 , t0 , x0 )) = Π(σ, (τ + t0 , φ(τ + t0 , t0 , x0 )) = Π(σ, Π(τ, (t0 , x0 )). This is the semi-group property for Π on the extended state space X. iii) continuity : functions, i.e.,

The continuity of Π follows directly from that of the constituent

(τ, (t0 , x0 )) −→ Π(τ, (t0 , x0 )) := (τ + t0 , φ(τ + t0 , t0 , x0 )) . Remark 2.3. This autonomous semi-dynamical system Π on the state space X is not very useful in practice since it has no compact invariant sets and hence cannot have an attractor. To see this note that if Π(t, A) = A for all t ∈ T+ then +

(t + t0 , φ(t + t0 , t0 , x0 )) = Π(t, (t0 , x0 )) ∈ A

for all t ∈ T and (t0 , x0 ) ∈ A, so A is unbounded in its first component and hence is not compact. Nevertheless, this example will give us insights on how to define invariant sets and attractors for a process. It is a simple (but a not very interesting) example of a skew-product flow, which will be introduced in the next chapter.

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Chapter 3

Skew product flows

Skew product flows are a more complicated formalism of nonautonomous dynamical systems than processes, but they have more detailed information built into them and so one can often obtain more specific results. Autonomous triangular systems will first be examined to motivate the definition of a skew product flow. 3.1

Autonomous triangular systems

Consider a system of autonomous ODEs on Rm × Rd with the triangular structure dp = g(p), p ∈ Rm , (3.1) dt dx = f (x, p), x ∈ Rd , p ∈ Rm . (3.2) dt Assume that forward existence and uniqueness of solutions holds as well as their continuity in initial values. Then the system of ODEs (3.1)–(3.2) generates an autonomous semi-dynamical system Π (t, (p0 , x0 )) = (p(t, p0 ), x(t, p0 , x0 )) m

d

on R × R .

Note that the p-system (3.1) is independent of the x system and generates an autonomous semi-dynamical system on Rm . We can consider the p-system as the driving system in the x-system, which is then nonautonomous dx = f˜p0 (x, t) = f (x, p(t, p0 )) (3.3) dt for each p0 ∈ Rm . We now want to know what the dynamics of this nonautonomous system on Rd looks like. Denote the solution mapping of the nonautonomous ODE (3.3) by ϕ(t, p0 , x0 ) given the 1) initial condition:

ϕ(0, p0 , x0 ) = x0 for each p0 and x0 .

We obviously also have 19

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2) continuity condition:

(t, p0 , x0 ) 7→ ϕ(t, p0 , x0 ) is continuous.

What kind of evolution property does ϕ satisfy? e.g., like a group or 2-parameter semi-group property: ∀s, t ∈ R+ , p0 ∈ Rm , x0 ∈ Rd .

ϕ(s + t, p0 , x0 ) = ??

Let us look at the corresponding autonomous semi-dynamical system on Π (t, (p0 , x0 )) = (p(t, p0 ), x(t, p0 , x0 )) m

d

on R × R . Its semi-group property reads: Π (s + t, (p0 , x0 )) = Π (s, Π (t, (p0 , x0 ))) , so from the definition of Π we have (p(s + t, p0 ), x(s + t, p0 , x0 )) = Π (s + t, (p0 , x0 )) = Π (s, Π (t, (p0 , x0 ))) = Π (s, (p(t, p0 ), x(t, p0 , x0 ))) = (p(s, p(t, p0 )), x (s, p(t, p0 ), x(t, x0 , p0 ))) , i.e., with the new initial condition (p(t, x0 ), x(t, p0 , p0 )) at time t. Now the second components are equal by the semi-group property of the driving system, i.e., p(s + t, p0 ) = p(s, p(t, p0 )). Comparing the first components we have x (s + t, p0 , x0 ) = x (s, p(t, p0 ), x(t, p0 , x0 )) . The evolution property of the solution mapping ϕ(t, p0 , x0 ) of the nonautonomous ODE (3.3) thus reads ϕ (s + t, p0 , x0 ) = ϕ (s, p(t, p0 ), ϕ(t, p0 , x0 )) , which is called the cocycle property. Remark 3.1. Note that the driving system is updated at time t to p(t, p0 ) when the system moves to ϕ(t, p0 , x0 ). Example 3.1. Consider a system of ODEs (3.2)–(3.1) with m = d = 1, i.e., dp dx = p(1 − p), = −x + p, dt dt with the explicit solutions p0 et p(t, p0 ) = 1 + p0 (et − 1) and Z t x(t, x0 , p0 ) = x0 e−t + e−t es p(s, p0 ) ds. 0

Let us restrict the p-system to the invariant set [0, 1]. (Note that p¯ = 0 and 1 are steady state solutions.) Then the p-system is a group, i.e., reversible dynamical system on [0, 1]. What happens to x(t)?

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Skew product flows

3.2

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Definition of a skew product flow

A skew product flow consists of an autonomous dynamical system (full group) on a base space P , which is the source of the nonautonomity in a cocycle mapping acting on a state space X. The autonomous dynamical system is often called the driving system. Suppose that (P, dP ) and (X, dX ) are complete metric spaces and consider the time set T. Definition 3.1. A skew product flow (θ, ϕ) on P × X consists of an autonomous dynamical system θ = {θt }t∈T acting on a complete metric space (P, dP ), which is called the base space or hull, i.e., ii) θs+t (p) = θs ◦ θt (p),

i) θ0 (p) = p,

iii) (t, p) 7→ θt (p) continuous

for all p ∈ P and s, t ∈ T, and a cocycle mapping ϕ : T+ × P × X → X acting on a complete metric space (X, dX ), which is called the state space, i.e., 1) initial condition:

ϕ(0, p, x) = x for all p ∈ P and x ∈ X,

2) cocycle property:

for all s, t ∈ T+ , p ∈ P and x ∈ X, ϕ(s + t, p, x) = ϕ(s, θt (p), ϕ(t, p, x)),

3) continuity:

Fig. 3.1

(t, p, x) 7→ ϕ(t, p, x) is continuous.

Under the cocyle property the full solution and the concatenated solutions coincide.

Remark 3.2. The base system θ serves as a driving system which makes the cocycle mapping nonautonomous. Skew product flows often have very nice properties when, in particular, the base space P is compact. This occurs when the driving system is, for example, periodic or almost periodic. It provides more detailed information about the dynamical behaviour of the system. George Sell, a pioneering researcher in the area, e.g., see [Sell (1971)], described the effect of a compact base space as being equivalent to compactifying time.

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3.2.1

Processes as skew product flows

Consider a process φ on a state space X with time set T, so φ : T+ ≥ × X → X. For each t ∈ T define θt : T → T for each t ∈ T by θt (t0 ) = t + t0 . The family {θt }t∈T forms a group under addition on T and is an autonomous dynamical system on P = T. In addition, define ϕ(t, t0 , x0 ) := φ(t + t0 , t0 , x0 ), +

where t ∈ T is the time that has elapsed since starting at t0 ∈ T in ϕ, while t + t0 and t0 are actual times in φ. Remark 3.3. Here and in what follows we use φ to denote a process and ϕ to denote a skew product flow. The 2-parameter semi-group property φ(s + t + t0 , t0 , x0 ) = φ (s + t + t0 , t + t0 , φ(t + t0 , t0 , x0 )) of the process φ becomes the cocycle property ϕ(s + t, t0 , x0 ) = ϕ (s, θt (t0 ), ϕ(t, t0 , x0 )) of the skew product flow ϕ because ϕ(s + t, t0 , x0 ) = φ(s + t + t0 , t0 , x0 ) and ϕ(s, θt (t0 ), ϕ(t, t0 , x0 )) = φ (s + t + t0 , t + t0 , φ(t + t0 , t0 , x0 )) . Here the base space P = T is locally compact, but not compact. 3.3

Examples

The following examples illustrate how the driving system of the skew product flow formalism often provides more specific, built-in information about the evolution of the system than the process formalism does. 3.3.1

Autonomous triangular system

As in Example 3.1 consider the system of scalar ODEs dp = p(1 − p), dt

dx = −x + p, dt

with the initial values p(0) = p0 ,

x(0) = x0 ,

(3.4)

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and denote the solutions by p(t) = p (t, p0 ) ,

x(t) = x (t, p0 , x0 ) .

Here t is the time since starting at t0 = 0. Note that the p-ODE is decoupled from the x-ODE. Its solutions form an autonomous dynamical system on P = [0, 1], but only a semi-dynamical system on R+ (For p0 < 0 the solutions blow up in finite positive time). Define θt (p0 ) = p(t, p0 ) for all t ∈ R and P = [0, 1]. These form a group under composition on P = [0, 1]. Define ϕ (t, p0 , x0 ) = x (t, p0 , x0 ). Then ϕ satisfies the cocycle property on X = R with respect to θt , i.e., ϕ (s + t, p0 , x0 ) = x (s + t, p0 , x0 ) = x (s, p(t, p0 ), x (t, p0 , x0 ))

uniqueness of solutions

= ϕ (s, θt (p0 ), ϕ (t, p0 , x0 )) . The second line is due to the existence and uniqueness of solutions starting at (θt (p0 ), ϕ (t, p0 , x0 )) at time t. Note that the base space P = [0, 1] is compact here. 3.3.2

Driving system need not be an autonomous ODE

Consider the scalar ODE dx = −x + cos t. dt This is like the example in Subsection 3.3.1 with p(t) = cos t. The main difference here is that this p(t) is given directly and not as the solution of a first order ODE.1 Nevertheless, we can define a driving system as the shift operator θt (f (·)) := f (t + ·),

t ∈ R,

on the function space (C(R, R), k · k∞ ) of continuous functions f : R → R with the uniform norm kf k∞ := max |f (t)|, t∈R

f ∈ C(R, R).

Note that the shift operator is an isomorphism on (C(R, R), k · k∞ ), i.e., kθt (f ) − θt (g)k∞ = kf (t + ·) − g(t + ·)k∞

= max |f (t + s) − g(t + s))| s∈R

= max |f (s) − g(s))| = kf − gk∞ . s∈R

Lemma 3.1. The mapping θt is continuous in (C(R, R), k · k∞ ) for each t ∈ R. 1 It

does, however, satisfy a second order ODE, but that is not relevant here.

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Proof. (Sketch)

This follows directly from the isometry property, i.e., kθt (fn ) − θt (f )k∞ = kfn − f k∞ .



For the base space P we take the closed subset of C(R, R) consisting of phase shifts of the cosine function, i.e., [ P := {cos(τ + ·)} , (3.5) 0≤τ ≤2π

which is invariant under the shift operator. This set P is not a linear subspace of C(R, R), but we can define the induced metric on it by dP (f, g) := kf − gk∞ ,

f, g ∈ P,

so (P, dP ) is a complete metric space. The compactness of (P, dP ) follows from the Ascoli Theorem since P is a closed subset of equi-bounded equi-continuous functions in (C(R, R), k · k∞ ). Corollary 3.1. The mapping θt is continuous in (P, dP ) for each t ∈ R. Remark 3.4. In view of the periodicity of the driving system, the parameter set P in (3.5) is homeomorphic to a unit circle. The driving system could alternatively be defined more simply and elegantly on such a circle. The reader should try to show this. We call the set P defined by (3.5) the hull of the function cos(·). This forms the base space of the driving system {θt }t∈R . Exercise 3.1. Show that (t, p0 ) 7→ θt (p0 ) is continuous in R × P . The cocycle mapping here is given by ϕ(t, p0 , x0 ) := x(t, p0 , x0 ), the solution mapping of the ODE dx = −x + θt (p0 ) dt with the initial value x(0) = x0 , i.e., Z t ϕ(t, p0 , x0 ) = x0 e−t + e−t es cos(τ0 + s) ds, 0

where p0 (·) = cos(τ0 + ·) corresponding to the phase τ0 ∈ [0, 2π], so (θt (p0 )) (·) = θt (cos(τ0 + ·)) = cos(τ0 + t + ·).

Exercise 3.2. Show that the mapping p0 7→ ϕ(t, p0 , x0 ) continuous. (Hint: The proof depends on uniform continuity and convergence under the integral sign.)

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3.3.3

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Nonautonomous difference equations

Consider a nonautonomous difference equation on X = R given by n ∈ Z,

xn+1 = fjn (xn ) ,

(3.6)

where the functions f1 , . . . , fN are given and the jn are chosen from {1, . . . , N } in some manner. Specifically, the jn are the components of a bi-infinite sequence s = (. . . , j−1 , j0 , j1 , j2 , . . .) . Let P := {1, . . . , N } be the collection of all such bi-infinite sequences. Then ∞ X dP (s, s0 ) := 2−|n| |jn − jn0 | Z

n=−∞

defines a metric on P . Lemma 3.2. (P, dP ) is a compact metric space. Define θn = θn , the n-fold composition of θ when n > 0 and of its inverse θ−1 when n < 0, where θ is the left shift operator of P , i.e., (θ(s))n = jn+1 ,

n ∈ Z.

Lemma 3.3. θ : P → P is Lipschitz continuous in (P, dP ). This follows from the definition and ∞ ∞ X X 0 = dP (θ(s), θ(s0 )) = 2−|n| jn+1 −jn+1 2−|k−1| |jk −jk0 | with k = n+1 n=−∞

≤2

∞ X k=−∞

k=−∞

2−|k| |jk −jk0 | = 2dP (s, s0 ) .

The cocycle mapping for n ≥ 1 here is

ϕ (n, s, x0 ) := fjn−1 ◦ · · · ◦ fj0 (x0 ) .

It is clear that the mapping x0 7→ ϕ (n, s, x0 ) is continuous due to the continuity of the composition of continuous functions. What about the continuity of the mapping s 7→ ϕ (n, s, x0 )?

(k)

Hint: If s(k) → s in (P, dP ), then after a finite k the first n indices jl = jl for l = 0, 1, . . . , n − 1 for k large enough, i.e., after a finite number of steps the first n indices are all the same. Remark 3.5. Note that for each index sequence s ∈ P , the corresponding nonautonomous difference equation (3.6) generates particular process. In contrast, the skew product flow (θ, ϕ) here includes the dynamics of all such processes. In this sense its dynamics can be considered to be richer.

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3.4

An Introduction to Nonautonomous Dynamical Systems and their Attractors

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An Introduction to Nonautonomous Dynamical Systems and their Attractors

Skew product flows as autonomous semi-dynamical systems

Just as a process can be reformulated as an autonomous semi-dynamical system on a product space, so can a skew product flow. Theorem 3.1. A skew product flow (θ, ϕ) on P × X is an autonomous semidynamical system Π on X = P × X. Proof.

Define Π : T+ × X → X by Π (t, (p0 , x0 )) := (θt (p0 ), ϕ(t, p0 , x0 )) .

The initial condition property of an semi-dynamical system is obtained by setting t = 0 and using the initial condition properties of θ and ϕ: Π (0, (p0 , x0 )) = (θ0 (p0 ), ϕ(0, p0 , x0 )) = (p0 , x0 ) . The semi-group property follows from that of θ and the cocycle property of ϕ: Π (s + t, (p0 , x0 )) = (θs+t (p0 ), ϕ(s + t, p0 , x0 )) = (θs ◦ θt (p0 ), ϕ(s, θt (p0 ), ϕ(t, p0 , x0 ))) = Π (s, (θt (p0 ), ϕ(t, p0 , x0 ))) = Π (s, Π (t, (p0 , x0 ))) . Finally, the continuity of the mapping (t, p0 , x0 ) 7→ Π (t, (p0 , x0 )) = (θt (p0 ), ϕ(t, p0 , x0 )) is inherited from that of its constituent functions.



Remark 3.6. Unlike the semi-dynamical system on the product space X = T × X generated by a process, the semi-dynamical system on the product space X = P ×X generated by a skew product flow can have compact invariant sets, in particular when the base space P is compact. In this case the product semi-dynamical system may be able to provide useful information.

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Chapter 4

Entire solutions and invariant sets

Invariant sets and bounded entire solutions provide much useful information about the dynamical behaviour of a dynamical system. Entire solutions are sometimes called complete or full solutions in the literature. 4.1

Autonomous case

Consider an autonomous semi-dynamical system π on a complete metric state space (X, dX ) with the time set T. Definition 4.1. An entire solution of a semi-dynamical system π on a state space X with the time set T is a mapping e : T → X with the property that for all s, t ∈ T with t ≥ s.

e(t) = π(t − s, e(s))

(4.1)

Note that t − s ∈ T+ , the time set on which the semi-dynamical system π is defined. However, the entire solution e is defined for all t ∈ T, not just in T+ . Example 4.1. Consider the continuous time semi-dynamical system on X = R defined by the solutions of the initial value problem dx = −x3 , x(0) = x0 , (4.2) dt i.e., x0 1 for all t > − 2 . x(t, x0 ) = p 2 2x 1 + 2x0 t 0 These solutions exist only on the semi-infinite interval (− 2x1 2 , ∞). The time t = 0

− 2x1 2 is a vertical asymptote. 0 The only entire solution is e(t) ≡ 0.

The semi-dynamical system here is defined by π(t, x0 ) = x(t, x0 ) for all t ≥ 0 and x0 ∈ R (even though all solutions can be extended backwards at least for a short time depending on the initial value x0 . 27

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x

4 3.5 3 2.5 2 1.5

x0

1

t

0

0.5

t = − 2x1 2 0 a vertical asymptote

0 −0.5 −1

−0.5

Fig. 4.1

0

0.5

1

1.5

2

Typical solution of the ODE (4.2).

It is clear that steady state, i.e., constant solutions are entire solutions, but they are not the only possibility. Example 4.2. The initial value problem dx = 4x(1 − x), dt has the explicit solution

x(0) = x0 ,

(4.3)

x0 e4t . 1 − x0 + x0 e4t Its steady state solutions are x ¯(t) ≡ 0 and x ¯(t) ≡ 1 for all t ∈ R. These are obviously entire solutions. In addition all solutions with x0 ∈ (0, 1) exist for all time, so are also entire solutions. x(t, x0 ) =

x

3 2.5 2 1.5 x=1

1 0.5

t

x=0

0

0 −0.5 −1 −1.5 −2 −1

−0.5

Fig. 4.2

0

0.5

1

1.5

2

2.5

3

Typical solutions of the ODE (4.3).

For x0 > 1 the solutions exist for all t ≥ 0, but have a vertical asymptote for a finite negative time depending on x0 . Similarly, for x0 < 0 the solutions exist for all t ≤ 0 but have a vertical asymptote at a finite positive time that depends on x0 . Note that the solution mapping x(t, x0 ) in the above ODE defines an autonomous semi-dynamical system (semi-group) on the state space X = R+ , but not on the

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whole of R. However, it defines an autonomous dynamical system (group) on X = [0, 1]. The steady state solutions of a semi-dynamical system, i.e., with π(t, x ¯) = x ¯ for all t ∈ R, are obvious examples of entire solutions. The singleton subset A = {¯ x} then satisfies π(t, {¯ x}) = {¯ x} for all t ∈ T+ . This is a special case of the following definition. Definition 4.2. A subset A of X is called invariant (with respect to a semidynamical system π) if π(t, A) = A

for all t ∈ T+ ,

where π(t, A) :=

[ a∈A

{π(t, a)}.

Thus π maps A onto itself for each t ∈ T+ . The set A = [0, 1] in Example 4.2 is example of such a set. There is a weaker version of this property, where π just maps A into itself. Definition 4.3. A subset A of X is called positively invariant (with respect to a semi-dynamical system π) if π(t, A) ⊂ A

for all t ∈ T+ .

Example 4.3. The set A = [0, 1] in Example 4.2 is invariant, while the sets A = [0, 1 + r] for r > 0 are only positively invariant. Invariant sets are made up of entire solutions. Lemma 4.1. Let A be an invariant set w.r.t. a semi-dynamical system π. Then for every a ∈ A there exists an entire solution ea : T → A with ea (0) = a. Proof: Invariant means that π(t, A) = A for all t ∈ T+ . In particular, given a ∈ A, then π(t, a) ∈ A for t ∈ T+ . Define ea (t) := π(t, a) for t ∈ T+ . In particular, π(1, A) = A, so there exists an a−1 ∈ A such that π(1, a−1 ) = a. Define ea (t) := π(1 + t, a−1 ) for t ∈ [−1, 0] ∩ T. Obviously, then ea (−1) = π(0, a−1 ) = a−1 ,

ea (0) = π(1, a−1 ) = a

and ea (t) = π(1 + t, a−1 ) ∈ π(1 + t, A) = A for all t ∈ [−1, 0] ∩ T.

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We repeat this argument for t ∈ [−n − 1, −n] ∩ T and construct a mapping ea : T → A with ea (0) = a and ea (t) ∈ A for all t ∈ T.

It remains to show that ea (t) satisfies the relationship (4.1). This follows from the definition of ea (t) and the semi-group property of π.  Invariant subsets are important as most of the long term dynamics is characterised by them. 4.2

Nonautonomous case

Now consider a 2-parameter semi-group or process φ on the metric state space X and time set T. An entire solution of a process is defined analogously to an entire solution of an autonomous semi-dynamical system. Definition 4.4. An entire solution of a process φ on a state space X with the time set T is a mapping e : T → X with the property that e(t) = φ(t, t0 , e(t0 )) for all (t, t0 ) ∈ T2≥ := {(t, t0 ) ∈ T × T : t ≥ t0 }. Steady state solutions are entire solutions, but there may be other interesting bounded entire solutions. Example 4.4. All solutions of the ODE dx = −x + cos t dt are entire solutions, but the only bounded entire solution is 1 1 cos t + sin t. 2 2 Moreover, x ¯(t) is a periodic solution, in fact, the only periodic solution. This ODE has no steady state solutions. x ¯(t) =

Nonautonomous invariant sets The analogue of an invariant set for an autonomous semi-dynamical system is too restrictive for a process, i.e., a subset A of X such that φ(t, t0 , A) = A

for all (t, t0 ) ∈ T2≥ .

Such a subset contains steady state solutions if there are any, but excludes almost everything else such as the periodic solution in the previous example. How then should we define an invariant set for a nonautonomous process?

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For this it is useful to consider the autonomous semi-dynamical system Π on X = T × X defined in terms of a process φ on X with time set T (see Section 2.4 of Chapter 2) by Π(τ, (t0 , x0 )) = (τ + t0 , φ(τ + t0 , t0 , x0 )) . Proposition 4.1. Let A ⊂ X = T × X be a nonempty invariant set of the autonomous semi-dynamical system Π on X, i.e., Π(τ, A) = A for all τ ∈ T+ . S Then A = t∈T {t} × At , where At is a nonempty subset of X for each t ∈ T. Proposition 4.2. Let At , t ∈ T, be the nonempty subsets of X given in Proposition 4.1. Then φ (t, t0 , At0 ) = At

for all (t, t0 ) ∈ T2≥ .

