Approximation Procedures in Nonlinear Oscillation Theory [Reprint 2012 ed.] 9783110885712, 9783110141320

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de Gruyter Series in Nonlinear Analysis and Applications 2

Editors A. Bensoussan (Paris) R. Conti (Florence) A. Friedman (Minneapolis) K.-H. Hoffmann (Munich) M. A. Krasnoselskii (Moscow) L. Nirenberg (New York) Managing Editors J. Appell (Würzburg) V. Lakshmikantham (Melbourne, USA)

Nikolai A. Bobylev Yuri M. Burman Sergey Κ. Korovin

Approximation Procedures in Nonlinear Oscillation Theory

W DE G Walter de Gruyter · Berlin · New York 1994

Authors Nikolai A. Bobylev Institute for Control Problems of RAS ul. Profsoyuznaya 65 117806 Moscow, Russia

Yuri M. Burman Technion Israel Institute of Technology Department of Mathematics 32000 Haifa, Israel

Sergey K. Korovin Department of Computational Mathematics and Cybernetics Moscow State University Vorob'evy Gory 119899 Moscow, Russia

1991 Mathematics Subject Classification: Primary: 47-02, 47 H 15 Secondary: 41 A99, 4 2 A 9 9 , 4 5 G 1 0 , 4 7 H 1 0 , 4 7 H 11, 65J99, 93A30, 9 3 C 1 0 , 9 4 C 0 5 Keywords: Oscillation, integral operator, rotation of the vector field, harmonic balance method, projection method, method of mechanical quadratures, method of finite differences

© Printed on acid-free paper which falls within the guidelines of the AINSI to ensure permance and durability. Library of Congress Cataloging-in-Publication

Data

Bobylev, N. A. Approximation procedures in nonlinear oscillation theory / Nikolai A. Bobylev, Yu. M. Burman, Sergey K. Korovin. p. cm. — (De Gruyter series in nonlinear analysis and applications ; 2) Includes bibliographical references. ISBN 3-11-014132-9 (acid-free) 1. Nonlinear oscillations. 2. Approximation theory. I. Burman, Yu. M. (Yuri M.). 1967II. Korovin, Sergey K „ 1945. III. Title. IV. Series. QA867.5.B63 1994 003',85—dc20 94-16854 CIP

Die Deutsche Bibliothek —

CIP-Einheitsaufnahme

Bobylev, Nikolai Α.: Approximation procedures in nonlinear oscillation theory / Nikolai A. Bobylev ; Yuri M. Burman ; Sergey K. Korovin. — Berlin ; New York: de Gruyter, 1994 (De Gruyter series in nonlinear analysis and applications ; 2) ISBN 3-11-014132-9 NE: Bûrman, Yûrî M.:; Korovin, Sergej K.:; G T

ISSN 0941-813 Χ © Copyright 1994 by Walter de Gruyter & Co., D-10785 Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Typeset with T g X : Lewis & Leins, Berlin. Printing: Gerike G m b H , Berlin. Binding: Lüderitz & Bauer G m b H . Berlin. Cover design: Thomas Bonnie, Hamburg.

Preface

Oscillatory processes now play a significant role not only in radio engineering, mechanics, optics, acoustics, chemical kinetics, etc., which are in the scope of mathematical sciences, but also in biological, economic and social phenomena. The general laws which govern the various processes is the subject of oscillation theory. In oscillation theory mainly periodical oscillations, almost periodical oscillations, quasi-periodic oscillations, stochastic and chaotic oscillations are studied. This book is devoted to periodical processes, the simplest kind of oscillatory processes. During the last decades the qualitative methods of oscillation theory have been developing quickly and came of age: the terminology is generally accepted, various approaches to the investigation of oscillatory processes were developed and their scope was determined. Some applications were explored. On the other hand numerical approximation methods for periodical oscillations are far from being complete and are now under active study. Most of them either deal with specific classes of oscillatory systems or are heuristic. This book is an attempt to study systematically projection procedures of approximate construction of oscillatory regimes in nonlinear systems. The book discusses general theorems and their applications. Most of the theorems proved in the book are applied to systems of automatic control. Other applications are mentioned only to illustrate a point. To understand most of the material in the book, the reader needs to know only basic facts of functional analysis and ordinary differential equations. Concepts beyond this background will be introduced and discussed in the course of the text.