We will prove these Propositions at the end of the section. Proposition 4.2 gives us a hint how to define an invariant set in the nonautonomous case. Definition 4.5. A family A = {At , t ∈ T} of nonempty subsets At of X is called invariant with respect to a process or φ-invariant if φ (t, t0 , At0 ) = At

for all (t, t0 ) ∈ T2≥ .

For example, for a steady state solution, i.e., with φ (t, t0 , x ¯) ≡ x ¯ for all (t, t0 ) ∈ T2≥ , the family of identical singleton subsets At ≡ {¯ x} for all t ∈ T is φ-invariant. In Example 4.4, the family of singleton sets   1 1 At = cos t + sin t , t ∈ R, 2 2 formed by the periodic solution is φ-invariant. The proof of the following Lemma is analogous to that of Lemma 4.1 in the autonomous case. Lemma 4.2. Let A = {At , t ∈ T} be a φ-invariant family of subsets of X. Then for any a0 ∈ At0 and any t0 ∈ T there exists an entire solution ea0 ,t0 : T → X of φ such that ea0 ,t0 (t0 ) = a0 and ea0 ,t0 (t) ∈ At

for all t ∈ T.

Proof of Proposition 4.1 The set A ⊂ X = T × X is nonempty by assumption, so there is at least one point (t0 , a0 ) ∈ A, which means that a0 ∈ At0 . Hence At0 is nonempty. Then At1 is nonempty for every t1 > t0 , since φ(t1 , t0 , a0 ) ∈ At1 . This follows by Π-invariance of A, i.e., Π(t1 − t0 , A) = A and the fact that Π(t1 − t0 , (t0 , a0 )) = (t0 + (t1 − t0 ), φ(t0 + (t1 − t0 ), t0 , a0 )) = (t1 , φ(t1 , t0 , a0 )) ∈ A

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so φ(t1 , t0 , a0 ) ∈ At1 . Hence At1 is nonempty for t1 > t0 . Now for t1 < t0 , by invariance Π(t0 − t1 , A) = A, so there exists a (t∗ , a∗ ) ∈ A such that Π(t0 − t1 , (t∗ , a∗ )) = (t0 , a0 ), i.e., t0 − t1 + t∗ = t0 and φ(t0 − t1 + t∗ , t∗ , a∗ ) = a0 , which means that t∗ = t1 and φ(t0 , t∗ , a∗ ) = a0 . Hence a∗ ∈ At1 , so At1 is nonempty for t1 < t0 .  Proof of Proposition 4.2 Let t1 > t0 and a0 ∈ At0 . By the invariance Π(t1 − t0 , A) = A we have Π(t1 − t0 , (t0 , a0 )) = (t1 , φ(t1 , t0 , a0 )) ∈ (t1 , At1 ) , which means that φ(t1 , t0 , a0 ) ∈ At1 . Hence φ (t1 , t0 , At0 ) ⊆ At1 . Now let a1 ∈ At1 . By the onto nature of the above invariance there exists a point (t∗ , a∗ ) ∈ A such that Π (t1 − t0 , (t∗ , a∗ )) = (t1 , a1 ) or (t1 − t0 + t∗ , φ(t1 − t0 + t∗ , t∗ , a∗ )) = (t1 , a1 ), which means that t1 − t0 + t∗ = t1 and hence that t0 = t∗ . In addition, φ(t1 − t0 + t∗ , t∗ , a∗ ) = φ(t1 , t0 , a∗ ) = a1 . Obviously, a∗ ∈ At∗ = At0 since t∗ = t0 . This means that φ (t1 , t0 , At0 ) ⊇ At1 . Combining both directions gives φ (t, t0 , At0 ) = At

for all (t, t0 ) ∈ T2≥ . 

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Pullback attractors

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Chapter 5

Attractors

Let (X, dX ) be a complete metric space and let H(X) be the collection of all nonempty compact subsets of X. The distance between two points a, b ∈ X is given by dX (a, b) = dX (b, a) (symmetric!). We define the distance between a point a ∈ X and a nonempty compact subset B in X by distX (a, B) := inf dX (a, b). b∈B

Remark 5.1. The mapping b 7→ d(a, b) is continuous for a fixed, in fact |dX (a, b) − dX (a, b0 )| ≤ d(b, b0 ), and the subset B is nonempty and compact, so the inf can be replaced by min here, i.e., it is actually attained. Then we define the distance of a compact subset A from a compact subset B by distX (A, B) := sup distX (a, B) = sup inf dX (a, b), which is sometimes written distance of A f rom B.

a∈A ∗ HX (A, B)

a∈A b∈B

and called the Hausdorff separation or semi-

Remark 5.2. The function a 7→ distX (a, B) is continuous for fixed B and the set A is compact, so the sup here can be replaced by max. The Hausdorff separation, distX (A, B) satisfies the triangle inequality distX (A, B) ≤ distX (A, C) + distX (C, B). However, distX (A, B) is not a metric, since it can be equal to zero without the sets being equal, i.e., distX (A, B) = 0 if A ⊂ B. Define

HX (A, B) := max {distX (A, B), distX (B, A)} . This is a metric on H(X) called the Hausdorff metric. Theorem 5.1. (H(X), HX ) is a complete metric space. 35

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5.1

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An Introduction to Nonautonomous Dynamical Systems and their Attractors

Attractors of autonomous systems

Let π be an autonomous semi-dynamical system on a complete metric space (X, dX ) with time set T+ . To facilitate the comparison with attractors of nonautonomous systems, which we introduce below, we recall some definitions and results about autonomous attractors from Chapter 1. Definition 5.1. A nonempty compact subset A of X is called an attractor of the semi-dynamical system π if • i) π-invariant A is π-invariant, i.e., π(t, A) = A for all t ∈ T+ , • ii) A attracts nonempty bounded subsets of X by π, i.e., distX (π(t, D), A) → 0

as t → ∞

for all bounded subsets D of X. Example 5.1. Consider the semi-dynamical system generated by the ODE on R1
dx = x 1 − x2 , dt

(5.1)

which has steady state solutions 0 and ±1. Its attractor A = [−1, +1]. Attractors may have a very complicated geometry and be difficult to determine clearly, e.g., a fractal set. More convenient to describe and determine is an absorbing set, which usually has simple geometry such as a box or sphere. x

2 1.5 1

1 0.5 t

0

0 −0.5

−1

−1 −1.5 −2 −0.5

0

Fig. 5.1

Typical solutions of the ODE (5.1).

0.5

1

1.5

2

Definition 5.2. A nonempty subset B of X is called an absorbing set for an semidynamical system π if for every nonempty bounded subset D of X there exists a finite time TD ≥ 0 such that π(t, D) ⊆ B

for all t ≥ TD .

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Example 5.2. In Example 5.1 the attractor is A = [−1, +1] and any subset Br = [−1 − r, 1 + r] with r > 0 is an absorbing set. Example 5.3. The Lorenz equations in R3 are famous in meteorology as a system of ODEs with a strange chaotic attractor. They are given by dx dy dz = −σx + σy, = −xz + rx − y, = xy − bz (5.2) dt dt dt with parameters σ, r, b > 0. Consider the Lyapunov-like function  1 2 V (x, y, z) = x + y 2 + (z − σ − r)2 . 2 Then along solutions of the system of ODEs (5.2) d V (x(t), y(t), z(t)) = x(t) (−σx(t)+σy(t))+y(t) (−x(t)z(t)+rx(t)−y(t)) dt + red(z(t)−σ − r) (x(t)y(t)−bz(t)) 1 1 1 2 = −σx(t)2 −y(t)2 − b (z(t)−σ−r) −red bz(t)2 + b(σ+r)2 . 2 2 2  Let λ := min σ, 1, 12 b . Then 1 d V (x(t), y(t), z(t)) ≤ −2λV (x(t), y(t), z(t)) + b(σ + r)2 . dt 2 We can integrate this differential inequality to obtain
b V (x(t), y(t), z(t)) ≤ V (x0 , y0 , z0 )e−2λt + (σ + r)2 1 − e−2λt 4λ b (σ + r)2 =: R∗ 4λ 1 := 2λ ln VD , where VD := max(x0 ,y0 ,z0 )∈D V (x0 , y0 , z0 ). ≤ 1+

for t ≥ TD

This means that after time TD the value of V (x(t), y(t), z(t)) will be bounded by R∗ . This forms a closed and bounded ball in R3 , i.e., an absorbing set for the Lorenz equations. If we know that there is an absorbing compact set, then there is an attractor inside this absorbing set. We recall Theorem 1.3. Theorem 5.2. Suppose that the compact set B is an absorbing set for a semidynamical system π. Then π has an attractor in B given by \[ A= π(s, B). t≥0 s≥t

If, in addition, the absorbing set B is positively invariant, i.e., π(t, B) ⊂ B for all t ≥ 0, then \ A= π(t, B). t≥0

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If the absorbing compact set B is not positively invariant and the state space X = Rd , then we can find a compact larger set B ∗ which is both absorbing and positively invariant, namely [ π(t, B) = π([0, TB ], B), B ∗ := 0≤t≤TB

where TB is the time for π to absorb the set B into itself, i.e., π(TB , B) ⊆ B.

Firstly, note that B ∗ is compact as the continuous image of the compact subset [0, TB ] × B in R × Rd . Secondly, B ∗ is an absorbing set because of its definition and the absorbing property of B, i.e., for any bounded set D of Rd there is a TD such that π(TD , D) ⊆ B ⊆ B ∗ . Proposition 5.1. B ∗ is π-positively invariant. Proof. Let b∗ ∈ B ∗ . Then there exists a t ∈ [0, TD ] and a bt ∈ B such that b∗ = π(t, bt ). Then for any s ∈ T+ , π(s, b∗ ) = π(s, π(t, bt )) = π(s + t, bt ),

where π(s + t, bt ) ∈ B ∗ if s + t ≤ TB by the definition of B ∗ and π(s + t, bt ) ∈ B if s + t ≥ TB by the absorbing property of B. Hence, π(s, b∗ ) ∈ B ∗

for all

s ∈ T+ .

However, b∗ ∈ B ∗ was chosen arbitrarily, so π(s, B ∗ ) ⊆ B ∗ for all s ∈ T+ .



Remark 5.3. The above construction also holds in a general complete metric space X, when the positive invariant set B is just closed and bounded. Then the invariant set B ∗ is also closed and bounded. As mentioned in Remark 1.5 some additional compactness property of π is then needed to obtain the existence of an attractor. 5.2

What is a nonautonomous attractor?

Let φ be a process on X with the time set T. It is fairly clear that an attractor for φ should be a family A = {At , t ∈ T} of nonempty compact subsets At of X, which is φ-invariant, i.e., φ(t, t0 , At0 ) = At

for all (t, t0 ) ∈ T+ 2.

There is, however, a problem with convergence. Example 5.4. The scalar ODE dx + x = cos t dt has no steady state solutions.

(5.3)

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We can use an integrating factor µ(t) := et to obtain
d t e x(t) = et cos t. dt Then we use complex integration and take the real part a + b. Proof. Suppose that γ < a + b. Multiply the I equation by 2I to obtain SI 2 d 2 I = −2(a + b)I 2 + 2γ ≤ 2(γ − a − b)I 2 dt N because I 2 ≥ 0 and 0≤

S ≤ 1. N

Hence I 2 (t) ≤ I 2 (0)e−2|γ−a−b|t → 0

as

t → ∞.

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If γ = a + b, the I equation gives d SI (N − I)I I = −(a + b)I + γ = −(a + b)I + γ dt N N I2 = (γ − a − b)I − γ N I2 I2 = −γ < −γ − < 0 N q for all I > 0, so 0 ≤ I(t) ≤

I(0)q − →0 q − + γI(0)t

as t → ∞.

In both cases   b (t) − I(t) → 0 S(t) − Sb1 (t) = N (t) − N

as t → ∞,

b (t) and both terms on the right converge to zero with increassince Sb1 (t) ≡ N ing time. The nonautonomous equilibrium solution is thus globally asymptotically stable when γ ≤ a + b. Finally, let γ > a+b and suppose that I ≤ εN in the I equation, so S ≥ (1−ε)N . Then SI d I = −(a + b)I + γ ≥ −(a + b)I + γ(1 − ε)I ≥ (γ(1 − ε) − a − b)I, dt N so I(t) is strictly increasing if 0 < I ≤ εN , provided ε < 1 − a+b γ . In particular, the b nonautonomous equilibrium solution (S1 (t), 0) is unstable when γ > a + b. 6.2.1

Nontrivial limiting solutions

Now consider the possibility of a nontrivial equilibrium, i.e., Ib 6= 0. The limiting b = Sb + I, b in the SI equations (6.6)–(6.7). Then the population will be used here, N equation for I can be rewritten I b b dIb = −(a + b)Ib + γ (N − I), b dt N that is as 1 dIb = (γ − a − b)Ib − γ Ib2 . b dt N This is a Bernoulli equation which can be solved with the substitution V = Ib−1 , to give the linear ODE dV 1 + (γ − a − b)V = γ . b dt N It has the explicit solution V (t) = e−(γ−a−b)(t−t0 ) V0 + γe−(γ−a−b)t

Z

t

t0

e(γ−a−b)s ds. b (s) N

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The pullback limit as t0 → −∞ exists only if γ − a − b > 0 and is then equal to Z t (γ−a−b)s e −(γ−a−b)t b ds. V (t) = γe b (s) N −∞ This gives a nontrivial nonautonomous equilibrium b =N b (t) − I(t), b S(t)

b = I(t)

e(γ−a−b)t γ

Rt −∞

e(γ−a−b)s b (s) N

ds

,

(6.9)

where b (t) = ae−at N

Z

t

q(s)eas ds.

−∞

b b and N b (t) = S(t) b + I(t) b defined here satisfy the SI Remark 6.1. The S(t), I(t) equations (6.6)–(6.7). b b Lemma 6.2. The nonautonomous equilibrium solution (S(t), I(t)) is globally ± asymptotically stable in Σ2 when γ > a + b. Proof. A general solution of the SI equations (6.6)–(6.7) will be compared with the nonautonomous equilibrium solution. From above 2 b (t) = e−2a(t−t0 ) N (t0 ) − N b (t0 ) → 0 as t → ∞. N (t) − N Now consider two solutions I(t) and Ib of the I equations with I(0) > 0, i.e., dI 1 = (γ − a − b)I − γ I 2 dt N and dIb 1 = (γ − a − b)Ib − γ Ib2 , b dt N respectively. These are Bernoulli equations which can be solved with the substitution V = I −1 , to give the linear ODE dV 1 + (γ − a − b)V = γ . dt N Writing Vb = Ib−1 , it follows that ∆V := V − Vb satisfies the linear ODE   1 d∆V 1 + ρ∆V = γ − , b dt N N where ρ := γ − a − b, which is positive here.

(6.10)

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This gives   b| 1 d |N − N 1 (∆V )2 + 2ρ(∆V )2 = 2γ ∆V ≤ 2γ − |∆V | b b dt N N NN 2 2γ b ||∆V | ≤ ρ(∆V )2 + 4γ |N − N b |2 , ≤ − 2 |N − N − (q ) ρ(q )4 by Young’s inequality, so d 4γ 2 b (0)|2 e−2at . |N (0) − N (∆V )2 + ρ(∆V )2 ≤ dt ρ(q − )4 Integrating gives 2

2 −ρt

∆V (t) = (∆(0) e

4γ 2 b (0)|2 e−ρt |N (0) − N + ρ(q − )4

Z

t

e−2as eρs ds

0

Z t 4γ 2 2 −ρt b ≤ ∆V (0) e + |N (0) − N (0)| e e(ρ−2a)s ds ρ(q − )4 0
4γ 2 b (0)|2 e−ρt − e−2at , |N (0) − N ≤ ∆V (0)2 e−ρt + − 4 ρ(ρ − 2a)(q ) 2 −ρt

so

∆V (t) → 0

as t → 0.

(Note if ρ = 2a the argument holds with a slight change in the use of Young’s inequality.) This says that 1 1 − →0 b I(t) I(t)

as

t→0

b →0 I(t) − I(t)

as

t→0

and hence

b > 0 for all t. Now |N (t) − N b (t)| → 0 as t → 0, so since I(t) > 0 and I(t)   b = (N (t) − I(t)) − N b (t) − I(t) b S(t) − S(t) b (t) + I(t) − I(t) b → 0 as t → 0. ≤ N (t) − N This completes the proof of Lemma 6.2. 6.3

The SI equations with variable interaction

Consider the SI equations for a constant limiting population N = S + I ≡ 1, but with a time-variable interaction term γ = γ(t). It will be assumed that γ ∈ C(R, [γ − , γ + ]), where 0 < γ − ≤ γ + < ∞. The equations now have the form

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dS = a − aS + bI − γ(t)SI, dt dI = −(a + b)I + γ(t)SI. dt On adding the equations reduce to

(6.11) (6.12)

dN = a − aN, dt which has the globally asymptotic solution N (t) ≡ 1, so the analysis can be restricted to the compact set Σ2 = {(S, I) : S, I ≥ 0, S + I ≡ 1} ⊂ R2 . ¯ I) ¯ = (1, 0) for all The equations (6.11)–(6.12) have an equilibrium solution (S, parameter values. ¯ I) ¯ = (1, 0) is globally Lemma 6.3. The nonautonomous equilibrium solution (S, + asymptotically stable in Σ2 when γ ≤ a + b. It is unstable when γ − > a + b. Proof. Suppose that γ + < a + b. Multiply the I equation by 2I to obtain d 2 I = −2(a + b)I 2 + 2γ(t)SI 2 ≤ −2(a + b)I 2 + 2γ + I 2 ≤ 2(γ + − a − b)I 2 dt because 0 ≤ S ≤ 1 and γ + > 0. Hence I 2 (t) ≤ I 2 (0)e−2|γ

+

−a−b|t

→0

as

t → ∞.

The case γ + = a + b is the same as in Lemma 6.1 with γ replaced by γ + . Now let γ − > a + b and suppose that I ≤ ε in the I equation, so S ≥ (1 − ε). Then d I = −(a + b)I + γ(t)SI ≥ −(a + b)I + γ − (1 − ε)I ≥ (γ − (1 − ε) − a − b)I, dt so I(t) is strictly increasing if 0 < I ≤ εN , provided ε < 1 − a+b γ − . In particular, the equilibrium solution (1, 0) is unstable when γ − > a + b. 6.3.1

Nontrivial limiting solutions

Now consider the possibility of a nontrivial equilibrium, i.e., with I 6= 0, when γ − > a + b. It cannot exist as a steady state, so a nonautonomous equilibrium will be sought. The equation (6.12) for I can be rewritten dI = −(a + b)I + γ(t)I(1 − I), dt that is as dI = (γ(t) − a − b)I − γ(t)I 2 . dt

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This is a Bernoulli equation which can be solved with the substitution V = I −1 , to give the linear ODE dV + (γ(t) − a − b)V = γ(t). dt It has the explicit solution

V (t) = e

−

Rt t0

ρ(s) ds

V0 + e

−

Rt t0

ρ(s) ds

Z

t

Rτ

γ(τ )e

t0

ρ(s) ds

dτ

t0

=e =e

−

−

Rt t0

Rt t0

ρ(s) ds

ρ(s) ds

Z

t

V0 +

γ(τ )e t0 Z t

V0 +

Rτ t0

γ(τ )e−

R ρ(s) ds− tt ρ(s) ds 0

Rt τ

ρ(s) ds

dτ

dτ,

t0

where ρ(t) := γ(t) − a − b > 0. Now Z t Z t
ρ(s) ds = (γ(s) − a − b) ds ≥ γ − − a − b (t − t0 ) ≥ 0. t0

t0

The pullback limit as t0 → −∞ exists only if γ − − a − b > 0 and is then equal to Z t Rt b V (t) = γ(τ )e− τ ρ(s) ds dτ. −∞

These integrals exists due to the assumptions on the coefficients. The nontrivial nonautonomous equilibrium Z t −1 Rt b = 1 − I(t), b b = γ(τ )e− τ ρ(s) ds dτ . S(t) I(t) −∞

For γ a positive constant, they integrate out to give the nontrivial steady state solution a+b γ−a−b S¯ = , I¯ = . γ γ The proof of the following lemma is very similar to that of Lemma 6.2, but is given for completeness. b b Lemma 6.4. The nonautonomous equilibrium solution (S(t), I(t)) is globally ± − asymptotically stable in Σ2 when γ > a + b. Proof. Consider an arbitrary solution I(t) of the I equation (6.12) with I(0) > 0 and the solution, i.e., dI = (γ(t) − a − b)I − γ(t)I 2 dt and dIb = (γ(t) − a − b)Ib − γ(t)Ib2 , dt

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respectively. These are Bernoulli equations which can be solved with the substitution V = I −1 to give the linear ODE dV + (γ(t) − a − b)V = γ(t). dt Writing Vb = Ib−1 , it follows that ∆V := V − Vb satisfies the linear ODE d∆V + ρ(t)∆V = 0, dt

(6.13)

where ρ(t) := γ(t) − a − b ≥ γ − − a − b > 0. This gives ∆V (t)2 = ∆V (0)2 e−2

Rt 0

ρ(s) ds

≤ ∆V (0)2 e−2(γ

−

−a−b)t

,

so ∆V (t) → 0

as

t → 0.

This says that 1 1 − →0 b I(t) I(t)

as

t→0

b →0 I(t) − I(t)

as

t→0

and hence

b > 0 for all t. Similarly since I(t) > 0 and I(t) b = I(t) − I(t) b → 0 S(t) − S(t)

as

t → 0.

This completes the proof of Lemma 6.4. 6.3.2

Critical case

The situation for a + b ∈ (γ − , γ + ] depends on how the function γ(t) varies in time. Specifically, it will depend on the asymptotic behaviour of the integral 1 t − t0

Z

t

γ(s) ds t0

as either t → ∞ or t0 → −∞. The limits usually will not exists, so the upper and lower limits should be used. These exist and lie between γ − and γ + . In fact, here the disease free steady state (1, 0) undergoes a nonautonomous b b bifurcation to the nonautonomous equilibrium solution (S(t), I(t)), see [Kloeden and P¨ otzsche (2015)].