Contents

Preface

ν

Chapter I: Basic Concepts

1

§ 1 Equations for oscillatory systems 1.1 First order systems 1.2 Linear periodic systems 1.3 Higher order systems

1 1 2 4

§ 2 The shift operator and first return function 2.1 Shift operator 2.2 Periodic solutions of autonomous systems

8 8 9

§ 3 Integral and integrofunctional operators for periodic problem 3.1 Completely continuous operators 3.2 Spaces of functions 3.3 Linear integral operators and their properties 3.4 A superposition operator 3.5 The Hammerstein operator 3.6 Integral and integrofunctional equations 3.7 The Fréchet derivative 3.8 Periodic problem for systems of automatic control

. . . .

11 11 12 13 13 14 14 16 17

§ 4 The harmonic balance method 4.1 Forced oscillations 4.2 Free oscillations 4.3 Oscillations in the systems of automatic control 4.4 Harmonic balance method and general theory of projection methods

20 20 20 21 22

§ 5 The method of mechanical quadratures 5.1 The Galerkin method with perturbations 5.2 Quadrature processes 5.3 The method of mechanical quadratures in looking for periodic solutions

25 25 26 27

§ 6 The collocation method 6.1 The operator scheme of the collocation method 6.2 The collocation method in approximation of oscillatory regimes 6.3 On choice of interpolation nodes

29 29 31 32

.

viii

Contents

§ 7 The method of finite differences 7.1 Formulae of numeric differentiation 7.2 Discretization of the differential equations 7.3 Reduction of the method of finite differences to the Galerkin method

33 33 33 34

§ 8 Factor methods 8.1 Discrete convergence 8.2 Discrete compactness 8.3 Discrete convergent sequences of operators 8.4 The method of mechanical quadratures 8.5 The method of finite differences

37 37 38 38 39 40

Chapter II: Existence theorems for oscillatory regimes

43

§ 1 Smooth manifolds and differential forms 1.1 Smooth manifolds 1.2 Tangent spaces 1.3 Orientation 1.4 Manifolds with boundaries 1.5 Exterior forms 1.6 Outer product 1.7 Differential forms 1.8 Integration of differential forms 1.9 Outer differentiation

43 43 44 46 46 47 47 48 49 51

§ 2 Degree of a mapping 2.1 The Sard theorem 2.2 Lemmas about one-dimensional manifolds 2.3 The degree of a mapping 2.4 The degree of a mapping (the second approach) 2.5 Relation between two definitions of the degree of a mapping 2.6 Properties of the degree of a mapping 2.7 The degree of continuous mappings

51 52 52 53 57 60 61 64

. .

§ 3 Rotation of vector fields 3.1 Vector fields 3.2 Homotope fields. Criteria of homotopy 3.3 Rotation of a vector field 3.4 Properties of the rotation

65 65 65 66 66

§ 4 Completely continuous vector 4.1 Finite-dimensional approximations 4.2 Soundness of the definition of the rotation 4.3 Properties of the rotation

73 73 75 77

fields

Contents

ix

§ 5 Fixed point principles and solution of operator equations 5.1 The Brouwer theorem 5.2 The Browder theorem 5.3 The Schauder theorem 5.4 The Leray-Schauder principle 5.5 The contraction mapping principle 5.6 Operator equations in products of Banach spaces

80 80 81 86 87 88 90

§ 6 Forced oscillations in systems with weak nonlinearities 6.1 Systems with bounded nonlinearities 6.2 Systems of automatic control

94 94 95

§ 7 Oscillations in systems with strong nonlinearities. Directing functions method 7.1 Points of Γ-irreversibility 7.2 Directing functions 7.3 The full system of directing functions 7.4 Regular directing functions 7.5 Construction of directing functions

98 98 100 103 105 105

Chapter III: Convergence of numerical procedures

107

§ 1 Projection methods 1.1 The Galerkin method procedure 1.2 The nondegenerate case 1.3 Topological principle of convergence of the Galerkin method . . 1.4 Convergence of the Galerkin method with perturbations . . . . 1.5 Projection procedures in the Hilbert space 1.6 Estimates of the convergence rate