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6.4

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The SIR model with variable population

The SIR equations with variable population(6.1)–(6.3) will be analysed here for the b (t) given by (6.5) asymptotically stable limiting population N dS SI , = aq(t) − aS + bI − γ b (t) dt N dI SI = −(a + b + c)I + γ , b (t) dt N

(6.14) (6.15)

dR = cI − aR. (6.16) dt These have a nonautonomous disease free equilibrium as for the SI equations for all parameter values. The dynamics beyond it becomes unstable and can be very complicated. It is convenient here to eliminate the S variable and just consider the equations b −I −R for the I and R variables in (6.14)–(6.16). Essentially, replace S by S = N in the I equation (6.15) to obtain the IR system I(I + R) dI , (6.17) = (γ − a − b − c)I − γ b (t) dt N dR = cI − aR, (6.18) dt b (t), i.e., with solutions in time dependent triangular sets where 0 ≤ I, R ≤ N  b (t) , T2 (t) := (I, R) : I, R ≥ 0, 0 ≤ I + R ≤ N hence in the larger common compact subset of R2  T2 (t) ⊂ (I, R) : I, R ≥ 0, 0 ≤ I + R ≤ g + .

6.4.1

Disease free nonautonomous equilibrium solution

Note that I(t) = R(t) ≡ 0 is a solution. The S equation (6.14) then reduces to dS = aq(t) − aS dt which is exactly the same as the equation for the total population, so it has pullback limiting Z t b (t) = ae−at Sb1 (t) = N q(s)eas ds. −∞

The proof of the following lemma is very similar to that of Lemma 6.1, using reduced IR equations (6.17)–(6.18) for the stability proof and the original I equation (6.15) for the instability proof. Lemma 6.5. The disease free nonautonomous equilibrium solution (Sb1 (t), 0, 0) is globally asymptotically stable in Σ± 3 when γ ≤ a + b + c. It is unstable when γ > a + b − c.

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Nontrivial dynamics

The situation is much more complicated for γ > a + b − c when the disease free nonautonomous equilibrium solution is unstable. For this the reduced IR equations (6.17)–(6.18) will be used and will be rewritten as dI I(I + R) , = ρI − γ b (t) dt N

dR = cI − aR, dt

(6.19)

where ρ := γ − a − b − c > 0. Since R ≥ 0 the solution of the I equation in (6.19) satisfies the differential inequality dI I2 . ≤ ρI − γ b (t) dt N This inequality written as an equality is a Bernoulli equation, which analogously with the derivation of the pullback solution (6.9) has the pullback limiting solution Z t eρt + −at b b I (t) = R t , N (t) = ae q(s)eas ds. ρs γ −∞ Nbe (s) ds −∞ By a monotonicity argument, any pullback limiting solution of the reduced IR equations (6.19) thus has the upper bound Z t + −at b b b I(t) ≤ I (t), R(t) ≤ be eas Ib+ (s) ds. −∞

Such a pullback limiting solution arises as a nonautonomous bifurcation as ρ becomes positive. However, it need not be unique, especially for larger ρ. Indeed, [Stone, Shulgin and Agur (2001)] show that the dynamics can become chaotic when the population is constant and the interaction parameter γ(t) is time periodic. Remark 6.2. To proceed further with the analysis requires an appropriate concept of a nonautonomous attractor. The asymptotically stable nonautonomous equilibria constructed above by pullback convergence is an ideal candidate, but do not cover many of the situations that may arise.

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Chapter 7

Attractors for processes

Let φ be a process on a complete metric space (X, d) with the time set T. We have seen that we should consider an invariant set of a nonautonomous dynamical system to be a family A = {At , t ∈ T} of nonempty compact subsets At of X such that φ(t, t0 , At0 ) = At

for all (t, t0 ) ∈ T+ 2.

Otherwise we lose some very interesting and useful behaviour. In Example 5.4 we saw that the bounded entire solution of the linear ODE (5.3) forms a family of singleton sets   1 At = (cos t + sin t) , t ∈ R, 2 to which all other solutions converge in the pullback sense i.e., as t0 → −∞ with t held fixed, as well as in the forward sense, i.e., as t → ∞ with t0 held fixed. This is because t − t0 → ∞, so e−(t−t0 ) → 0, in both cases. In Chapter 6 we saw other examples of such nonautonomous equilibria solutions. In general, however, we can have pullback convergence without forward convergence or forward convergence without pullback convergence. Example 7.1. The scalar nonautonomous ODE dx = 2tx dt has the explicit solution x(t, t0 , x0 ) = x0 et x0 et

2

−t20

2

= x0 et e

−t20

→0

2

−t20

as

(7.1) , for which

t0 → −∞

(fixed t).

Here there is a pullback attractor with singleton sets At = {0} for all t ∈ R. There is no forward attraction here. Example 7.2. The scalar nonautonomous ODE dx = −2tx dt 55

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3

x

2

1 t 0 0 −1

−2

−3 −2

Fig. 7.1

−1.5

−1

−0.5

0

0.5

x0 e−t

1.5

2

Typical solutions of the nonautonomous ODE (7.1).

has the explicit solution x(t, t0 , x0 ) = x0 e−t 2

1

+t20

t20

2

= x0 e e−t → 0

2

+t20

as

, for which

t→∞

(fixed t0 ).

Here there is a forward attractor with singleton component sets At = {0} for all t ∈ R. There is no pullback attraction here. Thus we have two types of nonautonomous attractors for processes, one with pullback convergence and one with forward convergence. Definition 7.1. A family A = {At , t ∈ T} of nonempty compact subsets of X, which is φ-invariant, is called a • pullback attractor if it pullback attracts all bounded subsets D of X, i.e., lim distX (φ(t, t0 , D), At ) = 0,

t0 →−∞

(fixed t)

• forward attractor if it forward attracts all bounded subsets D of X, i.e., lim distX (φ(t, t0 , D), At ) = 0,

t→∞

(fixed t0 ).

Remark 7.1. To enable us to handle non-uniformities, which are common in nonautonomous systems, we often replace a “bounded set” D by a “family of appropriately chosen bounded sets” D = {Dt , t ∈ T}. The limits in the above definition are then for the expression distX (φ (t, t0 , Dt0 ) , At ) . Note that the bounded sets in the family D = {Dt , t ∈ T} may have to be restricted in some way to ensure that the absorbing property holds in given examples. For example, they could be uniformly bounded , i.e., if there is a common bounded subset B of X such that Dt ⊆ B for all t ∈ T. More generally, the subsets Dt could be allowed to grow but not too quickly as t → ±∞. See Definition 9.4 of a tempered family of bounded subsets and the discussion in Section 9.3 in the context of skew product flows.

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Similarly, we say that a pullback attractor A = {At , t ∈ T} is uniformly bounded S if t∈T At is bounded or, equivalently, if there is a common bounded subset B of X such that At ⊆ B for all t ∈ T. We have the following characterisation of a uniformly bounded pullback attractor.

Proposition 7.1. A uniformly bounded pullback attractor A = {At , t ∈ T} of a process φ is uniquely determined by the bounded entire solutions of the process, i.e., x0 ∈ At0

⇐⇒

∃ a bounded entire solution e(·) with e(t0 ) = x0 .

Proof. (Sufficiency) Pick t0 ∈ T and x0 ∈ At0 arbitrarily. Then, due to the φ-invariance of the pullback attractor there exists an entire solution e(·) with e(t0 ) = x0 and e(t) ∈ At for all t ∈ T. (See Lemma 4.1 in Chapter 4). Moreover, e(t) ∈ At ⊆ B for all t ∈ T, so the entire solution e(·) is a uniformly bounded entire solution. (Necessity) Let e(·) be a bounded entire solution of the process φ. Then the set De := {e(t), t ∈ T} is a bounded subset of X. Since A pullback attracts bounded subsets of X, for each t ∈ T we have 0 ≤ distX (e(t), At ) ≤ distX (De , At ) → 0

as t0 → −∞,

from which we conclude that for each t ∈ T,

distX (e(t), At ) = 0

i.e., e(t) ∈ At for every t ∈ T since the At are compact sets. 7.1



Existence of a pullback attractor

In analogy to the autonomous case we can show that the existence of a pullback absorbing compact set implies the existence of a pullback attractor. Similarly, the compactness of the absorbing set can be replaced by closed and boundedness provided the process satisfies some additional compactness property. To handle non-uniformities in the dynamics we also consider a pullback absorbing family of sets instead of a single set and assume that it absorbs a family of nonempty bounded subsets, which will be assumed to be uniformly bounded here (but this can be weakened as indicated in Remark 7.1). Definition 7.2. A family B = {Bt , t ∈ T} of nonempty subsets of X is called a pullback absorbing family process φ on X if for each t ∈ T and every uniformly bounded family D = {Dt , t ∈ T} nonempty bounded subsets of X there exists a Tt,D ∈ T+ such that φ (t, t0 , Dt0 ) ⊆ Bt

for all t0 ≤ t − Tt,D .

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Definition 7.3. A family B = {Bt , t ∈ T} of nonempty subsets of X is called positive invariant with respect to a process φ or φ-positive invariant if φ (t, t0 , Bt0 ) ⊆ Bt

for all t ≥ t0 .

Theorem 7.1 (Existence of a pullback attractor). Suppose that a process φ on a complete metric space (X, d) has a φ-positive invariant pullback absorbing family B = {Bt , t ∈ T} of compact sets. Then φ has a global pullback attractor A = {At , t ∈ T} with component subsets determined by At =

\

for each t ∈ T.

φ (t, t0 , Bt0 )

t0 ≤t

(7.2)

Moreover, if A is uniformly bounded then it is unique. Proof The set At is nonempty and compact as the nested intersection of compact subsets since

φ (t, t00 , Bt0 ) = φ t, t0 , φ t0 , t00 , Bt00 ⊆ φ (t, t0 , Bt0 ) , t00 ≤ t0 . Let B = {Bt , t ∈ T} be the pullback absorbing family and let At be defined by (7.2). Clearly, At ⊆ B t

for each t ∈ T.

In fact, At ⊆ φ(t, t0 , Bt0 ) ⊆ Bt for any t0 ≤ t. Step 1.

First, we show for any t ∈ T that lim distX (φ (t, t0 , Bt0 ) , At ) = 0.

t0 →−∞

Suppose not. Then there exist sequences k → ∞ and xk ∈ φ (t, tk , Btk ) ⊆ Bt

with tk → −∞

such that distX (xk , At ) ≥ ε0 ,

for all

k,

for some ε0 > 0. The subset {xk }k∈N of Bt is relatively compact, so there is a convergent subsequence kj → ∞ and a point x0 ∈ Bt such that xkj → x0 as kj → ∞. Now   xkj ∈ φ t, tkj , Btkj ⊆ φ (t, tk , Btk ) for all all kj ≥ k and each k. This implies that x0 ∈ φ (t, tk , Btk )

for all k ∈ N,

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so x0 ∈

\

φ (t, tk , Btk ) ,

k≥0

hence x0 ∈ At , which is a contradiction. Hence the assertion of Step 1 has been proved. Step 2. From the above convergence, for every ε > 0 and t ∈ T there exists Tε,t ≥ 0 such that distX (φ (t, t0 , Bt0 ) , At ) < ε for all t0 ≤ t − Tε,t . Let D be any bounded subset of X. By the pullback absorption property of B = {Bt , t ∈ T} we have φ (t, t0 , D) ⊆ Bt

for all t0 < t with t − t0 large enough. Let t000 < t00 < t. Then by the 2-parameter semi-group property
φ (t, t000 , D) = φ (t, t00 , φ (t00 , t000 , D)) ⊆ φ t, t00 , Bt00

provided t00 − t000 is large enough (for the last inclusion). Step 3.

The φ-invariance of the family A = {At , t ∈ T} will now be shown.

By (7.2) the set Ft0 (t) := φ (t, t0 , Bt0 ) ⊆ Bt for all t0 ≤ t and by definition \ At = Ft0 (t). t0 0.

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The proof follows from the fact that the Bernoulli equation (8.3) can be transformed into the linear differential equation dv + 2av = 2b(t) dt with the substitution v = x−2 (recall that x = 0 is an equilibrium of the Bernoulli equation (8.3)). This linear differential equation can be integrated to give 1 1 = 2 e−2a(t−t0 ) + 2 xa (t, t0 , x0 )2 x0

t

Z

b(s)e−2a(t−s) ds.

(8.5)

t0

Holding t and x0 fixed in (8.5) and taking the pullback limit as t0 → −∞ gives the pullback limit solution φ¯a (t) =

lim xa (t, t0 , x0 )

t0 →−∞

given by (8.4). This is itself an entire solution of the differential equation (8.3) and thus satisfies (8.5) with x0 = φ¯a (t0 ). Specifically 1 1 = ¯ e−2a(t−t0 ) + 2 2 ¯ φa (t) φa (t0 )2

t

Z

b(s)e−2a(t−s) ds.

(8.6)

t0

To show that φ¯a (t) is asymptotically stable for all x0 > 0 subtract (8.5) from (8.6) to obtain 1 1 = − xa (t, t0 , x0 )2 φ¯a (t)2



1 1 − 2 x0 φ¯a (t0 )2



e−2a(t−t0 ) → 0

as t → ∞,

from which the result follows. φ¯ (t) is uniformly asymptotically Lyapunov stable in R+ , since φ¯a (t) ∈ hpIn fact, p a i a/b1 , a/b0 , so for every  > 0 and x0 > 0 there exists T (x0 , ) ≥ 0 such that 

1 1 − 2 2 ¯ x0 φa (t0 )



e−2a(t−t0 ) ≤



b1 1 − 2 a x0



e−2a(t−t0 ) < 

(8.7)

for t ≥ T (x0 , ) + t0 . The corresponding result holds for the solution −φ¯a (t) for all x 0 ∈ R− . Here the family A = {Aa (t), t ∈ T} of time dependent sets   Aa (t) = −φ¯a (t), φ¯a (t) ,

t ∈ R,

(8.8)

is the pullback attractor of the Bernoulli equation (8.3). It is also uniformly Lyapunov asymptotically stable and thus a forward atttractor.

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A dissipative ODE

Consider now the scalar nonlinear ODE dx = f (x) + cos t, (8.9) dt where f : R → R is continuously differentiable and satisfies the dissipativity condition xf (x) ≤ K − L|x|2 ,

∀x ∈ R,

with constants K ≥ 0 and L > 0. The vector field in the ODE (8.9) f¯(x, t) = f (x) + cos t also satisfies the dissipativity condition uniformly in t ∈ R: xf¯(x, t) = xf (x) + x cos t ≤ K − L|x|  ≤

2 K+ L

2




 −

 +

L 2 2 |x| + | cos t|2 2 L



L 2 ¯ − L|x| ¯ 2 |x| = K 2

with ¯ := K + 2 , ¯ := L . K L L 2 Thus the solution of the ODE (8.9) with initial value x(t0 ) = x0 satisfies d 2 ¯ − 2L|x(t)| ¯ |x(t)|2 = 2x(t)f¯(x(t), t) ≤ 2K , dt so  ¯  ¯ K K ¯ ¯ |x(t)|2 ≤ |x0 |2 e−2L(t−t0 ) + ¯ 1 − e−2L(t−t0 ) ≤ 1 + ¯ =: R2 L L for t − t0 ≥ T (x0 ), where  1 T (x0 ) := ¯ max 0, ln |x0 |2 . 2L For a family D = {Dt : t ∈ T} of bounded subsets and the initial values x0 ∈ Dt0 at time t0 we take TD,t0 := sup T (x0 ) < ∞. x0 ∈Dt0

Then the familiy B of identical sets Bt ≡ BR := [−R, R], where R is defined above, is pullback (and also forward) absorbing as well as positive invariant. Thus the process generated by the ODE (8.9) has a pullback attractor A with subsets At ⊆ BR for every t ∈ R. However, we cannot say anything more about the sets At without extra information about the function f (x) in the ODE (8.9).

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8.5

One-sided dissipative Lipschitz condition

Let us now assume that the function f : R → R in the ODE (8.9) is continuously differentiable and satisfies the one-sided dissipativity Lipschitz condition (x − y)(f (x) − f (y)) ≤ −L|x − y|2 ,

∀x, y ∈ R,

with a constant L > 0. An example is the scalar function f (t, x) = −x − x3 + 1 since
(x − y)(f (x) − f (y)) = −(x − y)2 − (x − y) x3 − y 3 = −(x − y)2 − (x − y)2 x2 + xy + y 2 2

2

= −(x − y) − (x − y)



1 x+ y 2

2


3 + y2 4

!

≤ −(x − y)2 . Then f also satisfies the dissipativity condition above. Taking y = 0 gives x(f (x) − f (0)) ≤ −L|x|2 , so xf (x) ≤ −L|x|2 + xf (0) ≤

2 L ˆ − L|x| ˆ 2 |f (0)|2 − |x|2 = K L 2

with ˆ = L. ˆ = 2 |f (0)|2 , L K L 2 Thus by Theorem 7.1 the ODE (8.9) has a pullback attractor A with component subsets At ⊆ BR := [−R, R], where R is defined by ˆ+ ¯ K K R2 = 1 + ¯ = 1 + ˆ L L

2 ˆ L

=1+

2

= 1+

2 L2 |f (0)|2 L 2

+

4
L 2 2

ˆ 2K 4 + ˆ ˆ L L2

=1+


8 |f (0)|2 + 2 . 2 L

Now the difference of two solutions of the ODE (8.9) satisfies d (x(t) − y(t)) = (f (x(t)) − f (y(t))) , dt so d 2 2 |x(t) − y(t)| = 2 (x(t) − y(t)) (f (x(t)) − f (y(t))) ≤ −2L |x(t) − y(t)| dt which implies that 2

2

|x(t) − y(t)| ≤ |x0 − y0 | e−2L(t−t0 ) ,

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and hence that |x(t) − y(t)| ≤ |x0 − y0 | e−L(t−t0 ) . Thus inside the interval BR the ODE (8.9) satisfies the uniform strict contractivity condition |x(t) − y(t)| ≤ 2Re−L(t−t0 ) . (8.10) This uniform strict contractivity condition allows to say much more about the pullback attractor A with component subsets At in BR . If we did not know the result of Theorem 7.2, then we could use the following result. Proposition 8.1. When the function f (x) in the ODE (8.9) satisfies the onesided dissipativity Lipschitz condition, then the pullback attractor consists of a single entire solution and is also forward attracting. Proof. Suppose that the pullback attractor does not consist of singleton component sets. Then there are x0 , x00 ∈ Aτ for some τ ∈ R such that |x0 − x00 | ≥ ε0 for some ε0 > 0. Since the pullback attractor A is φ-invariant for the process φ generated by the ODE (8.9) for every t0 < τ there exist points x0 (t0 ), x00 (t0 ) ∈ At0 such that φ (τ, t0 , x0 (t0 )) = x0 , φ (τ, t0 , x00 (t0 )) = x00 . By the uniform contractivity condition (8.10) we then obtain ε0 ≤ |x0 − x00 | ≤ |φ (τ, t0 , x0 (t0 )) − φ (τ, t0 , x00 (t0 ))|

for

≤ 2Re−L(τ −t0 ) ≤

1 ε0 2

1 ln 2R. 2L This is a contradiction, so we must have x0 = x00 , i.e., the components At are singleton sets, so the pullback attractor consists of a single entire solution. t0 ≤ τ −

Let this entire solution be denoted by e(·) so At = {e(t)} for all t ∈ R. Then, for any other solution x(·) of the ODE (8.9) the uniform strict contractivity condition (8.10) gives |x(t) − e(t)| ≤ |x(t0 ) − e(t0 )| e−L(τ −t0 ) → 0 as t → ∞. Thus the family of sets At = {e(t)} is also a forward attractor.  Remark 8.1. The one-sided dissipativity Lipschitz condition is a very special case. It means that an autonomous ODE dx = f (x) dt has a unique globally asymptotically stable steady state solution. A nonlinear example is f (x) = −x − x3 + 1. The nonautonomous ODE (8.9) obtained by adding cos t to the right hand side is uniformly strictly contracting and has a nonautonomous attractor consisting of singleton sets, which is both pullback and forward attracting.

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Chapter 9

Attractors of skew product flows

Let (P, dP ) and (X, dX ) be complete metric spaces. Recall that a skew product flow (θ, ϕ) on P × X with the time set T consists of autonomous dynamical system θ = {θt }t∈T on the base space P , which is called the driving system, and a cocycle mapping ϕ on the state space X w.r.t. θ, which describes the dynamics of the state variable. 9.1

Invariant sets and attractors

We saw in Chapter 3 that a process is a special case of a skew product flow with P = T and θt the left shift operator defined by θt (t0 ) = t − t0 . In addition, we saw that a skew product flow (θ, ϕ) is an autonomous semi-dynamical system Π on the product space X = P × X.

These facts are useful when we consider invariant sets and attractors of skew product flows. They help to motivate the following definitions. Definition 9.1. A family A = {Ap , p ∈ P } of nonempty subsets Ap of X is said to be ϕ-invariant for a skew product flow (θ, ϕ) on P × X if ϕ (t, p, Ap ) = Aθt (p)

for all p ∈ P and t ∈ T+ .

It is called ϕ-positive invariant if ϕ (t, p, Ap ) ⊆ Aθt (p)

for all p ∈ P and t ∈ T+ .

The essential difference with the definition for a process is that here we use the state of the driving system instead on the initial time. Similarly to processes we have two types of attractors for skew product flows, pullback and forward attractors. (Later we will consider a third type that comes from the corresponding autonomous semi-dynamical system Π on the state space X = P × X). Definition 9.2. A family A = {Ap , p ∈ P } of ϕ-invariant nonempty compact subsets of X is called a pullback attractor if it pullback attracts families D = 73

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{Dp , p ∈ P } of nonempty bounded subsets of X, i.e.,
lim distX ϕ(t, θ−t (p), Dθ−t (p) ), Ap = 0 t→∞

for each p ∈ P.

It is called a forward attractor if it forward attracts all bounded subsets D of X, i.e.,
lim distX ϕ(t, p, Dp ), Aθt (p) = 0 for each p ∈ P. t→∞

Also, as for a process, the existence of a pullback attractor for skew product flow is ensured by that of a pullback absorbing family. As in Remark 7.1 for processes we first restrict to uniformly bounded families of bounded subsets, but this will be generalised later in this chapter to tempered families. Definition 9.3. A family B = {Bp , p ∈ P } of nonempty subsets of X is called a pullback absorbing family for a skew product flow (θ, ϕ) on P × X if for each t ∈ P and every uniformly bounded family D = {Dp , p ∈ P } of nonempty bounded subsets of X there exists a Tp,D ∈ T+ such that
for all t ≥ Tp,D . ϕ t, θ−t (p), Dθ−t (p) ⊆ Bp Remark 9.1. Note the difference in the definition of pullback attraction here. For a process we start earlier at time τ and finish at the fixed time t, whereas for a skew product flow the driving system starts at the earlier state θ−t (p) and ends at time t latter at the fixed state θ0 (p) = p. Essentially the driving system in a skew product flow is used instead of the actual value of time, while the time variable in the cocycle mapping just indicates the time since starting at a given state of the driving system. The proof of the following theorem here is similar to that of Theorem 7.1, at least for a positively invariant absorbing family, so will be omitted. For a proof see [Kloeden and Rasmussen (2011), Theorem 3.20]. As with Theorem 7.1, the assumption that the pullback absorbing family consists of compact sets restricts its usefulness to finite dimensional state spaces such as Rd . An analogous result holds for more general spaces when the absorbing family consists of closed and bounded sets and the cocycle mapping satisfies a pullback asymptotic compactness property. Theorem 9.1 (Existence of a pullback attractor). Let (P, dP ) and (X, dX ) be complete metric spaces and suppose that a skew product flow (θ, ϕ) on P × X with the time set T has a pullback absorbing family B = {Bp , p ∈ P } of nonempty compact sets. Then (θ, ϕ) has a pullback attractor A = {Ap , p ∈ P } with component subsets determined by \[
Ap = ϕ t, θ−t (p), Bθ−t (p) for each p ∈ P. (9.1) t≤0 s≥t

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If B is ϕ-positively invariant then \
Ap = ϕ t, θ−t (p), Bθ−t (p) t≤0

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for each p ∈ P.