107 107 107 Ill 113 115 117

§2 Factor methods 2.1 Regular and compact convergence 2.2 The a posteriori estimate lemma 2.3 The convergence of the factor method (the nondegenerate case) . 2.4 The topological principle of convergence of the factor method . . 2.5 Convergence of the factor method for equations with a linear main part 2.6 Additional remarks

119 119 119 121 124

§ 3 Convergence of the harmonic balance method and the collocation method in the problem of periodic oscillations 3.1 Local convergence of the harmonic balance method 3.2 The topological principle of convergence 3.3 Convergence of the harmonic balance method in the case of nondifferentiable nonlinearities

129 130 148 148 153 154

χ

Contents

3.4 Convergence of the harmonic balance method in the problem of forced oscillations of the systems of automatic control 3.5 Local convergence of the collocation method 3.6 A global convergence of the collocation method 3.7 The harmonic balance and the collocation methods for periodic oscillations of multi-circuit systems of automatic control . . . . 3.8 The feasibility of the harmonic balance and the collocation methods

156 158 162 164 167

§ 4 Convergence of the method of mechanical quadratures 4.1 Convergent quadrature processes 4.2 Local convergence of the method of mechanical quadratures . . 4.3 The global convergence of the method of mechanical quadratures 4.4 Convergence of the method of mechanical quadratures for general nonlinear systems

171 171 174 179

§5 Convergence of the method of finite differences 5.1 Convergent formulae of numeric differentiation 5.2 Discretization of differential equations 5.3 Network spaces and connecting mappings 5.4 A convergence theorem for differentiable nonlinearities 5.5 Auxiliary statements 5.6 Proof of Theorem 5.1 5.7 The topological principle of convergence 5.8 Additional remarks

188 188 190 191 192 192 194 197 198

. . . .

181

§ 6 Numerical procedures of approximate construction of oscillatory regimes in autonomous systems 198 6.1 Specific features of the problem 198 6.2 The method of the functional parameter 199 6.3 The method of additional constraints 201 6.4 The degeneracy dimension 205 6.5 The convergence theorem 205 6.6 The harmonic balance method in search for oscillations of autonomous systems of automatic control 208 6.7 The functional characteristic 213 6.8 The topological principle of convergence 214 6.9 Additional remarks 215 §7 Affinity theory 7.1 Formulation of the problem 7.2 Domains with the same core 7.3 An affinity theorem for nonautonomous systems 7.4 The affinity theorem for nonautonomous systems of automatic control 7.5 Autonomous systems. The topological index of a cycle

215 215 217 217 220 223

Contents 7.6 Affinity theorem for autonomous systems of automatic control 7.7 Additional remarks § 8 Effective convergence criteria for numerical procedures 8.1 Use of affinity theory for proving the convergence of numerical procedures 8.2 The directing functions method and convergence of numerical procedures 8.3 Stability of oscillatory regimes and convergence of numerical procedures

xi . .

224 232 235 235 235 237

§ 9 Effective estimates of the convergence rate for the harmonic balance method 9.1 A posteriori error estimates 9.2 Quasi-linear systems 9.3 Systems of automatic control 9.4 Stability analysis for the periodic solution

240 240 241 245 248

Notes on the References

251

References

257

Index

271

Chapter I Basic Concepts

This Chapter is an introduction to the book. It will provide the main concepts, definitions, and description of numerical procedures.

§ 1 Equations for oscillatory systems 1.1 First order systems. A system of differential equations such as dx — =f(t,x)

(xeRN)

(1.1)

is called a nonautonomous first order system. Here χ = {χι, . . . , XN} is a quantity characterizing the state of the system, t is the time, and/(i, x) = { f \ (t, x¡,..., χ Ν), • • •, f N ( í , x\, . • •, The space to which χ belongs is called a state space. If the right-hand side of equation (1.1) is periodic in the time, i.e. for some Τ > 0 f ( t + T , x ) = f ( t , χ),

then it is natural to look for periodic motions of this dynamic system which period (in t) is equal to that of the right-hand side, i.e. solutions x j,t) of (1.1) such that Xjf{t

+ T) = x^t).

(1.2)

If the right-hand side of equation (1.1) does not depend on t then the system is called autonomous. It is described by the differential equation dx — =f(x)

(xeRN).