Moreover, A is unique if the components sets are uniformly bounded. The possible existence and properties of forward attraction and forward attractors will be discussed later in this and the following chapters. Note for now that if the pullback attractor is uniformly pullback attracting, i.e., if
lim sup distX ϕ(t, θ−t (p), Dθ−t (p) ), Ap = 0 for each p ∈ P, t→∞ p∈P

then it is uniformly forward attracting, since writing q = θ−t (p),

sup distX ϕ(t, θ−t (p), Dθ−t (p) ), Ap = sup distX ϕ(t, q, Dq ), Aθt (q) . p∈P

q∈P

In this case this uniform pullback/forward attractor is called a uniform (nonautonomous) attractor . 9.2

Examples

We consider some examples of pullback attractors of skew product flows, first for continuous time and then for discrete time. 9.2.1

Continuous-time example

Consider again the scalar ODE dx = −x + cos t. dt As in Example 5.4 define P to be the hull of the functions cos(·), i.e., [ P = cos(τ + ·), 0≤τ ≤2π

which is a compact metric space with the metric induced by the supremum norm ρ (p1 , p2 ) = sup |p1 (t) − p2 (t)| . t∈R

In addition, let θt : P → P be the left shift operator θt (cos(·)) = cos(t + ·), so θ−τ (cos(·)) = cos(−τ + ·). This shift operator is continuous in the above metric. Now write p0 for an element of P . The ODE

dx = −x + p0 (t) dt has a unique solution x(t) = x(t, p0 , x0 ) with initial value x(0) = x0 given by Z t −t −t x(t) = x0 e + e es p0 (s) ds. 0

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Then, replacing p0 (·) by θ−τ (p0 (·)) for some τ > 0 we have Z τ x (τ, θ−τ (p0 ), x0 ) = x0 e−τ + e−τ es θ−τ (p0 (s)) ds 0

= x0 e−τ +

Z

0

eρ p0 (ρ) dρ.

−τ

with the change of variable ρ = s − τ , so dρ = ds.

We take pullback convergence and obtain the single limit point Z 0 lim x(t, θ−τ (p0 ), x0 ) = eρ p0 (ρ) dρ. τ →∞

−∞

Remark 9.2. This pullback limit holds for x0 ∈ D for any bounded subset D of X because the x0 e−t term is bounded above by kDke−t , where kDk := supx0 ∈D kx0 k 0.

In this case it is called a tempered family. This may also be too general. Often we need to restrict to particular classes of such families called universes. Definition 9.5. A universe Datt of a skew product flow (θ, ϕ) on P ×X is collection of families D = {Dp , p ∈ P }nof nonemptyobounded subsets Dnp of X whichois closed (1)

under inclusion, i.e., D(1) = Dp , p ∈ P (1)

and Dp

(2)

⊆ Dp

for all p ∈ P .

(2)

∈ Datt if D(2) = Dp , p ∈ P

∈ Datt

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The definition of a pullback attractor then needs to be modified. Definition 9.6. A ϕ-invariant family A = {Ap , p ∈ P } of nonempty compact subsets of X is called a pullback attractor with respect to a universe Datt of a skew product flow (θ, ϕ) if
lim dist ϕ(t, θ−t (p), Dθ−t (p) ), Ap = 0 (9.2) t→∞

for all p ∈ P and all D = {Dp , p ∈ P } ∈ Datt . Such a universe Datt is often called the basin of attraction system of the pullback attractor A. Obviously, A ∈ Datt . In fact, Ap ⊂ int Datt (p) for each p ∈ P , where [ Datt (p) := Dp ⊂ X D={Dp , p∈P }∈Datt

for each p ∈ P .

Universes are also useful for discussing local pullback or forward attractors. As a simple example, consider the autonomous semi-dynamical system on X = R1 generated by the scalar ODE x = x(1 − x). (9.3) dt This can be formulated as a skew product flow on a base space P consisting of just one point. x

3 2.5 2 1.5 x=1

1 0.5

t

x=0

0

0 −0.5 −1 −1.5 −2 −1

−0.5

Fig. 9.1

0

0.5

1

1.5

2

2.5

3

Typical solutions of the ODE (9.3).

The compact invariant set [0, 1] is but it is a local attractor  not a 1global attractor, 1 in the nonnegative subspace R+ = x ∈ R ; x ≥ 0 . The basin of attraction system here consists of all families consisting of a single bounded subset of R1+ . 9.4

Upper semi-continuity of the set-valued mapping p 7→ Ap

It follows [Aubin and Franksowka (1990)] from the compactness of the sets Ap and the continuity of the single-valued mapping (t, p, x) 7→ ϕ (t, p, x) that the set-valued

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mapping t 7→ ϕ (t, p, Ap ) is continuous for each p ∈ P . Hence, by the ϕ-invariance of the pullback attractor the set-valued mapping t 7→ Aθt (p) = ϕ (t, p, Ap ) is continuous for each p ∈ P . Theorem 9.2. Let A = {Ap , p ∈ P } be a pullback attractor of a skew product flow (θ, ϕ) on P × X with the time set T and suppose that [ A(P ) := Ap p∈P

is compact. Then the set-valued mapping p 7→ Ap is upper semi-continuous in the sense that distX (Aq , Ap ) → 0 as q → p. On the other hand, if P is compact and the set-valued mapping p 7→ Ap is upper semi-continuous, then A(P ) is compact. Proof. Since A(P ) is compact, the pullback attractor component sets are uniformly bounded, so the pullback attractor is unique. Suppose that the set-valued mapping p 7→ Ap is not upper semi-continuous. Then there exists ε0 > 0 and a sequence pn → p0 in P such that distX (Apn , Ap0 ) ≥ 3ε0 for all n ∈ N. Since the sets Apn are compact, there exists points apn ∈ Apn for each n ∈ N such that distX (apn , Ap0 ) = distX (Apn , Ap0 ) ≥ 3ε0 for all n ∈ N. (9.4)

By pullback attraction, for every bounded subset B of X we have distX (ϕ (t, θ−t (p0 ), B) , Ap0 ) ≤ ε0 for all t ≥ TB,p0 ,ε0 . We will use this for the bounded set B = A(P ). By ϕ-invariance of the pullback attractors, there exists bn ∈ Aθ−t (pn ) (with a fixed t here) for each n ∈ N such that ϕ (t, θ−t (p0 ), bn ) = an . Now the bn are contained in the compact set A(P ), so there is a convergent subsequence bn0 → ¯b ∈ A(P ). Finally, by the continuity of p 7→ θ−t (p) and (p, x) 7→ ϕ(t, p, x) we have

dX ϕ (t, θ−t (pn0 ), bn0 ) , ϕ t, θ−t (p0 ), ¯b ≤ ε0 for all large enough n0 . Thus
distX Apn0 , Ap0 = distX (ϕ (t, θ−t (pn0 ), bn0 ) , Ap0 )

≤ dX ϕ (t, θ−t (pn0 ), bn0 ) , ϕ t, θ−t (p0 ), ¯b

+ distX ϕ t, θ−t (p0 ), ¯b , Ap0 ≤ 2ε0 , which contradicts the inequality (9.4). Thus p 7→ Ap is upper semi-continuous. The remaining assertion of the theorem holds because the image of a compact subset under an upper semi-continuous set-valued mapping with compact values is compact. (See [Aubin and Franksowka (1990)] for properties of set-valued mappings.) 

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The set-valued mapping p 7→ Ap need not be continuous

We have just shown that, under certain conditions, the set-valued mapping p 7→ Ap is upper semi-continuous. Here we show that it need not be continuous.

Fig. 9.2

Supercritical bifurcation in the autonomous ODE (9.5) with parameter p.

Consider that autonomous scalar ODE
dx = −x x4 − 2x2 + 1 − p dt with a parameter p ∈ P = [−2, 2].

(9.5)

The steady state x ¯p = 0, which exists for all p ∈ P , under goes a subcritical bifurcation at p = 0. It has the steady state solutions • x ¯p = 0 for p < 0, • x ¯p = 0, ±

p

1+

√

p √ p and ± 1 − p for 0 < p < 1,

p √ • x ¯p = 0 and ± 1 + p for 1 ≤ p.

Fig. 9.3

Global attractor Ap in the autonomous ODE (9.5) with parameter p.

The global attractors (autonomous) are  {0}   Ap =

for p < 0,

 √ p √   p − 1 + p, 1 + p for p ≥ 0.

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Now consider the trivial nonautonomous dynamical system with the constant driving system θt (p) ≡ p for each p ∈ P = [−2, 2] and all t ∈ R. The pullback attractor A has these Ap as its component sets: since θt (p) ≡ p, the dynamics just stays on the same Ap for all t ∈ R. For p → 0− , i.e., from below, we have Ap → A0 in the Hausdorff semi-distance  √ √  distR (Ap , A0 ) = distR {0}, [− 2, 2] = 0,

whereas  √  √ √ distR (A0 , Ap ) = distR [− 2, 2], {0} = 2 6= 0 for all p < 0. The convergence is only upper semi-continuous convergence and not lower semicontinuous convergence and hence not continuous convergence. 9.5

A weak form of forward convergence

Pullback attractors are, in general, not forward attractors. We saw this for processes and it is also true for skew product flows. But if the base space P is compact, then we have the following partial result on forward convergence when we have a pullback attractor. Essentially, the set A(P ) contains all of the limiting dynamics in the forwards sense, but it is not an attractor as it is not invariant. (More will be said on this topic in Chapter 11.) Theorem 9.3. Assume, in addition to the asumptions of Theorem 9.1, that the base space P is compact and that the pullback absorbing family B = {Bp , p ∈ P } is uniformly bounded by a compact set, i.e., there exists a compact set C of X with Bp ⊆ C for each p ∈ P . Then the pullback attractor A = {Ap , p ∈ P } of the skew product flow (θ, ϕ) on P × X satisfies lim sup distX (ϕ (t, p, D) , A(P )) = 0

t→∞ p∈P

(9.6)

for every bounded subset D of X. Proof. Since Ap is contained in Bp for each p ∈ P so we have Ap ⊆ C for each p ∈ P . Hence A(P ) ⊆ C, which means that A(P ) is compact, so the pullback attractor is unique. Suppose that the limit (9.6) does not hold. Then there exist a ε0 > 0 and sequences tj → ∞ in T, pˆj ∈ P and xj ∈ C such that distX (ϕ (tj , pˆj , xj ) , A(P )) ≥ 3ε0 .

(9.7)

Set pj = θtj (ˆ pj ). By the compactness of P there exists a convergent subsequence pˆj 0 → p0 ∈ P .

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From the pullback attraction for τ ≥ 0 large enough distX (ϕ (τ, θ−τ (p0 ), C) , Ap0 ) ≤ ε0 . The cocycle property gives ϕ (tj , pj , xj ) = ϕ τ, θ−τ (pj ), ϕ tj − τ, θ−tj (pj ), xj





for tj > τ . By the pullback absorbing property of B it follows that
ϕ tj − τ, θ−tj (pj ), xj ⊆ Bθ−τ (pj ) ⊆ C. Since C is compact there exists a convergent (sub)sequence   zj 00 = ϕ tj 00 − τ, θ−tj00 (pj 00 ), xj 00 → z0 ∈ C. The continuity of p 7→ θt p and (p, x) 7→ ϕ(t, p, x) then implies that distX (ϕ (τ, θ−τ (pj 00 ), zj 00 ) , ϕ (τ, θ−τ (p0 ), z0 )) ≤ ε0

00

for j ≥ N (ε0 ), i.e., large enough. Hence     distX ϕ tj 00 , θ−tj 00 (pj 00 ), xj 00 , Ap0

    ≤ distX ϕ tj 00 , θ−tj 00 (pj 00 ), xj 00 , ϕ (τ, θ−τ (p0 ), z0 ) + distX (ϕ (τ, θ−τ (p0 ), z0 ) , Ap0 ) = distX (ϕ (τ, θ−τ (pj 00 ), zj 00 ) , ϕ (τ, θ−τ (p0 ), z0 )) + distX (ϕ (τ, θ−τ (p0 ), z0 ) , Ap0 ) ≤ 2ε0 .

Therefore     2ε0 > distX ϕ tj 00 , θ−tj00 (pj 00 ), xj 00 , Ap0 = distX (ϕ (tj 00 , pˆj 00 , xj 00 ) , Ap0 ) ≥ distX (ϕ (tj 00 , pˆj 00 , xj 00 ) , A(P )) > 3ε0 , which is a contradiction, so the asserted convergence must hold.



Example 9.1. Suppose that θt : P → P is T -periodic, i.e., there is a T > 0 such that θT (p) = p for all p ∈ P , and that P consists of single θ trajectory, i.e., there is a p0 ∈ P such that P = {θt (p0 ), 0 ≤ t ≤ T }. Hence P is compact. In addition, suppose that the pullback attractor has singleton component sets Ap = {ap }, so t 7→ aθt is T -periodic. Then A(P ) = {aθt (p0 ) , 0 ≤ t ≤ T } represents the geometric curve in X of the periodic solution forming the pullback attractor. Theorem 9.3 says that other solutions converge forwards in time to this geometric limit cycle. However, the family {Ap , p ∈ P } need not be a forward attractor since the convergence to the point at the same time instant on the curve, i.e., dX (φ(t, p, x0 ), aθt (p) ) → 0, need not hold t → ∞. (It will hold if, for example, the system is uniformly strictly attracting as in the examples in Section 9.2).

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Another type of attractor for skew product flows

We consider some more results on the relationship between the attractors of a skew product flow and its representation as an autonomous semi-dynamical system. Let (θ, ϕ) be a skew product flow on P × X with a time set T and let Π be the corresponding autonomous semi-dynamical system on X = P × X, i.e., defined by Π (t, (p0 , x0 )) = (θt (p0 ), ϕ(t, p0 , x0 )) . We use the metric dX on X defined by dX ((p1 , x1 ), (p2 , x2 )) = dP (p1 , p2 ) + dX (x1 , x2 ) . Proposition 9.1. (Uniform convergence of global attractors) Suppose that A = {Ap , p ∈ P } is a uniform attractor for the skew product flow (ϕ, θ), i.e., uniformly attracting in both forward and pullback senses. Also, assume that P and the set [ A(P ) = Ap p∈P

are compact in X. Then the union A=

[ p∈P

{p} × Ap

is the global attractor of the autonomous semi-dynamical system Π. Proof The Π-invariance of A follows by the ϕ-invariance of A and the θ-invariance of P via [ Π (t, A) = {θt (p)} × ϕ (t, p, Ap ) p∈P

=

[ p∈P

{θt (p)} × Aθt (p) =

[ q∈P

{q} × Aq = A

since by the θ-invariance of P for each θt (p) there is a q ∈ P with q = θt (p). Since A is a pullback attractor and A(P ) is compact, the set-valued mapping p 7→ Ap is upper semi-continuous. This means that the mapping p 7→ F (p) := {p} × Ap is also upper semi-continuous. Hence F (P ) = A is a compact subset of X = P × X. See Aubin and Frankowska [Aubin and Franksowka (1990)] for the properties of set-valued mappings. Moreover, by the definition of the metric dX on X, we have distX (Π (t, (p, x)) , A) = distX ((θt (p), ϕ(t, p, x)) , A) ≤ distX (θt (p), ϕ(t, p, x)) , {θt (p)} × Aθt (p)




= distP (θt (p), θt (p)) + distX ϕ(t, p, x), Aθt (p) {z } | =0


= distX ϕ(t, p, x), Aθt (p) → 0

as t → ∞




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since A is also a forward attractor.



Without the assumption forward attraction, the pullback attractor need not give a global attractor of the autonomous semi-dynamical system Π. Proposition 9.2. Suppose that A = {Ap , p ∈ P } is a pullback attractor of the skew product flow (θ, ϕ) on P × X and that P and A(P ) = ∪p∈P Ap are compact in X. Then A = ∪p∈P {p} × Ap is the maximal invariant compact subset in X of the autonomous semi-dynamical system Π. Proof. The compactness and Π-invariance of A follow in the same way as in the previous proof. S To prove that the compact invariant set A is maximal. Let C = p∈P {p} × Cp be another compact Π-invariant subset in X = P × X. Then C = {Cp , p ∈ P } is a ϕ-invariant family of compact subsets of X. By pullback attraction distX (Cp , Ap ) = distX ϕ(t, θ−t (p), Cθ−t (p) ), Ap




≤ distX (ϕ(t, θ−t (p), C(P )), Ap ) → 0

as t → ∞,

where C(P ) =

[

Cp

p∈P

is a compact subset of X. Thus Cp ⊆ Ap and C=

[ p∈P

{p} × Cp ⊆

[ p∈P

i.e., C is a subset of A. Hence A is maximal.

{p} × Ap = A, 

Remark 9.4. The set A in the above proposition need not be a global attractor of the autonomous semi-dynamical system Π on X = P × X. For an example in terms of processes see Section 10.3, while Cheban, Kloeden and Schmalfuß [Cheban, Kloeden and Schmalfuß (2001)] give an example in terms of skew product flows. In the other direction we have Proposition 9.3. If P is compact and the autonomous semi-dynamical system Π in X has a global attractor A = ∪p∈P {p} × Ap , then the family A = {Ap , p ∈ P } is a pullback attractor for the skew product flow (θ, ϕ).

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Proof. The sets P and K := ∪p∈P Ap are compact by the compactness of A. Moreover, A ⊆ P × K, which is a compact set. Now distX (ϕ(t, p, x), K) = distP (θt (p), P ) + distX (ϕ(t, p, x), K) | {z } =0

= distX ((θt (p), ϕ(t, p, x)) , P × K) = distX (Π (t, (p, x)) , P × K) ≤ distX (Π (t, P × D) , P × K) , ≤ distX (Π (t, P × D) , A) → 0

x ∈ D, as t → ∞,

for any bounded subset D of X (hence P × D is bounded in X), since A is the global attractor of Π. Replacing p by θ−t (p) here we have lim distX (ϕ(t, θ−t (p), D), K) = 0.

t→∞

This says that the skew product flow (θ, ϕ) is asymptotically compact, which is a sufficient condition for the existence of a pullback attractor A0 = {A0p , p ∈ P } with A0p ⊆ K.

From Proposition 9.2 above, A0 = ∪p∈P {p} × A0p is a maximal invariant compact subset of X, but so is the global attractor. This means that A0 = A and hence that A0 = A. 

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Forward attractors and attracting sets

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Chapter 10

Limitations of pullback attractors of processes

Pullback attractors have been extensively investigated in past decades. Essentially, they describe the behaviour of a system “from the past” and, in general, have little to say about the future behaviour of the system. This will be illustrated here with some simple examples involving difference equations, i.e., discrete-time systems. We will also be seen that the skew product flow formulation often provides more detail about the behaviour of a nonautonomous dynamical system than does the process formulation due to its built-in driving system. The future behaviour of the system is described by forward attractors or, more generally, forward omega limit sets. These have only become better understood in recent years and will be discussed in the last chapters of the book. Both pullback and forward attractors or omega limit sets, when they exist, are needed to provide a full description of the dynamical behaviour of a nonautonomous system. In the simple examples here the pullback attractors are with respect to the pullback attraction of bounded sets or families of uniformly bounded sets. 10.1

An autonomous difference equation

We start with an autonomous scalar difference equation xn+1 =

λxn , 1 + |xn |

(10.1)

where λ is a real-valued parameter with λ ≥ 0. Clearly, x∗ = 0 is always a steady state solution for all λ ≥ 0. Other steady state solutions are obtained by solving x∗ =

λx∗ 1 + |x∗ |

for x∗ 6= 0, i.e., solving 1 + |x∗ | = λ or |x∗ | = λ − 1. 89

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Such solutions exist only if λ ≥ 1. In fact, the point λ∗ = 1 is a bifurcation point, where two new steady states x∗λ = ± (λ − 1) arise for λ > λ∗ = 1. There are thus two cases: (λ ≤ 1) : x∗ = 0 is the only steady state solution and it is globally asymptotically stable. The global (autonomous) attractor is A = {0}. x

1.5

1

0.5 n

0 0

−0.5

−1

−1.5 −5

Fig. 10.1

0

5

10

Typical solutions (interpolated) of the difference equation (10.1) with λ ≤ 1.

(λ > 1) : x∗ = 0 is the unstable steady state solution. The other steady states x∗λ = ± (λ − 1) are locally asymptotically stable with basins of attractors given by ±x0 > 0. The global attractor, i.e., which attracts all x0 in R, in this case is   Aλ = − |x∗λ | , |x∗λ | = [1 − λ, λ − 1] . 1.5

x

1 λ−1 0.5

x

0

A

n

λ

0

−0.5 1− λ −1

−1.5 −5

0

5

10

t

Fig. 10.2

Typical solutions (interpolated) of the difference equation (10.1) with λ > 1.

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10.2

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A piecewise autonomous difference equation

Consider the same difference equation (10.1) with the parameter λ switching its value, specifically ( λ for n ≥ 0 λ n xn xn+1 = , where λn = (10.2) 1 + |xn | λ−1 for n ≤ 0 for some λ ≥ 1. .5

x 1.5

1

1

0.5

0.5

0

0

0.5

−0.5

−1

−1

n

0

−1.5

.5 −5

Fig. 10.3

−4

−3

−2

−1

00

2

4

6

8

10

Typical solutions (interpolated) of the switching system (10.2).

This is a nonautonomous difference equation which results from switching between the two autonomomous difference equations at n = 0. The only bounded entire solution is the zero solution x ¯n ≡ 0

for all

n∈Z

and the pullback attractor consists of the identical singleton sets An ≡ {0}

for all

n ∈ Z.

Remark 10.1. The zero solution appears to be stable for n ≤ 0 (it is indeed asymptotically stable for the autonomous difference equation acting   for those times), but it is then unstable for n > 0. For n ≥ 0, the set − |x∗λ | , |x∗λ | looks like a global attractor (it is indeed the global attractor for the autonomous difference equation acting for these times), but it cannot be an attractor for this piecewise system because it is not invariant for n < 0. S the set of pullback attractor points n∈Z An = {0} is contained in  Obviously  − |x∗λ | , |x∗λ | , but the pullback attractor gives little information about the future

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An Introduction to Nonautonomous Dynamical Systems and their Attractors

asymptotic dynamics after the switch at n∗ = 0. In particular, this example has no uniformly bounded forward attractor.1 Remark 10.2. One often talks about asymptotically autonomous systems in one or both directions as n → ±∞ and analyses the “limiting“ asymptotic dynamics of the limiting autonomous systems separately.

10.3

Fully nonautonomous system

Let us now consider the nonautonomous difference equation xn+1 =

λ n xn , 1 + |xn |

(10.3)

where {λn }n∈Z is the monotonically increasing sequence given by λn = 1 +

0.9n , 1 + |n|

for all

n ∈ Z.

Obviously, λn < 1 for n < 0 and λn > 1 for n > 0. Note that 0.9 → 0.1 +1

λn = 1 −

n → −∞,

as

1 |n|

for n < 0, while λn = 1 +

0.9 ¯ → 1.9 =: λ +1

1 n

as

n → +∞

for n > 0. The convergence here is monotone. Moreover, λ never reaches the limit ¯ λ. x

1.5

1

0.5 n

0 0

−0.5

−1

−1.5 −10

Fig. 10.4

−5

0

5

10

15

Typical solutions (interpolated) of the system (10.3) with monotonically increasing λn .

1 If x ∗ n is any unbounded entire positive solution converging to xλ as n → ∞ then the sets An = [−xn xn ] are forward attracting, but grow unboundedly as n → −∞.

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As above, the zero solution is the only bounded entire solution. It appears to be “asymptotically stable” for n < 0 and to become unstable for n > 0. The pullback attractor consists of the identical singleton sets An ≡ {0}

n ∈ Z.
¯ However, in this case the forward limit points ± λ − 1 are not solutions of the difference equation at all. In particular, they are not entire solutions, so they cannot belong to an attractor, either autonomous or nonautonomous. This example has no uniformly bounded forward attractor (but see the previous footnote). for all

Remark 10.3. Note that we have used the process formulation of the nonautonomous system in the above analysis without actually explicitly saying so. In this and the previous example the pullback attractor provides only partial information about the full dynamics, in particular, about the future dynamics, of the difference equations. 10.4

More information through skew products

Consider the above nonautonomous difference equation (10.3) λ n xn , xn+1 = 1 + |xn | where λ = {λn }n∈Z is a given strictly increasing sequence with 0 < λ− < · · · < λn < λn+1 < · · · < λ+

where 0 < λ− := lim λn < 1 < λ+ := lim λn , n→−∞

n→∞

that is, with 0 < λ− < 1 < λ+ . Let Λ be the space of real valued bi-infinite sequences ρ = {ρn }n∈Z consisting + − as well as of the constant sequences ρ± with components ρ− and ρ+ n ≡ λ n ≡ λ k all shifts θ λ, k ∈ Z, of the given sequence λ = {λn }n∈Z , where θ is the left shift operator on the space of real valued bi-infinite sequences. Then, Λ is a compact metric space with the metric X 2−|n| |ρn − λ0n | dΛ (ρ, ρ0 ) := n∈Z

and θ : Λ → Λ is continuous, as the cocycle mapping defined by ϕ(n, ρ, x0 ) := xn . The pullback attractor A = {Aρ }ρ∈Λ for this skew product flow has component sets ( {0} for ρ ∈ Λ \ {ρ+ }, Aρ = [1 − λ+ , λ+ − 1] for ρ = ρ+ .

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The forward omega limit points end up in the set [   Aρ = 1 − λ+ , λ+ − 1 , ρ∈Λ

so the pullback attractor for the skew product flow under consideration contains all of the future asymptotic behaviour too. In contrast, the pullback attractor of the process for the difference equation (10.3) corresponding to the given sequence λ consists just of the zero solution. Remark 10.4. The skew product formulation in this example provides more detailed information about the behaviour of the nonautonomous dynamical system than does the process formulation. Essentially, it represents a family of processes, one for each of the sequences in the base space Λ of the driving system. Moreover, since the base space Λ is compact, it also provides information about the future limiting behaviour too.

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Chapter 11

Forward attractors

Autonomous systems depend only on the elapsed time, so their attractors and limit sets exist in current time. Similarly, the pullback limit defines an individual set of a nonautonomous pullback attractor at each instant of current time. The forward limit in the definition of a nonautonomous forward attractor is different as it is the limit to the asymptotically distant future. In particular, the limiting objects forward in time do not have the same dynamical meaning in current time as in the autonomous or pullback cases. Until recently it was not known how to construct the component sets of a forward attractor. Here we show how this can be done, but unlike the pullback case the result is only a necessary condition. The sets so constructed must satisfy additional conditions to be forward attracting. Remark 11.1. In this chapter and the following chapter we restrict attention to processes defined on the state space X = Rd . 11.1

Pullback attractors recalled

We consider a continuous-time process φ on Rd and recall that a φ-invariant family A = {At , t ∈ R} of nonempty compact subsets of Rd is called a pullback attractor if it pullback attracts all tempered families D = {Dt , t ∈ R} of nonempty bounded subsets of Rd , i.e., lim distRd (φ(t, t0 , Dt0 ), At ) = 0, (fixed t). (11.1) t0 →−∞

Recall that the subsets Dt in a tempered family D cannot grow too quickly as t → ±∞, See Definition 9.4. The existence of a pullback attractor is ensured by that of a pullback absorbing family. Theorem 11.1. Suppose that a process φ on Rd has a φ-positively invariant pullback absorbing family B = {Bt , t ∈ R} of nonempty compact subsets of Rd , i.e., for each t ∈ R and every family D = {Dt , t ∈ R} of nonempty tempered bounded subsets of Rd there exists a Tt,D ∈ R+ such that φ (t, t0 , Dt0 ) ⊆ Bt for all t0 ≤ t − Tt,D . 95

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Then φ has a global pullback attractor A = {At , t ∈ R} with component subsets determined by At =

\

for each t ∈ R.

φ (t, t0 , Bt0 )

t0 ≤t

Moreover, if A is uniformly bounded, i.e., if

S

t∈R

(11.2)

At is bounded, then it is unique.

The uniqueness of the pullback attractor is assured if, for example, the pullback S absorbing family B is uniformly bounded, i.e., if t∈R Bt is bounded. Theorem 11.1 is essentially a necessary and sufficient condition for the existence of a pullback attractor, since it can be shown that every pullback attractor has a positively invariant pullback absorbing family. 11.2

Nonautonomous forward attractors in Rd

Recall that a φ-invariant family A = {At , t ∈ R} of nonempty compact subsets of Rd is called a forward attractor if it forward attracts all families D = {Dt , t ∈ R} of nonempty tempered bounded subsets of Rd , i.e. lim distRd (φ(t, t0 , Dt0 ), At ) = 0,

t→∞

(fixed t0 ).

(11.3)

In contrast to the pullback limit in (11.1), the limit here involves a “moving” target. Definition 11.1. A φ-invariant family A = {At , t ∈ R} of nonempty compact subsets of Rd is said to be Lyapunov stable for a process φ if for each t0 ∈ R and ε > 0 there exists δ = δ(t0 , ε) > 0 such that distRd (φ(t, t0 , x0 ), At ) < ε

(11.4)

d

for all t ≥ t0 and x0 ∈ R with distRd (x0 , At0 ) ≤ δ. Proposition 11.1. A forward attractor A = {At , t ∈ R} of a process φ in Rd is Lyapunov asymptotically stable, i.e., both forward attracting in the sense of (11.3) and Lyapunov stable. This follows from the invariance of A, the continuity with respect to the Hausdorff metric of the set-valued mapping C 7→ φ(t, t0 , C) in compact sets uniformly on bounded time intervals t ∈ [t0 , T ] for each fixed t0 and the forward attracting property (11.3). The proof is left to the reader. that it uses the compactness  Note d of the closed and bounded subset Bδ [At0 ] = x ∈ R : distRd (x0 , At0 ) < δ in Rd . Recall that a family A = {At , t ∈ R} is uniformly bounded if there exists an R > 0 such that At ⊂ BR := {x ∈ Rd : kxk ≤ R} for each t ∈ R and define B1 [A] := {x ∈ Rd : distRd (x, A) ≤ 1} for any nonempty compact subset A of Rd . Proposition 11.2. A uniformly bounded forward attractor A = {At , t ∈ R} of a process φ in Rd has a φ-positively invariant family B = {Bt , t ∈ R} of nonempty compact subsets with At ⊂ Bt for each t ∈ R, which is forward absorbing.

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Proof. It can be assumed without loss of generality that R is large enough so that B1 [At ] ⊂ BR for each t ∈ R. By forward attraction of the set BR , for each t0 ∈ R there exists T (t0 , 1) ≥ 0 such that distRd (φ(t, t0 , BR ), At ) ≤ 1,

t ≥ t0 + T (t0 , 1).

Set t0 = 0 and define t1 to be the first integer with t1 ≥ t0 + T (t0 , 1). Then φ(t1 , t0 , B1 [At0 ]) ⊂ φ(t1 , t0 , BR ) ⊂ B1 [At1 ].

Repeat this construction for tk < tk+1 for k = 0, 1,2, . . . and define Bt := φ (t, tk , B1 [Atk ]) ,

tk ≤ t < tk+1 , k = 0, 1, 2, . . . .

These sets are obviously nonempty and compact. In particular, Btk = B1 [Atk ] and φ(t, t0 , Bt0 ) ⊂ Bt for all t ≥ t0 ≥ 0. The construction of Bt proceeds differently for t < 0. Define r0 := 1. Then the continuity with respect to the Hausdorff metric of the set-valued mapping C 7→ φ(t, t0 , C) in compact sets uniformly on bounded time intervals t ∈ [t0 , T ] for each fixed t0 with ε = 1, there exists r−1 := δ(−1, r0 ) = δ(−1, 1) > 0 such that φ(0, −1, Br−1 [A−1 ] ∩ BR ) ⊂ Bε [A0 ] = B1 [A0 ] = B1 [A0 ] ∩ BR .

Repeat this construction in the interval [−k, −k + 1] for k = 1,2, . . . and define
Bt := φ t, −k, Br−k [A−k ] ∩ BR ] , −k ≤ t < −k + 1, k = 1, 2, . . . with radius r−k = δ(−k, 1) > 0 according to Proposition 11.1, respectively. These sets are obviously nonempty and compact with B(−k) = Br−k [A−k ] ∩ BR and φ(t, t0 , Bt0 ) ⊂ Bt for all t0 ≤ t ≤ 0. Since r0 = 1, the positive invariance property holds for all t ∈ R. The family B = {Bt , t ∈ R} is by construction forward absorbing provided the time is taken large enough to reach one of the sets B1 [A−k ] with k ∈ N, which is possible by the forward attraction property.  11.3

Construction of possible forward attractors

It is possible to construct the component subsets of candidates for forward attractors with the same expression (11.2) as for a pullback attractor. This is based on the observation that a φ-positively invariant family of nonempty compact subsets of a process φ on Rd contains a maximal φ-invariant family of nonempty compact subsets. Essentially, it is formed by all the entire trajectories in the φ-positively invariant family. Theorem 11.2. Suppose that a process φ on Rd has a φ-positively invariant family B = {Bt , t ∈ R} of nonempty compact subsets of Rd . Then φ has a maximal φ-invariant family A = {At , t ∈ R} in B of nonempty compact subsets determined by \ At = φ (t, t0 , Bt0 ) for each t ∈ R. (11.5) t0 ≤t

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In view of Proposition 11.2 the component sets of all forward attractors can be constructed in this way. The proof of Theorem 11.2 is straightforward. The expression (11.5) is clear, since the sets {φ (t, t0 , Bt0 ) , t0 ≤ t} with t fixed are nonempty and compact by the continuity of φ and the assumed compactness of the subsets Bt0 . They are also nested since by the two-parameter semigroup property and the positive invariance, φ (t, t0 , Bt0 ) = φ (t, t1 , φ (t1 , t0 , Bt0 )) ⊂ φ (t, t1 , Bt1 ) for (t, t0 ), (t, t1 ), (t1 , t0 ) ∈ R2≥ := {(s, t) ∈ R2 : s ≥ t}. Remark 11.2. This pullback construction is used only inside the φ-positively invariant family B. It is equivalent to lim distRd (φ(t, t0 , Bt0 ), At ) = 0,

t0 →−∞

(fixed t).

(11.6)

Theorem 11.2 does not, however, imply that the subsets given by (11.5) form a pullback attractor, since nothing has been assumed so far about what is happening outside of the subsets Bt in B. With the additional assumption that B is pullback absorbing, then Theorem 11.1 holds and the family A = {At , t ∈ R} is a pullback attractor. The situation with forward attractors is somewhat more complicated than for pullback attractors. This is a consequence of the different nature of the forward limit in (11.3) and the pullback limit in (11.1). For example, the forward attractor need not, however, be unique, even when it is uniformly bounded. Example 11.1. Consider the nonautonomous scalar ODE x˙ = −2tx 2

(11.7)

+t20

with explicit solutions x(t, t0 , x0 )= x0 en−t . o (r) + (r) For any r ∈ R the family B = Bt , t ∈ R of nonempty compact subsets ( −t2 re [−1, 1] : t ≤ 0, (r) Bt = r[−1, 1] : t ≥ 0, is φ-positive invariant (and also forward absorbing). The corresponding subsets 2 (r) determined by (11.5) are At = re−t [−1, 1] for t ∈ R. Remark 11.3. The previous example shows that the forward attractor need not (r) be unique since for any r ∈ R+ , the sets At are the component sets of a forward attractor A(r) . Interestingly, the unique solution of the ODE (11.7) for each initial value x(t0 ) = x0 is bounded and entire, i.e., defined for all t ∈ R. In fact, every solution of this ODE is globally attracting in the forward sense and thus forms a forward attractor with component subsets consisting of singleton points of the solution.

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Let A(1) = n Forward oattractors are, n however, o asymptotically equivalent. (1) (2) (2) At , t ∈ R and A = At , t ∈ R be two forward attractors with the same attraction universe. Then, by invariance and forward attraction,       (2) (2) (1) (1) = distRd φ t, t0 , At0 , At distRd At At → 0 as → ∞ and similarly with the superscripts interchanged. Hence in the Hausdorff metric   (1) (2) → 0 as t → ∞. HRd At At 11.4

Forward asymptotic behaviour

The φ-invariant family A = {At , t ∈ R} with subsets At given by the expression (11.5) in Theorem 11.2 does not necessarily reflect the full limiting dynamics in the forward time direction of the process. Recall the discussion in Chapter 10. In particular, a pullback attractor need not be forward attracting in the sense of the forward convergence in (11.3). Example 11.2. Consider the process φ on R1 generated by the switching ODE ( −x : t ≤ 0, x˙ = f (t, x) := (11.8)
x 1 − x2 : t > 0.

This nonautonomous system is asymptotically autonomous in both time directions with the limiting autonomous systems equal to the component systems holding on the whole time set R. x 2

2

1.5

1.5

1

1

0.5

0.5

0

0

1

t 0

−0.5

−0.5

−1

−1

−1.5

−1.5

−2 −1.5

Fig. 11.1

−1

−0.5

−1

−2 0 0

0.5

1

1.5

Typical solutions of the switching system (11.8).

In the linear autonomous system the steady state solution 0 is asymptotically stable, while in the nonlinear autonomous system it is unstable and the steady state solutions ±1 are (locally) asymptotically stable. These autonomous systems have global autonomous attractors given by {0} and [−1, 1], respectively. The family B of constant sets Bt = [−2, 2] is φ-positively invariant. The corresponding family A constructed by the pullback method in (11.5) has identical

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component subsets At ≡ {0}, t ∈ R, corresponding to the zero entire solution and is the only bounded entire solution of the process. Remark 11.4. Here A = {At , t ∈ R} is, in fact, the pullback attractor of the nonautonomous system generated by the switching ODE (11.8), but that is not of interest to us here. More importantly, it is obviously not forward asymptotically attracting. Indeed, the set [−1, 1] of forward omega-limit points starting in the family B is not invariant for the process. (These are the limit points of convergent subsequences of sequences of points of the form φ(tn , t0 , bn ), where bn ∈Bt0 for some t0 with tn → ∞, see below) In particular, the boundary points ±1 are not entire solutions of the process, so cannot belong to a nonautonomous attractor, forward or pullback, since these consist of entire solutions. To ensure that the family A constructed by the pullback method in (11.5) is forward attracting we need the set of omega limit points of trajectories in it to coincide with the set of omega limit points of trajectories starting in the family B, rather than just to be a proper subset. In addition, the family B should be forward absorbing. 11.5

Omega limit points of a process

The asymptotic dynamics of the process φ on Rd inside the uniformly bounded φ-positively invariant family B = {Bt , t ∈ R} of nonempty compact subsets Bt of Rd will be investigated more closely here. For each t0 ∈ R, the forward omega limit set with respect to B is defined by \ [ φ (s, t0 , Bt0 ), ωB,t0 := t≥t0 s≥t

which is nonempty and compact as the intersection of nonempty nested compact subsets. In particular, lim distRd (φ(t, t0 , Bt0 ), ωB,t0 ) = 0,

t→∞

(fixed t0 ).

(11.9)

Since At0 ⊂ Bt0 and At = φ(t, t0 , At0 ) ⊂ φ(t, t0 , Bt0 ), it follows that lim distRd (At , ωB,t0 ) = 0,

t→∞

(fixed t0 ).

(11.10)

Let B be the bounded subset of R such that Bt ⊂ B for all t ∈ R. Moreover, φ(t, t0 , Bt0 ) ⊂ Bt ⊂ B for each t ≥ t0 since B is φ-positively invariant, so ωB,t0 ⊂ ωB,t00 ⊂ B,

t0 ≤ t00 ,

where the final inclusion is from the uniform boundedness of B. Hence the sets [ \ ωB,t0 ωB,t0 , ωB+∞ := ωB−∞ := t0 ∈R

t0 ∈R

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are nonempty compact sets with ωB−∞ ⊂ ωB+∞ ⊂ B. From (11.9) it is clear that
lim distRd At , ωB−∞ = 0. (11.11) t→∞

Consider now the set of omega limit points for dynamics starting inside the family of sets A = {At , t ∈ R} (note the sets here need not be nested), which is defined by \ [ \ [ ωA := At = φ(t, t0 , At0 ) t0 ∈R t≥t0

t0 ∈R t≥t0

which is nonempty and compact as a family of nested compact sets. Obviously, ωA ⊂ ωB−∞ ⊂ ωB+∞ ⊂ B. The inclusions here may be strict. Example 11.3. For the process generated by the ODE (11.8), ωA = {0}, while ωB−∞ = ωB+∞ = [−1, 1]. The ODE (11.8) can be changed to include a second switching at, say, time t = 10 to the nonlinear ODE x˙ = x(1 − x2 )(x2 − 4). Then ωB−∞ = [−1, 1], while ωB+∞ = [−2, 2]. 11.6

Conditions ensuring forward convergence

The existence of omega limit points in ωB+∞ that are not in ωA means that A cannot be forward attracting from within B, i.e., starting at points x0 ∈ Bt0 for t0 ∈ R. The converse also holds. Theorem 11.3. A is forward attracting from within B or a process φ on Rd , i.e., the forward convergence in (11.3) holds, if and only if ωA = ωB+∞ . Proof. Suppose that the forward convergence (11.3) does not hold. Then there is an ε0 > 0 and a sequence tn → ∞ as n → ∞ such that distRd (φ(tn , t0 , Bt0 ), Atn ) ≥ ε0 for all n ∈ N. Since the sets here are compact there exist points φn ∈ φ(tn , t0 , Bt0 ) such that distRd (φn , Atn ) = distRd (φ(tn , t0 , Bt0 ), Atn ) ≥ ε0 . Then for any points an ∈ Atn , kφn − an k ≥ distRd (φn , Atn ) ≥ ε0 . In particular, kφn − an k ≥ ε0 for all n ∈ N. Now the φn and an belong to the compact set B, so there are convergent subsequences φnj → φ¯ ∈ B and anj → a ¯ ∈ B. It follows that kφ¯ − a ¯k ≥ ε0 . From the +∞ definitions φ¯ ∈ ωB,t0 ⊂ ωB and a ¯ ∈ ω
A . Since the an and hence
a ¯ were otherwise +∞ ¯ arbitrary, it follows that distRd φ, ωA ≥ ε0 , so distRd ωB , ωA ≥ ε0 and hence ωA 6= ωB+∞ . 

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Forward attraction also holds when the rate of pullback convergence from within B to construct the component sets At of A in Theorem 11.2 is uniform. In fact, it suffices that the convergence rate is eventually uniform. Theorem 11.4. For a process φ on Rd the family A = {At , t ∈ R} with subsets At constructed by (11.5) is forward attracting from within B, i.e., the forward convergence (11.3) holds, if the rate of pullback convergence from within B to the component sets At of A is eventually uniform in the sense that for every ε > 0 there exist τ (ε) ∈ R and T (ε) > 0 such that for each t ≥ τ (ε) distRd (φ(t, t0 , Bt0 ), At ) < ε

(11.12)

holds for all t0 ≤ t − T (ε). Proof. Given any t0 ∈ R and ε > 0, it is clear that the inequality (11.12) holds for every t ≥ max{t0 , τ (ε)} + T (ε), i.e., the forward convergence (11.3) holds.  Theorem 11.4 is concerned only with the rate of pullback convergence starting from inside the positively invariant family B. It makes no assumptions about what is happening outside of B such as pullback or forward absorption. Remark 11.5. The family A for the simple switching system generated by the ODE (11.8) is not forward attracting. By considering any time t > 0, it is not hard to see that the rate of pullback convergence is not eventually uniform. 11.7

Asymptotically autonomous systems

The ODE (11.8) generates a simple process φ on R1 formed by switching between two autonomous systems. It is a special case of an asymptotically autonomous system, see, e.g., Hale [Hale (1988)], Kato et al. [Kato, Martynyuk and Sheshakov (1996)] and Lasalle [Lasalle (1976)]. These have additional structure that provides an easier way to check the condition of Theorem 11.3 for forward convergence. theorem in [Kloeden and Simsen (2015)]. Theorem 11.5. Let φ be a uniformly bounded process on Rd with a φ-positively invariant family B = {Bt , t ∈ R} of nonempty compact subsets of Rd with all component subsets contained in a compact set B. Let π be an autonomous semi-dynamical system on Rd with a global attractor A∞ in B. In addition, suppose that φ(t + τ, τ, xτ ) → π(t, x0 )

as τ → +∞

(11.13)

uniformly in t ∈ R+ whenever xτ ∈ Bτ and xτ → x0 as τ → +∞. +∞ Then ωB,t ⊂ A∞ for each t0 ∈ R and 0 lim distRd (At , A∞ ) = 0,

t→+∞

(11.14)

where A = {At , t ∈ R} is the family of nonempty compact subsets of Rd with all component subsets defined by pullback characterisation (11.5) from within B.