(1.3)

Even though equation (1.3) looks simpler than equation (1.1) (the right-hand side of equation (1.3) depends only on the variable χ while the right-hand side of equation (1.1) is a function of the variables χ and t), periodic solutions of equation (1.3) are usually much more difficult to find than for equation (1.1) because the periods of periodic solutions of equation (1.3) are not known a priori while in the nonautonomous case the period is always equal to that of the right-hand side of the equation multiplied by an integer.

2

I Basic Concepts

1.2 Linear periodic systems. If the right-hand side of equation (1.1) is linear in χ then equation (1.1) has the form dx — = A{t)x.

(1.4)

Here A(t) is an Ν χ Ν matrix with Γ-periodic elements a¡j(t): /

flii(í)

A(f)=í

...

:

··.

\am(t)

...

aw(t)\ :

J.

(1.5)

aNN(t)J

The solution of the Cauchy problem for equation (1.4), or the solution of this equation with given initial datum *(0)=*0

(1.6)

x(t) = X(t)xQ

(1.7)

is given by the formula

where X(t) is a fundamental matrix of solutions of equation (1.4). It follows from this formula that initial data of Γ-periodic solutions of equation (1.4) can be obtained from the equation (/ — X(T))x = 0.

(1.8)

The matrix X(T) in the left-hand side of equation (1.8) is called monodromy matrix of equation (1.4). It follows from equation (1.8) that equation (1.4) has nontrivial periodic solutions if and only if 1 is a multiplier of equation (1.4). Consider also an inhomogeneous equation ^-=A(t)x+f{t)

(1.9)

where f ( t ) is a Γ-periodic vector function. A solution of the Cauchy problem for equation (1.9) with initial datum *(0) = xq is given by the formula x(t) = X(t)xo+

/

/ X(t)X~\s)f(s)ds. (1.10) Jo If the solution x(t) defined by formula (1.10) is Γ-periodic, then the initial datum of this solution satisfies the equation x0 = X(.T)x0+

[ X(T)X~\s)f(s)ds. (1.11) Jo Hence if 1 is not a multiplier of equation (1.4) then the initial datum XQ of the Γ-periodic solution x(t) of equation (1.9) is given by the formula x0 = (I-

Χ(Γ))"1 í Jo

X(T)X~1(s)f(s)

ds

(1.12)

§ 1 Equations for oscillatory systems

3

and the Γ-periodic solution x ( t ) itself, by the formula

x ( t ) = X ( t ) ( I - X i T ) ) - 1 [ X ( T ) X - l ( s ) f ( s ) d s + f X { t ) X - \ s ) f { s ) d s . (1.13) Jo Jo Formula (1.13) is conveniently written in the form *(/)= [

Jo

G(t, s)f(s)ds

(1.14)

where G(t

s) =

i - X(T))-lX-\s) when 0 < s < t < T , ' X ( t ) ( I - X ( T ) ) - l X ( T ) X ~ l ( s ) when 0 < t < s < T .

^' '

The kernel G i t , 5) of the integral operator A defined by the right-hand side of equation (1.14) is called Green's function of the differential operator

Lx = — — A ( t ) x dt associated with Γ-periodic boundary conditions. The operator A is the inverse operator of L in some natural classes of Γ-periodic functions which will be discussed later in the book. One can easily prove that Green's function possesses the following properties: (1) G ( t + 0 , t ) - G ( t — 0, t) — / ; (2) G(0, s) — G ( T , 5);

(3) ^G(t,! = M t ) G ( t , s) dt

(0 < s < t < T ) .

There exists only one matrix function G ( t , s ) satisfying conditions (l)-(3). One important class of linear inhomogeneous equations (1.9) is made up by equations with a constant matrix A , i.e. by equations of the form ^

= Ax+/(/)·

(1.16)

For equation (1.16) the fundamental matrix of solutions and monodromy matrix are e A t and e A T , respectively. Hence the multipliers p¡ of equation (1.16) are related with eigenvalues λ, of the matrix A by the equation Pi

= ex¡T

(i=l,...,N).

(1.17)

It follows from this equation that 1 is not a multiplier of equation (1.16) if the eigenvalues of the matrix A are not equal to the numbers

„

, 2πί

, Ani

2km

0, ± — , ± — , . . . , ± — ,

....