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Proof. Fix t ∈ R+ and t0 ∈ R and define sn := tn − t, where the sequence tn → ∞ as n → ∞. By the 2-parameter semigroup property and the φ-positive invariance of B φ (tn , t0 , Bt0 ) = φ (tn , sn , φ (sn , t0 , Bt0 )) ⊂ φ (tn , sn , B(sn )) . Hence φ (t + sn , t0 , Bt0 ) ⊂ φ (t + sn , sn , Bsn ). By compactness of the sets involved, for each n ∈ N, there exists a bn ∈ Bsn ⊂ B such that distRd (φ (t + sn , sn , bn ) , A∞ ) = distRd (φ (t + sn , sn , Bsn ) , A∞ ) . By the compactness of the set B there is a convergent subsequence (labelled for convenience as the original one) bn → ¯b ∈ B as n → ∞. Hence,

distRd (φ (t + sn , sn , bn ) , A∞ ) ≤ distRd φ (t + sn , sn , bn ) , π t, ¯b

+ distRd π t, ¯b , A∞

≤ ε + distRd π t, ¯b , A∞ for any ε > 0 and n ≥ N (ε) uniformly in t ∈ R by the asymptotic autonomous condition (11.13). Since A∞ is the global attractor of π for any ε > 0 there exists a T (ε, B) ≥ 0 such that

distRd π t, ¯b , A∞ ≤ distRd (π (t, B) , A∞ ) < ε for all t ≥ T (ε, B). Combining the results gives lim distRd (φ (t, t0 , Bt0 ) , A∞ ) = 0,

t→+∞

(11.15)

+∞ for each t0 ∈ R. This means that ωB,t ⊂ A∞ for each t0 ∈ R. 0 The limit (11.14) then follows since At = φ (t, t0 , At0 ) ⊂ φ (t, t0 , Bt0 ), so

distRd (At , A∞ ) = distRd (φ (t, t0 , At0 ) , A∞ ) ≤ distRd (φ (t, t0 , Bt0 ) , A∞ ) .  Combining Theorem 11.3 and the limit (11.15) in the proof of Theorem 11.5 gives: Corollary 11.1. Suppose that the assumptions of Theorems 11.3 and 11.5 hold and that ωA = A∞ . Then A = {At , t ∈ R} is forward attracting from within B, i.e., the forward convergence in (11.3) holds. Condition 11.13 in Theorem 11.5 can be easily checked directly in terms of the continuous mappings f and fn of the autonomous and nonautonomous differences equation xn+1 = f (xn ),

xn+1 = fn (xn ),

n∈Z

provided the compact set B in Theorem 11.5 is positively invariant under f and the fn , i.e., f (B) ⊂ B and fn (B) ⊂ B, for all large n.

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Let φ be the process generated by the nonautonomous difference equation and π be the semi-group generated by the autonomous difference equation. Lemma 11.1. Condition (11.13) of Theorem 11.5 holds if
lim distRd fn (xn ), f (x0 ) = 0 for all xn → x0 , n→∞

(11.16)

provided the compact set B in Theorem 11.5 is positively invariant under f and the fn , i.e., f (B) ⊂ B and fn (B) ⊂ B for all large n. Proof. The proof is by induction. Condition (11.13) is true for N = 1 since it then reduces to (11.16). Suppose it is true for N ≥ 1 and consider the case with N + 1. By the twoparameter semi-group property φ(n + N + 1, n, xn ) = φ(n + N + 1, n + N, ϕ(n + N, n, xn )) = fn+N (ϕ(n + N, n, xn )) and by the semi-group property π(N + 1, x0 ) = π(1, π(N, x0 )) = f (π(N, x0 )). Hence φ(n + N + 1, n, xn ) − π(N + 1, x0 ) = fn+N (ϕ(n + N, n, xn )) − f (π(N, x0 )) = fn+N (zn+N ) − f (z),

where zn+N := φ(n + N, n, xn ) → z := π(N, x0 ) by the induction assumption. Relabelling the sequences with an index k = n + N , it follows that φ(n + N + 1, n, xn ) → π(N + 1, x0 ), so the result is then true for N + 1.



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Chapter 12

Omega-limit sets and forward attracting sets

In autonomous dynamical systems the attractor corresponds to the ω-limit set ωB of an absorbing set B. Although this set is defined as the limit as time goes to infinity, it actually exists for all time, since the time variable in an autonomous dynamical system π(t, x) is just the elapsed time from the start and not the actual time. In particular, the ω-limit set ωB here is an invariant set. The situation is rather different in a nonautonomous dynamical system described by a process φ(t, t0 , x) which depends on both the actual time and starting time and not only their difference. ω-limit sets can be defined here, but depend on the starting time, i.e., ωB,t0 [Lasalle (1976)]. These are increasing in t0 and the closure of their union ωB represents the set of all future limit points of the system. Haraux [Haraux (1991)] introduced the concept of a uniform attractor 1 as the smallest set that attracts all bounded sets uniformly in the initial time. This was extensively investigated in applications by Vishik and his coworkers [Chepyzhov and Vishik (2002); Vishik (1992)] It contains all forward ω-limit points, hence ωB , but can be a larger set as an example in the appendix of [Carvalho, Langa and Robinson (2013)] shows. However, neither the Haraux-Vishik uniform attractor nor ωB need be invariant or even positive invariant. Nevertheless, as will be shown in this chapter, they are nearly invariant in an asymptotic sense, i.e., asymptotically positive invariant and also asymptotically negative invariant under some increasingly stronger uniformity assumptions on the future dynamics. They thus resemble attractors as usually understood, i.e., with the properties of attraction, compactness and invariance. We saw in earlier chapters that skew product flows with a driving system on a compact base space allow much more to be said about the future asymptotic dynamics of the system. These are important, but special cases, which include many interesting situations. 1 This is not to be confused with the uniform (nonautonomous) attractor introduced in earlier, which is uniformly both a forward and pullback attractor.

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Forward attracting sets

Let φ be a continuous time process on Rd and suppose that φ is dissipative, i.e., it has a closed and bounded (hence compact) absorbing set B in Rd , which is eventually uniformly absorbing. Assumption 12.1. There exists a φ-positive invariant compact subset B in Rd for a process φ on Rd and a T ∗ ≥ 0 such that for any bounded subset D of Rd and every t0 ≥ T ∗ there exists a TD ≥ 0 for which φ(t, t0 , x0 ) ∈ B

∀t ≥ t0 + TD , x0 ∈ D.

It follows from this dissipativity assumption and the compactness of the set B that the ω-limit set \ [ φ (s, t0 , B) ωB,t0 := t≥t0 s≥t

is a nonempty compact set of R for each t0 ≥ T ∗ . Note that d

lim distRd (φ (t, t0 , B) , ωB,t0 ) = 0

t→∞

(12.1)

for each t0 ≥ T ∗ and that ωB,t0 ⊂ ωB,t00 ⊂ B for t0 ≤ t00 . Hence, the set [ ωB,t0 ⊂ B ωB := t0 ≥T ∗

is nonempty and compact. It contains all of the future limit points of the process starting in the set B at some time t0 ≥ T ∗ . In particular, y¯ ∈ ωB if there exist sequences b0,j ∈ B and t0,j ≤ t00,j with t0,j → ∞ as j → ∞ such that φ(t00,j , t0,j , b0,j ) → y¯ as j → ∞. The set ωB characterises the forward asymptotic behaviour of the nonautonomous system. It was called the forward attracting set of the nonautonomous system in [Kloeden and Yang (2016)] and is closely related to the Haraux-Vishik uniform attractor, but it may be smaller and does not require the generating ODE (12.5) to be defined for all time or the attraction to be uniform in the initial time [Kloeden and Lorenz (2016); Kloeden and Yang (2016)]. Example 12.1. The scalar ODE x˙ = −x + e−t with ωB = {0} shows that the set ωB need not be invariant or even positive invariant. See Figure 12.1. 12.1.1

Asymptotically positive invariance

The set ωB = {0} in the previous example appears to become more and more invariant the later one starts in the future. This motivated the concept of asymptotically positive invariance in the literature for nonautonomous differential equations [Kloeden (1975); Lakshmikantham and Leela (1967)]. Positive invariance says that a set is mapped into itself at future

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25

20

15

10

5

0 -4

Fig. 12.1

-3

-2

-1

0

1

2

3

4

Solutions of x˙ = −x + e−t with different initial conditions.

times, while asymptotical positive invariance says it is mapped almost into itself and closer the later one starts. Definition 12.1. A set A is said to be asymptotically positive invariant for a process φ on Rd if for every ε > 0 here exists a T (ε) such that φ (t, t0 , A) ⊂ Bε (A) , t ≥ t0 , (12.2) d for each t0 ≥ T (ε), where Bε (A) := {x ∈ R : distRd (x, A) < ε}. Theorem 12.1. Suppose that Assumption 12.1 hold for a process φ on Rd . Then ωB is asymptotically positive invariant. Proof. Since ωB,t0 ⊂ ωB ⊂ B and φ(t, t0 , ωB ) ⊂ φ(t, t0 , B) ⊂ B, by the convergence (12.1), for every ε > 0 and t0 ≥ T ∗ there exists T0 (t0 , ε) ∈ R+ such that distRd (φ(t, t0 , ωB ), ωB ) < ε for t ≥ t0 + T0 (t0 , ε). Suppose that for some ε0 > 0 there are sequences t0,j ≤ tj ≤ t0,j + T0 (t0,j , ε0 ) with t0,j → ∞ as j → ∞ such that distRd (φ(tj , t0,j , ωB ), ωB ) ≥ ε0 , j ∈ N. Since φ(tj , t0,j , ωB ) is compact there exists a bj ∈ ωB ⊂ B such that distRd (φ(tj , t0,j , bj ), ωB ) = distRd (φ(tj , t0,j , ωB ), ωB ) ≥ ε0 , j ∈ N. Define yj := φ(tj , t0,j , bj ). Since the points yj ∈ B, which is compact, there exists a convergent subsequence yjk → y¯ ∈ B. Moreover, y¯ ∈ ωB by the definition. However, distRd (yj , ωB ) ≥ ε0 , so distRd (¯ y , ωB ) ≥ ε0 , which is a contradiction.

Hence for any ε > 0 there exists a T (ε) large enough such that distRd (φ(t, t0 , ωB ), ωB ) < ε for t ≥ t0 ≥ T ε). It follows that ωB is asympotically positive invariant for the process φ on Rd .



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Asymptotically negative invariance

The concept of negative invariance of a set implies that any point in it can be reached in any even time from another point in it. The set ωB is generally not negative invariant, but under an additional uniformity assumption it is asymptotically negative invariant. Definition 12.2. A set A is said to be asymptotically negative invariant for a process φ if for every a ∈ A, ε > 0 and T > 0, there exist tε and aε ∈ A such that kφ (tε , tε − T, aε ) − ak < ε. To show that this property holds a further assumption about the future uniform behaviour of the process is needed. Assumption 12.2. There exists a T ∗ ≥ 0 such that the process φ on Rd is Lipschitz continuous in initial conditions in B on finite time intervals [t0 , t0 + T ] uniformly in t0 ≥ T ∗ , i.e., there exists a constant LB > 0 independent of t0 ≥ T ∗ such that kφ (t, τ, x0 ) − φ (t, τ, y0 )k ≤ kx0 − y0 k eLB (t−τ ) ≤ kx0 − y0 k eLB T (12.3) for all x0 , y0 ∈ B and T ∗ ≤ τ < t ≤ τ + T . Theorem 12.2. Suppose that the Assumptions 12.1 and 12.2 hold for a process φ on Rd . Then ωB is asymptotically negative invariant. Proof. To show this let ω ∈ ωB , ε > 0 and T > 0 be given. Then there exist sequences bn ∈ B and τn < tn with τn → ∞ and an integer N (ε) such that 1 kφ (tn , τn , bn ) − ωk < ε, n ≥ N (ε). 2 Define an := φ (tn − T, τn , bn ) ∈ B.
Since B is compact, there exists a convergent subsequence anj := φ tnj − T, τnj , bnj → ωε as nj → ∞. By definition, ωε ∈ ωB .

From Assumption 12.2 the process φ is continuous in initial conditions uniformly on finite time intervals of the same length, i.e., (12.3). Hence



b (ε).

φ tnj , tnj − T, anj − φ tnj , tnj − T, ωε < 1 ε, nj ≥ N (12.4) 2 By the 2-parameter semi-group property



φ tnj , tnj − T, anj = φ tnj , tnj − T, φ tnj , τnj , bnj = φ tnj , τnj , bnj . Then





ω − φ tnj , tnj − T, ωε ≤ ω − φ tnj , tnj − T, anj



+ φ tnj , tnj − T, anj − φ tnj , tnj − T, ωε


≤ ω − φ tnj , τnj , bnj



+ φ tnj , tnj − T, anj − φ tnj , tnj − T, ωε
0 and T > 0 there exists a δ(ε, T ) > 0 such that

ν

φ (t, t0 , b)) − φ0 (t, t0 , b)) < ε, t0 ≤ t ≤ t0 + T, b ∈ B.

for |ν| < δ(ε, T ) and t0 ≥ T ∗ .

0 Finally, the uniform attraction of the set ωB for the system φ0 is also needed for the following result. 0 Assumption 12.5. ωB uniformly attracts the set B under the process φ0 on Rd , i.e., for every ε > 0 there exists a T (ε), which is independent of t0 ≥ T ∗ , such that
0 distRd φ0 (t0 + t, t0 , B) , ωB < ε, t ≥ T (ε), t0 ≥ T ∗ .

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Then the upper semi-continuous convergence of the forward attracting sets in the parameter holds. Theorem 12.3. Suppose that Assumptions 12.3, 12.4 and 12.5 hold for the family of processes φν on Rd . Then
ν 0 distRd ωB , ωB →0 as ν → 0. Proof. A proof by contradiction will be used. Suppose for some sequence of parameters νj → 0 that the above limit is not true, i.e., there exists an ε0 > 0 such that
ν 0 ≥ ε0 , j ∈ N. distRd ωBj , ωB ν

ν

Since ωBj is compact, there exists ωj ∈ ωBj such that

ν 0 0 distRd ωj , ωB = distRd ωBj , ωB ≥ ε0 ,

j ∈ N.

(12.9)

By Assumption 12.5 there is a T = T (ε0 /8) such that for any t0 ≥ T ∗
1 0 < ε0 . distRd φ0 (t0 + T, t0 , B) , ωB 8 Then use Assumption 12.4 with this T to pick a νj < δ(ε0 /2, T ) to ensure that

ν

φ j (t0 , t0 − T, b)) − φ0 (t, t0 − T, b) < 1 ε0 , b ∈ B, t0  0. (12.10) 2 ν Fix such a νj and use the asymptotical negative invariance of ωBj to obtain the νj existence of an ωj,T ∈ ωB and a tjε  0 so that

ν j j




φ j tε , tε − T, ωj,T − ωj < 1 ε0 . 8 j Then, with t0 taken as tε above,

0 distRd ωj , ωB ≤ ωj − φνj tjε , tjε − T, ωj,T



+ φνj tjε , tjε − T, ωj,T − φ0 tjε , tjε − T, ωj,T
0
+ distRd φ0 tjε , tjε − T, ωj,T , ωB 1 1 3 1 ε0 + ε0 + ε0 = ε0 , 8 2 8 4 which contradicts the assumption (12.9).
0 such that

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for all t ≥ T ∗ , kxk ≥ R∗ , ν ∈ [0, ν ∗ ].

hf ν (t, x), xi ≤ −1

In this case, B is the closed and bounded ball B0 [R∗ ] of radius R∗ about the origin in Rd . In addition, Assumption 12.4 holds if, for example, the vector fields of the ODEs (12.11) uniform Lipschitz condition (12.6) uniformly in the parameter ν and converge continuously as the parameter ν → 0 uniformly in time on the set B, i.e., if b T ) > 0 such that for every ε > 0 and T > 0 there exists a δ(ε,

ν

f (t, x) − f 0 (t, x) < ε,

t0 ≤ t ≤ t0 + T, x ∈ B,

(12.12)

b T ) and any t0 ≥ T ∗ . for |ν| < δ(ε,

Consider the solutions φ0 (t, τ, x0 ) and φν (t, τ, x0 ) of the ODEs (12.5) and (12.11), respectively, in the set B. Then

ν

φ (t, τ, x0 ) − φ0 (t, τ, x0 ) ≤ ≤

Z

t

ν


f (s, φν (s, τ, x0 ))) − f 0 s, φ0 (s, τ, x0 ) ds

τ

Z

t

ν


f (s, φν (s, τ, x0 )) − f ν s, φ0 (s, τ, x0 ) ds

τ

t

Z τ

ν



f s, φ0 (s, τ, x0 ) − f 0 s, φ0 (s, τ, x0 ) ds

Z

t

+ ≤ LB

τ

ν

φ (s, τ, x0 ) − φ0 ((s, τ, x0 ) ds + εT,

when 0 ≤ t−τ ≤ T . Here LB is the common Lipschitz constant, while the continuous convergence(12.12) is used to estimate the integral in the second last line. Then Gronwall’s inequality gives

ν

φ (t, τ, x0 ) − φ0 (t, τ, x0 ) ≤ εT eLB (t−τ ) ≤ εT eLB T , when 0 ≤ t − τ ≤ T , i.e., the bound depends just on the length of the time interval and not the starting time. Then the assertion of Assumption 12.4 then holds with δ(ε, T ) = b −1 e−LB T , T ). δ(εT Example 12.5. Continuing with Example 12.3, but now with a parameterised family of mappings fnν : Rd → Rd , we note that Assumption 12.4 holds of the fn are Lipschitz on B uniformly in n ≥ N ∗ , i.e., there exists a constant L such that

max fnν (b) − fn0 (b) → 0 b∈B

as ν → 0 uniformly in n ≥ N ∗ .

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Concluding remark

The nonautonomity of a dynamical system allows a much wider variety of behaviour than is possible in an autonomous system. Essentially, the vector fields of the underlying differential and difference equations can vary arbitrarily in time. In order to obtain specific results about the behaviour of such systems, as we have seen, we need to restrict the arbitrariness of these changes in some way with some kind of uniformity assumption such as periodicity, almost periodicity or some eventual uniformity in attraction rates, etc. Although such assumptions restrict the classes of systems under consideration, these classes are, nevertheless, still of interest in a wide range of applications.

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Random attractors

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Chapter 13

Random dynamical systems

Random dynamical systems are intrinsically nonautonomous due to the changing nature of the driving noise. Continuous time random dynamical systems are generated by random ordinary differential equations (RODEs) that is, ordinary differential equations with random coefficients or stochastic processes in their coefficients, and by stochastic differential equations (SDEs), while random difference equations generate discrete time random dynamical systems. Their solutions are stochastic processes, which can be thought of as a family of time dependent paths labelled by a point ω in sample space Ω of the underlying probability space (Ω, F, P). A very fundamental stochastic process is the Wiener process or Brownian motion {Wt }t≥0 . Definition 13.1. A Wiener process {Wt }t≥0 is defined by the properties: 1) Wt ∼ N (0, t) for each t ≥ 0, i.e., Wt is Gaussian distributed with (i) W0 = 0 with probability 1; (ii) E(Wt ) = 0 for each t ≥ 0; (iii) E(Wt2 ) = t for each t ≥ 0; 2) the nonoverlapping increments of Wt are independent, i.e., Wt2 − Wt1 and Wt4 − Wt3 are independent random variables for all 0 ≤ t1 < t2 ≤ t3 < t4 . It follows from these properties that a Wiener process has continuous, in fact H¨ older continuous sample paths, which are not differentiable. Other important stochastic processes with continuous sample paths are the fractional Brownian motion and the Ornstein-Uhlenbeck process. A two-sided Wiener process Wt is defined for all t ∈ R and not just for R+ . Essentially, Wt and W−t for t ∈ R+ are two independent Wiener processes as in the definition above. 117

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13.0.1

A simple example

RODEs are essentially nonautonomous ODEs for each path labelled by ω ∈ Ω and can be solved using deterministic calculus and the standard methods and tricks for ODES. Let cos Wt (ω) be a specific sample path of a two-sided Wiener process Wt and consider the simple scalar RODE dx = −x + cos Wt (ω), (13.1) dt which obviously has no constant solution. However, the RODE (13.1) with an initial value x0 ∈ R1 at the initial time t0 ∈ R has the explicit solution Z t −(t−t0 ) −t es cos Ws (ω) ds. (13.2) x(t, t0 , x0 , ω) = x0 e +e t0

This is very similar to Example 5.4, but with bounded noise cos Wt (ω) instead of the deterministic cos t as the forcing term. It has no pathwise limit as t → ∞, but the difference of two solutions satisfies |x(t, t0 , x0 , ω) − y(t, t0 , y0 , ω)| ≤ |x0 − y0 | e−(t−t0 ) → 0

as t → ∞,

(13.3)

so all solutions converge pathwise to each other. On the other hand, the pullback limit, i.e., for t0 → −∞ with t held fixed, of the solution (13.2) does exist and is given by Z t lim x(t, ω) = x ¯(t, ω) := e−t es cos Ws (ω) ds. (13.4) t0 →−∞

−∞

Moreover, x ¯(t, ω) is itself a solution of the RODE (13.1), so the inequality (13.3) applies and gives |x(t, t0 , x0 , ω) − x ¯(t, ω)| ≤ |x0 − x ¯(t0 , ω)| e−(t−t0 ) → 0

as t → ∞.

(13.5)

This means that the pullback limit solution (13.4) attracts all other solutions for each sample path forwards in time. It is, pathwise, globally asymptotically stable with an exponential rate of attraction. It is the counterpart of a nonautonomous equilibrium solution in a nonautonomous ODE and is sometimes called a random equilibrium. Remark 13.1. This simple example suggests that random dynamical systems could be defined analogously to nonautonomous 2-parameter semi-groups with an additional parameter ω ∈ Ω to label the sample paths. Pullback attractors of such random dynamical systems would then be a candidate for the definition of a random attractor. In the above example the random attractor would consist of the singleton sets A(t, ω) = {¯ x(t, ω)} for t ∈ R and ω ∈ Ω. This approach is particularly useful when the system is also driven by deterministic forcing, e.g., periodic forcing, as well as random forcing as for example in the RODE dx = −x + cos t + cos Wt (ω), (13.6) dt

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where Wt is two-sided Wiener process. This has the explicit solution Z t Z t es cos s ds + e−t es cos Ws (ω) ds x(t, t0 , x0 , ω) = x0 e−(t−t0 ) + e−t t0

(13.7)

t0

and the pullback limit of the solution (13.7) is lim x(t, ω) = x ¯(t, ω) :=

t0 →−∞

1 (cos t + sin t) + e−t 2

Z

t

es cos Ws (ω) ds.

−∞

This is a unique bounded entire solution that attracts all other solutions in both the pullback and forwards senses. It suggests that the random 2-parameter semigroup x(t, t0 , x0 , ω) generated by the RODE (13.6) has a nonautonomous random attractor which consists of the singleton sets A(t, ω) = {¯ x(t, ω)} for t ∈ R and ω ∈ Ω. This approach has been used in the literature to define nonautonomous random dynamical systems and a nonautonomous random attractors. 13.1

Definition of a random dynamical system

Historically, the mathematical theory of random dynamical systems (RDS) has developed similarly to skew product flows, except that the driving system is represented by a metrical dynamical system (this terminology comes from ergodic theory) on a measure space and is measurable rather than continuous in the corresponding variable. This system represents the noise which drives the system, while a cocycle mapping represents the dynamics in the topological state space. This approach, while seemingly more complicated, has various mathematical advantages. For example, it allows the theory to be developed without direct reference to the detailed properties of a specific driving stochastic process. Instead the driving system θ in the continuous time case is defined in terms of shift operators θt on the canonical sample space Ω := C0 (R, R) of continuous functions ω : R → R with ω(0) = 0, i.e., with θt (ω(·)) := ω(t + ·) − ω(·),

t ∈ R.

(13.8)

The σ-algebra of Borel subsets of C0 (R, R) is taken as the σ-algebra of events F and P is the corresponding Wiener measure when noise is based on a Wiener process and similarly for other noises. Note that no other topological properties of the space C0 (R, R) are used here apart from those defining the Borel sets. Remark 13.2. Essentially, the detailed information of the specific noise process is built into and hidden in the underlying probability space (Ω, F, P), which allows a more general and abstract theory to be developed. The price that one pays for this generality is the difficulty in applying it to certain applications. Let (Ω, F, P) be a probability space, i.e., with sample space Ω, a σ-algebra F of admissible events (measurable subsets of Ω) and a probability measure P on F, and

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let T = R or Z, depending on whether a continuous time or discrete time system is being considered. Definition 13.2 (Random dynamical system). A random dynamical system (θ, ϕ) on Ω × Rd consists of a metric dynamical system θ on Ω, i.e., a group family of measure preserving transformations θt : Ω → Ω, t ∈ T, such that (i) θ0 = idΩ and θt ◦ θs = θt+s for all t, s ∈ T, (ii) the map (t, ω) 7→ θt (ω) is measurable and invariant with respect to P in the sense that θt (P) = P for all t ∈ T, d d and a cocycle mapping ϕ : T+ 0 × Ω × R → R such that

(a) initial condition: ϕ(0, ω, x) = x for all ω ∈ Ω and x ∈ Rd , (b) cocycle property: ϕ(s + t, ω, x) = ϕ(s, θt (ω), ϕ(t, ω, x)) for all s, t ∈ R+ , ω ∈ Ω and x ∈ Rd , (c) measurability: (t, ω, x) 7→ ϕ(t, ω, x) is measurable, (d) continuity: x 7→ ϕ(t, ω, x) is continuous for all (t, ω) ∈ R × Ω. The notation θt (T) = T for the measure preserving property of θt with respect to T is just a compact way of writing T(θt (A)) = T(A), t ∈ R, A ∈ F. A systematic treatment of the theory of random dynamical system is given in [Arnold (1998)], where technical details can be found. Example 13.1. Using the canonical sample space representation (13.8) of the noise, the RODE (13.1) takes the form dx = −x + cos(θt (ω)), dt and its cocycle mapping for an initial value x0 ∈ R1 and noise sample path ω ∈ Ω is given by Z t −t −t ϕ(t, x0 , ω) = x0 e + e es cos(θs (ω)) ds, 0

that is, ϕ(t, x0 , ω) = x(t, 0, x0 , ω). Here t is the time that has elapsed since starting with the initial noise state ω and initial value x0 ∈ R1 . The solution x(t, t0 , x0 , ω) of the RODE (13.1) given by (13.2) corresponds to ϕ(t, x0 , θt0 (ω)) with the initial noise state ω adjusted to θt0 (ω) corresponding to the starting time t0 . Remark 13.3. A random dynamical system (θ, ϕ) on Ω × Rd is not a skew product flow or an autonomous semi-dynamical system on Ω × Rd , since Ω (in the context considered here) is not a topological space. Nevertheless, skew product flows and random dynamical systems have many common properties, and concepts and results for one can be used with appropriate modifications for the other. The most significant modification concerns measurability and the use of families of random sets.

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Random attractors

The example in the previous section suggests that a random attractor can be defined similarly to pullback attractors or skew product flows in Chapter 9. This requires the following definitions. Definition 13.3. A family D = {D(ω), ω ∈ Ω} of nonempty subsets of Rd is called a random set if the mapping ω 7→ distRd (x, D(ω)) is F-measurable for all x ∈ Rd . A random set D = {D(ω), ω ∈ Ω} is called a random closed set if D(ω) is closed for each ω ∈ Ω and a random compact set if D(ω) is compact for each ω ∈ Ω. Definition 13.4. A random set D = {D(ω), ω ∈ Ω} is said to be tempered if  D(ω) ⊂ x ∈ Rd : kxk ≤ r(ω) , ω ∈ Ω , where the random variable r(ω) > 0 is tempered, i.e., sup r(θt (ω))e−γ|t| < ∞, t∈R

γ > 0, ω ∈ Ω .

The collection of all tempered random sets in Rd will be denoted by D. A random attractor of a random dynamical system is essentially a pullback attractor which is a random set. Definition 13.5 (Random attractor). A random closed set A = {A(ω), ω ∈ Ω} from D is called a random attractor of a random dynamical system (θ, ϕ) on Ω × Rd in D if A is a ϕ-invariant set, i.e., ϕ(t, ω, A(ω)) = A(θt (ω)),

t ≥ 0, ω ∈ Ω ,

and pathwise pullback attracting in D, i.e., lim distRd (ϕ(t, θ−t (ω), D(θ−t (ω))), A(ω)) = 0,

t→∞

ω ∈ Ω, D ∈ D .

The existence of a random attractor follows from that of a pullback absorbing tempered random set, i.e., a tempered random set B = {B(ω), ω ∈ Ω} with compact component sets for which there exists a TD,ω ≥ 0 such that ϕ(t, θ−t (ω), D(θ−t (ω)) ⊂ B(ω),

t ≥ TD,ω ,

(13.9)

for every tempered random set D = {D(ω), ω ∈ Ω}. Theorem 13.1 (Existence of random attractors). Let (θ, ϕ) be a random dynamical system on Ω × Rd such that ϕ(t, ω, ·) : Rd → Rd is a continuous for each fixed t > 0 and ω ∈ Ω. If there exists a tempered random set B = {B(ω), ω ∈ Ω} with compact component sets, then the random dynamical system (θ, ϕ) has a random pullback attractor A = {A(ω), ω ∈ Ω} with compact component sets defined for each ω ∈ Ω by \ [ A(ω) = ϕ(t, θ−t (ω), B(θ−t (ω))) . (13.10) s>0 t≥s

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Proof. The proof of Theorem 13.1 is essentially the same as for its counterpart Theorem 9.1 for deterministic skew product flows, see the proof of Theorem 3.20 in [Kloeden and Rasmussen (2011)]. The full proof in the random case is given in [Crauel and Flandoli (1994)]. The only new feature is that of measurability, i.e., to show that A = {A(ω), ω ∈ Ω} is a random set. This follows from the fact that the set-valued mappings ω 7→ ϕ(t, θ−t (ω), B(θ−t (ω))) are measurable for each t ∈ R+ 0 and that the union and intersection in (13.10) can be taken over a countable number of time instants (in the continuous time case). Remark 13.4. There is no analogous theorem for the existence of a random forward attractor, although when θ is ergodic, then P (ω ∈ Ω : distRd (ϕ(t, θ−t (ω), B(θ−t (ω)) , A(ω)) ≥ ε) = P (ω ∈ Ω : distRd (ϕ(t, ω, B(ω), A(θt (ω)) ≥ ε) for each ε > 0, so a (pathwise convergent, hence convergent in probability) random pullback attractor also converges in the forwards direction, but in the weaker sense of convergence in probability. This is due to the possibility of large deviations of individual sample paths. Example 13.2. If the random attractor consists of singleton sets, i.e., A(ω) = ¯ t (ω) := X ∗ (θt (ω)) {X ∗ (ω)} for some random variable X ∗ with values in Rd , then X d is a stationary stochastic process on R . In Example 13.1 above, X ∗ is given by the pullback limit, i.e., Z 0 ∗ X (ω) := es cos Ws (ω) ds, −∞

which is essentially the limit (13.4) at time t = 0. In this case the random attractor is also forward attracting in the stronger pathwise sense. The example above is a strictly contracting system. There is a random counterpart of Theorem 7.2, which gives the existence of a random attractor consisting of singleton subsets for RDS with the uniformly strictly contracting property, see [Caraballo, Kloeden and Schmalfuß (2004)]. 13.3 13.3.1

Examples of random attractors Random attractors for RODES

To apply the theory of random dynamical systems to RODEs, the RODEs need to be rewritten in the canonical form dx = f (x, θt (ω)), dt

x ∈ Rd ,

(13.11)

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with a the noise represented by a measure-preserving dynamical system θ = {θt }t∈R acting on a probability space (Ω, F, P) rather than by a specific noise process as in (13.1). This was done in Example 13.1 above, which has a simple random attractor consisting of singleton subsets. For a less trivial random attractor consider the Bernoulli ODE (8.3), i.e., dx = ax − b(t)x3 , dt see Subsection 8.3, with the coefficient function b(t) replaced by a random term b(θt (ω)) to give the RODE dx = ax − b(θt (ω))x3 , (13.12) dt where a > 0 and b : Ω → R is continuous and satisfies b(ω) ∈ [b0 , b1 ] for all ω ∈ Ω, where 0 < b0 < b1 < ∞. An example is b(θt (ω)) = 1 + cos2 (θt (ω)) with the noise paths coming from a Wiener process. It can be shown in the same way as in Subsection 8.3 with b(t) representing b(θt (ω)) that the random dynamical system generated by the RODE (13.12) has random equilbria solutions ±φ¯a (ω), where 1 , ω ∈ Ω, φ¯a (ω) = q R 0 2 −∞ b(θs (ω))e−2as ds is obtained by taking the pathwise limit, and that the random attractor A = {A(ω), ω ∈ Ω} here has the component interval sets   Aa (ω) = −φ¯a (ω), φ¯a (ω) , ω ∈ Ω. (13.13) 13.3.2

Random attractors for stochastic differential equations

Stochastic differential equations are not differential equations at all since the sample paths of their solutions, like those of the driving Wiener process, are not differentiable. They are in fact integral equations and require the stochastic calculus for their treatment rather than the familiar deterministic calculus. The general form of a scalar stochastic differential equation is dXt = f (t, Xt )dt + g(t, Xt )dWt , where f is called the drift coefficient and g the diffusion coefficient. In addition, Wt is a scalar Wiener process, which is assumed here to be two-sided, i.e., defined for all t ∈ R. This is only a symbolic representation of the stochastic integral equation Z t Z t Xt = Xt0 + f (s, Xs ) ds + g(s, Xs ) dWs , t0

t0

where the second integral is an Itˆo stochastic integral. The cocycle mapping is defined by ϕ(t, ω, x0 ) := Xtx0 (ω), where Xtx0 is the solution of the SDE with deterministic initial value x0 at time t = 0, and θt is

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the shift operator on the canonical sample space (Ω, F, P) defined in terms of the sample paths of the Wiener process. It is not difficult to show that the mapping ϕ so defined satisfies the cocycle property for all ω except those in a null event Ns,t ∈ F, i.e., with P(Ns,t ) = 0, for given s, t ∈ R+ 0 . This gives the crude cocycle property. However, Definition 13.2 requires the perfect cocycle property, for which the relationship holds for all ω except those in a null event N which is independent of s, t ∈ R+ 0 . The original sample space Ω is then replaced by Ω∗ := Ω \ N . It is much harder to show that the perfect cocycle property holds, see [Arnold (1998)]. 13.3.2.1

The Ornstein–Uhlenbeck process

The Ornstein–Uhlenbeck process provides a simple example of a random attractor, which can be computed explicitly. It is the unique stochastic stationary solution a linear Itˆ o stochastic differential equation (SDE) with additive noise and plays an important role in investigating random attractors in nonlinear SDE. Consider the SDE with linear drift and additive noise, dXt = −Xt dt + αdWt ,

(13.14)

where α is a positive constant, which has the explicit solution Z t Xt = Xt0 e−(t−t0 ) + αe−t es dWs . t0

Let Wt is a two-sided Wiener process, i.e., defined for all t ∈ R . Taking the pathwise pullback limit as t0 → −∞ gives the Ornstein–Uhlenbeck process Z t Ot = αe−t es dWs , (13.15) −∞

which is the unique stochastic stationary solution of the SDE (13.14). In particular, Ot (ω) = O∗ (θt (ω)) , where Z 0 ∗ O (ω) := es Ws (ω) ds. −∞

The SDE (13.14) is strictly contracting and all other solutions converge pathwise to the Ornstein–Uhlenbeck solution forwards in time. Subtracting any two solutions Xt and Yt of (13.14) gives d (Xt − Yt ) = −(Xt − Yt ), dt so, pathwise, |Xt − Yt | = |x0 − y0 |e−t → 0

as t → ∞.

Its random attractor A = {A(ω), ω ∈ Ω} thus has singleton sets A(ω) = {O∗ (ω)}.

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A nonlinear SDE with additive noise

Consider a nonlinear scalar SDE with additive noise dXt = f (Xt )dt + αdWt ,

(13.16)

where α is a positive constant. Suppose that the drift coefficient f : R → R is continuously differentiable and satisfies the one-sided dissipative Lipschitz condition hx1 − x2 , f (x1 ) − f (x2 )i ≤ −L|x1 − x2 |2 ,

x1 , x2 ∈ R ,

(13.17)

and also satisfies Assumption 13.1. Integrability condition: There exists λ0 > 0 such that Z t eλs |f (u(s))|2 ds < ∞ (13.18) −∞

holds for every λ ∈ (0, λ0 ] and any continuous function u : R → Rd with subexponential growth. (It can be assumed without loss of generality that L ≤ m0 ). The SDE (13.16) is really the integral equation Z t Xt = X0 + f (Xs ) ds + αWt . 0

In order to use the one-sided dissipative Lipschitz condition (13.17) we need to transform it into a RODE. Consider the difference Xt − Ot , where Ot is the Ornstein–Uhlenbeck stationary process (13.15) satisfying the linear stochastic differential equation (13.14). This difference is pathwise continuous since the paths are continuous and satisfy the integral equation Z t Xt − Ot = X0 − O0 + (f (Xs ) + Os ) ds, 0

so by the Fundamental Theorem of Calculus the difference is actually pathwise differentiable. This integral equation is thus equivalent to the pathwise RODE dx = f (x + Ot ) + Ot , dt where x(t, ω) := Xt (ω) − Ot (ω), or more simply,

d (Xt − Ot ) = f (Xt ) + Ot dt

for each ω ∈ Ω.

Now take the inner product with Xt − Ot and apply the one-sided Lipschitz condition to this RODE to obtain d |Xt − Ot |2 = 2 hXt − Ot , f (Xt ) − f (Ot )i + 2 hXt − Ot , f (Ot ) + Ot )i dt 4 ≤ −2L|Xt − Ot |2 + L|Xt − Ot |2 + |f (Ot ) + Ot )|2 . L

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Hence, |Xt − Ot |2 ≤ |Xt0 − Ot0 |2 e−L(t−t0 ) +

4e−Lt L

Z

t

t0

eLs |f (Os ) + Os |2 ds .

Pathwise pullback convergence (i.e., as t0 → −∞) then gives Z 4e−Lt t Ls 2 2 |Xt − Ot | ≤ RX (θt (ω)) := 1 + e |f (O(θs (ω))) + O(θs (ω))|2 ds L −∞ for all t ≥ TD and tempered families D = (Dω )ω∈Ω of the initial conditions. The integrability condition (13.18) and properties of the Ornstein–Uhlenbeck process 2 ensure that RX (θt (ω)) is finite. Thus, |Xt (ω) − Ot (ω)| ≤ RX (θt (ω)),

t ≥ TD ,

|Xt (ω)| ≤ |Ot (ω)| + RX (θt (ω)),

t ≥ TD .

or equivalently

The family of compact balls B(ω) centered on O0 (ω) with radius RX (ω) is a tempered pullback absorbing family of compact subsets. This means that this system has a random attractor A = {A(ω), ω ∈ Ω} by Theorem 13.1. The difference of any two solutions satisfies the differential inequality

d 1 |X − Xt2 |2 ≤ −2L|Xt1 − Xt2 |2 , dt t so the system is strictly contracting. This means all solutions converge pathwise to each other forwards in time and that the random attractor subsets are singleton sets A(ω) = {X ∗ (ω)}, i.e., the random attractor is formed by a stationary stochastic ¯ t (ω) = X ∗ (θt (ω)). process X

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Chapter 14

Mean-square random dynamical systems

Mean-square properties are of traditional interest in the investigation of stochastic systems in engineering and physics. This is quite natural since the Itˆo stochastic calculus is a mean-square calculus. At first sight, it is thus somewhat surprising that the classical theory of random dynamical systems is a pathwise theory, although this can be justified since stochastic differential equations can be transformed into pathwise random ordinary differential equations as in the example at the end of the last chapter. Such transformations, however, need not apply to mean-field stochastic differential equations of the form dXt = f (t, Xt , EXt , EXt2 ) dt + g(t, Xt , EXt , EXt2 ) dWt

(14.1)

which include expectations and higher moments of the solution in their coefficient functions. Let {Wt }t∈R be a two-sided scalar Wiener process defined on a probability space (Ω, F, P) and let {Ft }t∈R be the natural filtration generated by {Wt }t∈R , i.e., an increasing and right-continuous family of sub-σ-algebras of F, which contain all Pnull sets. Essentially, Ft represents the information about the randomness at time t ∈ R. Define the spaces

X := L2 (Ω, F; Rd ), Xt := L2 (Ω, Ft ; Rd ) for t ∈ R p with the norm kXkms := EkXk2 , where k · k is the Euclidean norm on Rd and note that Xt ⊂ Xs ⊂ X,

−∞ < t < s < ∞.

Given any initial condition Xs ∈ Xs , s ∈ R, a solution of (14.1) is a stochastic process {Xt }t≥s with Xt ∈ Xt for t ≥ s, satisfying the stochastic integral equation Z t Z t 2 Xt = Xs + f (u, Xu , EXu , EXu ) du + g(u, Xu , EXu , EXu2 ) dWu . s

s

Under appropriate assumptions on the coefficient functions, e.g., locally Lipschitz with a growth bound or a dissipativity property, the mean-field SDE (14.1) has a unique solution and generates a mean-square random dynamical system φ on the underlying phase space Rd with a probability set-up (Ω, F, {Ft }t∈R , P), defined by φ(t, s, Xs ) = Xt

for − ∞ < t < s < ∞, Xs ∈ Xs . 127

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Mean-square random dynamical systems

Mean-square random dynamical systems were defined in [Kloeden and Lorenz (2012)] as deterministic two-parameter semi-groups acting on a state space of meansquare random variables.  Consider the time set R, and define R2≥ := (t, s) ∈ R2 : t ≥ s . Definition 14.1. A mean-square random dynamical system (MS-RDS for short) φ on the underlying phase space Rd with the filtered probability space (Ω, F, {Ft }t∈R , P) is a family of mappings φ(t, s, ·) : Xs → Xt ,

(t, s) ∈ R2≥ ,

which satisfies: (1) Initial condition: φ = Xs for all Xs ∈ Xs and s ∈ R. (2) Two-parameter semigroup property: For all X ∈ Xu and all (t, s), (s, u) ∈ R2≥ φ(t, u, X) = φ(t, s, φ(s, u, X)). (3) Continuity: φ is continuous in its variables. Mean-square random dynamical systems are essentially deterministic with the stochasticity built into or hidden in the time-dependent state spaces. 14.2

Mean-square random attractors

The mean-square random attractor of a mean-square random dynamical system is defined here as the pullback attractor of a deterministic nonautonomous dynamical system as in Chapter 6. Definition 14.2. A family A = {A(t), t ∈ R} of nonempty compact subsets of X with A(t) ⊂ Xt for each t ∈ R is called a mean-square attractor of a MS-RDS φ if it pullback attracts all uniformly bounded families D = {D(t), t ∈ R} of subsets of {Xt }t∈R , i.e., lim distXt (φ(t, s, D(s)) , A(t)) = 0.

s→−∞

Uniformly bounded here means that there is an R > 0 such that kXkms ≤ R for all X ∈ D(t) and t ∈ R. The existence of pullback attractors follows from that of an absorbing family. Definition 14.3. A uniformly bounded family B = {B(t), t ∈ R} of nonempty closed and bounded subsets of {Xt }t∈R is called a pullback absorbing family for a MS-RDS φ if for each t ∈ R and every uniformly bounded family D = {D(t), t ∈ R} of nonempty subsets of {Xt }t∈R , there exists some T = T (t, D) ∈ R+ such that φ(t, s, D(s)) ⊆ B(t)

for s ∈ R with s ≤ t − T.

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Since closed and bounded sets in the infinite dimensional space X are not necessarily compact, the mappings defining the MS-RDS must be assumed to have some kind of compactness property. Theorem 14.1. Suppose that a MS-RDS φ has a positively invariant pullback absorbing uniformly bounded family B = {B(t), t ∈ R} of nonempty closed and bounded subsets of {Xt }t∈R and that the mappings φ(t, s, ·) : Xs → Xt are pullback compact (respectively, eventually or asymptotically compact) for all (t, s) ∈ R2≥ . Then, φ has a unique mean-square random attractor A = {A(t), t ∈ R} with component sets determined by \ A(t) = φ(t, s, B(s)), t ∈ R. s≤t

The main difficulty in applying this theorem is the lack of useful characterisations of compact sets of such spaces X of mean-square random variables. In the example below total boundedness, hence pre-compactness, will be shown directly. 14.3

Example of a mean-square attractor

Consider the nonlinear mean-field SDE
dXt = αXt + EXt − Xt EXt2 dt + Xt dWt

(14.2)

with a real-valued parameter α. The first and second moment equations of the mean-field SDE (14.2) are given by d EXt = (α + 1)EXt − EXt EXt2 , (14.3) dt d EXt2 = (2α + 1)EXt2 + 2(EXt )2 − 2(EXt2 )2 , (14.4) dt where Itˆ o’s formula (i.e., the chain rule in stochastic calculus ) with y = x2 was used to derive (14.4). These can be rewritten as the system of ODEs dx = x(α + 1 − y) dt dy = (2α + 1)y + 2x2 − 2y 2 dt under the restriction that x2 ≤ y. This system of ODEs has a steady state solution x ¯ = y¯ = 0 for all α, which corresponds to the zero solution Xt ≡ 0 of the mean-field p SDE (14.2). There also exist valid (i.e., with y ≥ 0) steady state solutions x ¯ = ± (α + 1)/2 , y¯ = α + 1 for α > −1 and x ¯ = 0, y¯ = α + 12 for α ≥ − 12 . It needs to be shown if there are solutions of the SDE (14.2) with moments corresponding to these steady state solutions. Theorem 14.2. The MS-RDS φ generated by (14.2) has a uniformly bounded positively invariant pullback absorbing family.

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Proof. Let α be arbitrary and define p  B(t) = X ∈ Xt : kXkms ≤ |α| + 2

for t ∈ R .

Using (2α + 1)EXt2 + 2(EXt )2 − 2(EXt2 )2 ≤ (2α + 3)EXt2 − 2(EXt2 )2 , which holds since (EXt )2 ≤ EXt2 , the second moment equation (14.4) gives the differential inequality d EXt2 ≤ (2α + 3)EXt2 − 2(EXt2 )2 . dt

(14.5)

Let D = {D(t), t ∈ R} be a uniformly bounded family of nonempty subsets of {Xt }t∈R , i.e., D(t) ⊂ Xt and there exists R > 0 such that kXkms ≤ R for all X ∈ D(t). Specifically, it will be shown that φ(t, s, D(s)) ⊂ B(t) for t − s ≥ T , where T is defined by   R2 . (14.6) T := log |α| + 2 Pick Xs ∈ D(s) arbitrarily and (t, s) ∈ R2≥ with t − s ≥ T . Motivated by the differential inequality (14.5), consider the scalar ODE y˙ = (2α + 3)y − 2y 2 ,

(14.7)

with the initial value y(s) ≤ R2 . A direct computation yields Z t  Z t  2 y(t) = y(s) exp (2α + 3) − 2y(u) du ≤ R exp (2α + 3) − 2y(u) du . s

s

From the definition of T in (14.6), it follows that mins≤u≤t y(u) ≤ |α| + 2. Furthermore, y = 0 and y = α + 32 are stationary points of the ODE (14.7). For this reason, mins≤u≤t y(u) ≤ |α| + 2 implies that y(t) ≤ |α| + 2. Then from (14.5), it follows that y(t) ≥ kφ(t, s, Xs )k2ms . This means that kφ(t, s, Xs )k2ms ≤ y(t) ≤ |α| + 2 ,

(14.8)

i.e., φ(t, s, Xs ) ∈ Bt for t − s ≥ T . Hence, B = {B(t), t ∈ R} is a pullback absorbing family for the MS-RDS φ. This family is clearly uniformly bounded and positively invariant for the MS-RDS φ. Theorem 14.3. The MS-RDS φ generated by (14.2) has a mean-square attractor A = {A(t), t ∈ R} with component sets A(t) = {0} when α < −1. Proof. Let α < −1 be arbitrary. Let D = {D(t), t ∈ R} be a uniformly bounded family of nonempty subsets of {Xt }t∈R with kXkms ≤ R for all X ∈ D(t) where R > 0.

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Let (Xs )s∈R be an arbitrary sequence with Xs ∈ D(s). The moment equations (14.3)–(14.4) can be written as
d E[φ(t, s, Xs )] = E[φ(t, s, Xs )] α + 1 − E[φ(t, s, Xs )]2 dt d E[φ(t, s, Xs )]2 = (2α + 1)E[φ(t, s, Xs )]2 + 2(E[φ(t, s)Xs ])2 − 2(E[φ(t, s)Xs ]2 )2 . dt Then  Z t
α + 1 − EXu2 du , E[φ(t, s, Xs )] = EXs exp s

which implies that (E[φ(t, s, Xs )])2 ≤ e(α+1)(t−s) R2 . Moreover, by the variation of constants formula, E[φ(t, s, Xs )]2 = e(2α+1)(t−s) EXs2 + 2

Z

≤ e(2α+1)(t−s) R2 + 2R2 ≤ e(2α+1)(t−s) R2 + 2e

t

s

  e(2α+1)(t−u) E[φ(u, s, Xs )]2 − (E[φ(u, s, Xs )]2 )2 du

Z

t

e(2α+1)(t−u) e(α+1)(u−s) du

s (α+1)(t−s)

(t − s)R2 ,

which implies that lims→−∞ kφ(t, s, Xs )kms = 0. Thus the family of sets A(t) = {0} is the mean-square attractor of (14.2) in this case. When α becomes larger than −1, the mean-square attractor bifurcates to include nontrivial solutions. Theorem 14.4. The MS-RDS φ generated by (14.2) has a nontrivial mean-square attractor when −1 < α < − 21 . Proof. In order to apply Theorem 14.1, we need to show that φ is pullback asymptotically compact, i.e., given a uniformly bounded family D = {D(t), t ∈ R} of nonempty subsets of Xt and sequences {tk }k∈N in (−∞, t) with tk → −∞ as k → ∞ and {Xk }k∈N with Xk ∈ D(tk ) ⊂ Xtk for each k ∈ N, then the subset {φ(t, tk , Xk )}k∈N ⊂ Xt is relatively compact. For this purpose, let  > 0 be arbitrary. We will construct a finite cover of {φ(t, tk , Xk )}k∈N with diameter less than . Choose and fix s ∈ R with s < t such that 2 , (14.9) 4e(2α+1)(t−s) (|α| + 2)2 < 2 and define Ysk := φ(s, tk , Xk ). Using (14.8) it can be assumed without loss of generality that E(Ysk )2 ≤ |α| + 2

for k ∈ N.

(14.10)

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For u ∈ [s, t], let Yuk := φ(u, s, Ysk ) and consider a family of functions f k : [s, t] → R defined by f k (u) := EYuk . By the Itˆo formula, Z t Z t
Ytk = eα(t−s) Ysk + eα(t−u) EYuk − Yuk E(Yuk )2 du + eα(t−u) Yuk dWu , s

s

from which it can be shown that {f k }k∈N is a uniformly equicontinuous sequence q α 1 of functions. Hence by the Arzel`a–Ascoli theorem, for δ :=  2 + 4 , there exists a finite index set J(δ) ⊂ N such that for any k ∈ N there exists nk ∈ J(δ) for which sup |EYuk − EYunk | < δ .

(14.11)

u∈[s,t]

To conclude the proof of asymptotic compactness, it is sufficient to show the inequality kYtk − Ytnk kms ≤ . Indeed, for u ∈ [s, t], a direct computation gives d E(Yuk − Yunk )2 = (1 + 2α)E(Yuk − Yunk )2 + 2(EYuk − EYunk )2 − 2(E(Yuk )2 )2 dt − 2(E(Yunk )2 )2 + 2EYuk Yunk (E(Yuk )2 + E(Yunk )2 ) . Note that

2(E(Yuk )2 )2 + 2(E(Yunk )2 )2 − 2EYuk Yunk (E(Yuk )2 + E(Yunk )2 ) ≥ 0 .

Hence, by the variation of constants formula, kYtk

−

Ytnk k2ms

≤e

(2α+1)(t−s)

kYsk

−

Ysnk k2ms

Z +2 s

t

e(2α+1)(t−u) (EYuk − EYunk )2 du ,

which together with (14.10) and (14.11) implies that 2δ 2 2 2 kYtk − Ytnk k2ms ≤ 4e(2α+1)(t−s) (|α| + 2)2 + < + = 2 . 2α + 1 2 2 Here (14.9) was used to obtain the preceding inequality. Thus the set {φ(t, tk , Xk )}k∈N is covered by the finite union of open balls with radius  centered at φ(t, tk , Xk ), where k ∈ J(λ), i.e., it is totally bounded. It follows that the MS-RDS is asymptotically compact. Hence by Theorem 14.1 it has a mean-square attractor A with component subsets A(t) for t ∈ R which contain the zero solution. It remains to show that the sets A(t) also contain points other than zero. Consider a uniformly bounded family of bounded sets defined by n o p D(t) := X ∈ Xt : EX = (α + 1)/2 , EX 2 = α + 1 . Recall that these values are a steady state solution of the moment ODEs (14.3)– (14.4). Hence φ(t, s, Xs ) ∈ D(t) for all t > s when Xs ∈ D(s). Then by pullback attraction distXt (φ(t, s, Xs ), A(t)) ≤ distXt (φ(t, s, D(s)), A(t)) → 0

as

s → −∞ .

The convergence here is mean-square convergence, and Eφ(t, s, Xs )2 ≡ α + 1 for all t > s. Thus, there exists a random variable Xt∗ ∈ A(t) ∩ D(t) for each t ∈ R, i.e., the mean-square attractor is nontrivial. It follows by the arguments in Chapter 4 that there is in fact a nonzero entire ¯ t ∈ A(t) ∩ D(t) for all t ∈ R. solution X

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Notes

1. This book, like part of the monograph Nonautonomous Dynamical Systems by [Kloeden and Rasmussen (2011)], is based on lectures given in Frankfurt am Main by the first author and later in Wuhan by both authors. The first part of the book overlaps with the material in [Kloeden and Rasmussen (2011)], but also contains some new work, in particularly on an alternative construction of pullback attractors for strictly contractive processes and the construction of forward attractors. 2. There are many books on dynamical systems, which usually means autonomous dynamical systems. [Hale (1988); Robinson (2001); Vishik (1992); Temam (1997)] are mainly concerned with attractors in autonomous systems. A standard reference on ordinary differential equations with an orientation to dynamical systems is [Hirsch, Smale and Devaney (2004)]. [Aulbach (1998)] is highly recommended for those who can read German. [Kato, Martynyuk and Sheshakov (1996)] and [Lasalle (1976)] are mainly concerned with stability issues. Properties of set-valued mappings are given in [Aubin and Franksowka (1990)]. 3. Applications of nonautonomous and random dynamical systems in the life sciences are given, e.g., in [Caraballo and Han (2016); Chesson (2017); Kloeden, and P¨ otzsche (2013a,b)]. See also the review article [Crauel and Kloeden (2015)]. Discrete time nonautonomous dynamical systems are treated in [Kloeden, P¨otzsche and Rasmussen (2012)] and also [P¨otzsche (2010)]. The SIR model is described in detail in [Brauer, van den Driessche and Wu (2008)]. Chapter 6 is based on the paper [Kloeden and Kozyakin (2011)]. See [Kloeden and P¨ otzsche (2015)] for bifurcations in the SIR models. 4. [Carvalho, Langa and Robinson (2013); Chepyzhov and Vishik (2002)] deal with nonautonomous attractors of infinite dimensional nonautonomous dynamical systems, e.g., those generated by parabolic partial differential equations. 5. The definition of process is due to [Dafermos (1971); Hale (1988)] and is the generalisation of the notion of semigroup corresponding to an autonomous system. 133

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The definition of a pullback attractor was motivated by that of a random attractor in [Crauel and Flandoli (1994)]. In some earlier papers [Kloeden and Schmalfuß (1996); Kloeden and Schmalfuss (1997a,b); Kloeden and Stonier (1998)] they were called cocycle attractors. The name pullback attractor was later introduced in [Kloeden (2000)] to distinguish them from forward attractors. The existence of pullback attractors has been proved in many versions in the literature with the original proofs being based on those for the existence of random attractors, see the notes in [Kloeden and Rasmussen (2011)] for more information. Attraction universes and tempered bounded sets were introduced in [Schmalfuß (1997)] for random dynamical systems. See also [Kloeden and Rasmussen (2011)]. The existence of pullback attractors for uniformly contracting processes was first used in the random context in [Caraballo, Kloeden and Schmalfuß (2004)] and later in the deterministic setting by [Kloeden and Lorenz (2013)]. The analysis of the pullback attractor of the nonautonomous Bernoulli ODE (8.3) is taken from [Kloeden and Siegmund (2005)], where bifurcations in nonautonomous systems are explored. The one-sided Lipschitz condition is well-known in numerical analysis. The one-sided dissipative Lipschitz condition is often called the uniform dissipativity condition. Properties of set-valued mappings are given in [Aubin and Franksowka (1990)]. 6. Skew product flows originated in ergodic theory and were extensively studied in connection with ordinary differential equations by [Sell (1967, 1971)]. Related work on limiting equations is presented in [Kato, Martynyuk and Sheshakov (1996)]. A class of ODEs with time variation more general than almost periodic is investigated in [Kloeden and Rodrigues (2011)]. These also generate skew product flows with a compact base space. 7. The material in the last three chapters is new and is based on papers in journals by the authors and their coworkers. 8. Chapter 10 on the limitations of pullback attractors is based on the paper [Kloeden, P¨ otzsche and Rasmussen (2012)]. See also [Cheban, Kloeden and Schmalfuß (2001)]. 9. The construction of forward attractor was given by [Kloeden and Lorenz (2016)], see also [Kloeden, Lorenz and Yang (2016)]. Nonautonomous omega limit sets were discussed in [Crauel and Flandoli (1994); Lasalle (1976); P¨otzsche (2010)]. They and a new concept of forward attracting set, which is a weaker version of the Haraux–Vishik uniform attractor [Chepyzhov and Vishik (2002); Haraux (1991); Vishik (1992)], were investigated in [Kloeden (2016); Kloeden and Yang (2016)]. Theorem 11.5 on asymptotically autonomous systems is a modification of a theorem in [Kloeden and Simsen (2015)]. Lemma 11.1 is taken from [Cui and Kloeden (2019)].

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10. Asymptotic positive invariance for ODEs was introduced in [Lakshmikantham and Leela (1967)]. See also [Kloeden (1975)]. Related concepts quasi-invariance and lifted invariance for skew product flows were introduced by [Miller (1965)] and [Botolan, Carvalho and Langa (2014)], respectively. Asymptotic positive and negative invariance for nonautonomous omega-limit sets were proposed in [Kloeden (2016)]. Properties of forward attracting sets and nonautonomous omega-limit sets such as asymptotical invariance and upper semi continuous dependence on a parameter were investigated in [Kloeden (2016)] and [Kloeden and Yang (2016)]. Corresponding results for a reaction diffusion equation on a time varying domain given in [Kloeden and Yang (2019)]. See also [Crauel, Kloeden and Yang (2011)]. 11. The theory of random dynamical systems is expounded in [Arnold (1998)]. See also [Crauel and Flandoli (1994); Crauel and Kloeden (2015)]. There are many books on stochastic differential equations at different levels of abstraction, see for example [Arnold (1978); Kloeden and Platen (1992)]. Random ordinary differential equations are developed in [Han and Kloeden (2017)]. 12. Mean-square random dynamical systems based on deterministic two-parameter semi-groups on a state space of random variables or random sets with the meansquare topology were introduced in [Kloeden and Lorenz (2012)]. Stochastic differential equations with nonlocal sample dependence, including expectation terms, were proposed in [Kloeden and Lorenz (2010)]. The example analysed in Chapter 14 comes from [Doan, Rasmussen and Kloeden (2015)].

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B. Schmalfuß, The random attractor of the stochastic Lorenz system, Zeitschrift f¨ ur Angewandte Mathematik und Physik, 48 (1997), 951–975. G. R. Sell, Nonautonomous differential equations and dynamical systems. Transactions of the American Mathematical Society, 127:241–283, 1967. G. R. Sell, Topological Dynamics and Ordinary Differential Equations. Van Nostrand Reinhold Mathematical Studies, London, 1971. L. Stone, B. Shulgin, and Z. Agur, Theoretical examination of the pulse vaccination policy in the SIR epidemic model, Math. Computer Modelling 31 (2001), 207–215. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, SpringerVerlag, 1997. M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations, Cambridge University Press, Cambridge, 1992.

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Index

Asymptotically autonomous systems, 92, 102 Asymptotically compact autonomous system, 11 pullback, 61 Asymptotically equivalent, 99 Attractor autonomous, 10, 36 global, 89 mean-square, 129 nonautonomous, 38 uniform, 105 Autonomous difference equation, 3 difference operator, 3 solution, 3 Autonomous dynamical systems local attractor, 10 Autonomous dynamical systems absorbing set, 10 asymptotically compact, 11 attractor, 10 continuous-time, 6, 7 difference equation, 3 differential equation, 6 discrete-time, 5 global attractor, 10 group property, 5 homeomorphism, 4 invariant set, 8 left shift operator, 5 omega limit set, 8 positive invariance, 10 right shift operator, 5 semi-dynamical system, 5, 8 semi-group property, 3

T -periodic, 83 IR model, 53 SIR model, 43 variable population, 53 nonautonomous, 44 nonautonomous equilibrium, 44 pullback limit, 44 variable population disease free nonautonomous equilibrium, 53 SI model variable interaction, 49 nonautonomous equilibrium, 50 nontrivial equilibrium, 50 nontrivial nonautonomous equilibrium, 51 variable population, 45 nonautonomous equilibrium, 46 nontrivial equilibrium, 48 Attractor mean-square, 128 random, 121 Invariance asymptotically positive, 107, 108 Absorbing set autonomous, 10, 36 nonautonomous, 57 positive invariant, 38 Abstract semi-dynamical system semi-group property, 8 Asymptotically negative invariant, 108 Asymptotically positive invariant, 107, 108 141

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An Introduction to Nonautonomous Dynamical Systems and their Attractors

An Introduction to Nonautonomous Dynamical Systems and their Attractors

state space, 7 Autonomous semi-dynamical systems absorbing set, 36 attractor, 36 entire solution, 27 invariant set, 29 positive invariance, 29, 38 steady state solution, 29 Autonomous triangular system, 19 driving system, 19 Basin of attraction system, 79 Bi-infinite sequence, 5, 25, 77 Canonical sample space, 120 Construction forward attractors, 97 Convergence in probability, 122 forward, 40, 56 pullback, 41, 56 uniform continuous, 110 Difference equations nonautonomous dynamical systems, 13 solution mapping, 14 Differential equations SIR model, 44 Bernoulii equation, 68 Bernoulli equation, 47 Lorenz equations, 37 mean-field, 127 random, 117 steady state solution, 8 stochastic, 117 Disspative, 106 Distance between compact subsets, 35 between a point and a set, 35 between two points, 35 semi-distance, 35 Entire solution autonomous, 27 complete solution, 27 full solution, 27 process, 30 Euler scheme, 13 Examples Attractor

nonautonomous difference equation, 92 Global attractor autonomous difference equation, 89 Mean-square attractor, 129 Pullback attractor piecewise autonomous difference equation, 90 Pullback attractors for processes dissipative ODE, 69 linear difference equation, 66 linear differential equation, 65 nonlinear differential equation, 67 Pullback attractors of skew product flows continuous-time, 75 discrete-time, 77 Random attractor nonlinear SDE with additive noise, 125 Random attractors RODES, 122 stochastic differential equations, 123 Skew product flow autonomous triangular system, 22 nonautonomous difference equations, 24 nonautonomous differential equations, 23 Existence of pullback attractors, 58 autonomous attractors, 10 mean-square attractor, 129 pullback attractors, 74 random attractors, 121 Forward attracting set, 106 Forward attracting set, 103 Forward attractor, 74, 96 construction, 97 process, 56 unbounded, 92 Forward convergence, 40 Group property, 4 Hausdorff metric, 35 Hausdorff separation, 35

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An Introduction to Nonautonomous Dynamical Systems and their Attractors

Index

Homeomorphism, 4, 77 Hull, 21, 24 Integrability condition, 125 Invariant set autonomous, 29 nonautonomous, 31 Left shift operator, 73 Lyapunov stable, 45 Lyapunov function, 37 Lyapunov stable, 96 asymptotic, 96 Mean-square attractor, 128 pullback absorbing family, 128 uniformly bounded, 128 Mean-square random dynamical system, 128 two-parameter semigroup property, 128 Natural filtration, 127 Nonautonomous dynamical systems skew product flow, 16 Nonautonomous dynamical systems 2-parameter semi-group, 14–16 difference equation, 13 process, 16 Nonautonomous equilibrium solution, 44 Omega limit set nonautonomous, 100 Omega-limit set autonomous, 8 One-sided dissipative Lipschitz condition, 70, 125 Ordinary differential equations autonomous, 6 nonautonomous, 15 Ornstein–Uhlenbeck process, 124 Periodic process, 40 Positive invariance nonautonomous, 104 Positive invariant family, 58 Positive invariant w.r.t a process φ, 58 Probability space, 119 Process as an autonomous system, 17 as skew product flow, 22

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entire solution, 30 forward attractor, 56, 96 invariant set, 31 pullback attractor, 56 Pullback attraction uniform, 75 Pullback absorbing family process, 57 skew product flow, 74 uniformly bounded, 82 Pullback absorbing set tempered random set, 121 Pullback asymptotically compact, 61 Pullback attraction skew product flow, 74 Pullback attractor existence, 74 process, 56 skew product flow, 73 uniformly bounded, 57 universe of attraction, 79 Pullback convergence, 41 eventually uniform, 102 Random attractor, 121 convergence in probability, 122 crude cocycle property, 124 invariant, 121 pathwise pullback attracting, 121 Random dynamical system, 120 cocycle mapping, 120 continuity, 120 measurability, 120 Random equilibrium, 118 Random ordinary differential equations, 117 Random set, 121 closed, 121 compact, 121 tempered, 121 RODEs, 117 Sample space, 119 Sample space canonical, 120 Semi-group property, 3 Skew product flow as an autonomous system, 26 compactifying time, 21 Skew product flow

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An Introduction to Nonautonomous Dynamical Systems and their Attractors

An Introduction to Nonautonomous Dynamical Systems and their Attractors

driving system, 73 Skew product flow, 19, 21, 134 base space, 21 cocycle property, 20, 21 driving system, 21 hull, 21, 24 invariant family, 73 positive invariant family, 73 pullback attractor, 73 triangular system, 19 driving system, 19 Stochastic differential equation, 117, 123 Subexponential growth, 78

Uniform attractor, 75 Uniform continuous convergence, 110 Uniform pullback attraction, 75 Uniform strict contraction property, 61, 71 Uniformly attracting, 110 Uniformly bounded family, 56 pullback attractor, 57 Universe of attraction, 79 skew productflow, 78 Upper semi-continuous set-valued mapping, 80

Tempered family, 78, 95 Two-sided Wiener process, 117

Wiener process, 117 two-sided, 117

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