(1.18)

Hence if the spectrum of the matrix A does not contain the points (1.18), then for any Γ-periodic function/(i) equation (1.16) has a Γ-periodic solution x ( t ) given

4

I Basic Concepts

by the formula

x ( t ) = f H ( t — s\ T ) f ( s ) d s Jo

(1.19)

where

H(t; T ) = ( I - eAT)~leAt

(0 < t < T ) .

(1.20)

It should be emphasized that unlike Green's function (1.15), Green's function (1.19) depends only on the difference of the arguments t and s .

1.3 Higher order systems. Consider a dynamic system described by the system of differential equations v (n)_

(t —J f ι yl,

vl , . . . , A

f ft y Λν (Ό_ ft —

ν Av'j ΛΝ,

, . . . ,

ν Αν'| , ΛΝ,

. ,.

Av'f t , . . . ν'

, Λ ^ ,

(«-I)

, V Λ[

, . .

, . . . ,

(/ι—1) ,

, Λ |

Aft

),

Aft

).

(1.21) . . .

,

This system can also be represented in a vector form:

JC(B) = f ( t , x , x ' , . . . , χ ( η ~ 1 ) )

(xef).

(1.22)

If the right-hand side / of equation (1.22) is Γ-periodic in t , then it would be natural to try to find Γ-periodic solutions x*(i) of equation (1.22). While the Γ-periodic solutions for the first order equations were chosen by boundary conditions (1.2), for the n-th order equations conditions analogous to (1.2) are

χΛΟ ) = x * ( T ) ,

· • · , ^ n _ 1 ) (0 ) = x ( r l \ T ) .

(1.23)

If the right-hand side of system (1.21) does not depend on t , then the system is called autonomous. The periodic solutions of the autonomous system

x(n) = f ( x , x ' , . . . , x { n ~ l ) )

(1.24)

are found by determining the a priori unknown period Γ* and a solution x*(i) of equation (1.24) such that x„(0 ) = x * ( T * ) ,

4n_1)(0)=4""1)(^)·

(1-25)

We describe two important classes of dynamic systems given by the higher order equations. The first describes oscillations of mechanical systems, the second one, oscillatory processes in systems of automatic control. Numerous oscillatory mechanical systems can be described by the equations

Ν ^ 2 ( a j k q k + djkqk + cjkqk) = ß f j ( t , q{, . . . , qN, q\, . . . , q N ) + F j ( t ) k=l (j=l,...,N). (1.26)

§ 1 Equations for oscillatory systems

5

Here ajk, cjk, djt are constant coefficients, E2 is called a completely continuous operator. The concept of a completely continuous operator is very important because many problems of oscillation theory are solved by using equations with completely continuous operators. Superpositions of operators are often considered in applications. If the operator A acts from E\ to £2 and the operator Β acts from E2 to £3, then the operator Β o A acts from E\ to £3, and its properties are determined by the properties of operators A and B. If Β is a completely continuous operator and A is continuous and bounded, then the operator Bo A is completely continuous. If Β is a continuous operator, then complete continuity of operator A implies complete continuity of the operator Β o A.

12

I Basic Concepts

3.2 Spaces of functions. The following spaces of functions will be taken up: 1. Space CN. The elements of that space are continuous Revalued functions x(t) defined in the segment [0, T], The norm in CN is defined by the equation \\x\\Clt = max |*(f)|.

(3.1)

2. Space EN- It is a subspace of the space CN consisting of functions x(t) satisfying the condition X(0) =

x(T).

(3.2)

3. Spaces CkN (k = 0, 1, . . . ). For any positive integer k elements of the space C N are k times differentiable RN-valued functions x(t) defined in the segment [0, T] with the norm k

max = Σ o regarded as an operator from Lp to any space Ü j (1 < r < oo).

3.8 Periodic problem for systems of automatic control. This Subsection will outline the properties of integral operators which appear in problems of periodic oscillations in the systems of automatic control. Consider a single-circuit system of automatic control described by the differential equation

L(j¡)x{t) = M{j^u(t).

(3.25)

Here u ( t ) is a Γ-periodic input, and x ( t ) is a Γ-periodic output. Assume that the numbers 0, ± 2 π i / T , . . . , ± 2 n k i / T , . . . are not roots of the polynomial

L i p ) = pl + a\pl~l

+ ·•· + «/

(3.26)

and its degree I is higher than the degree m of the polynomial

M i p ) = boPm + b i P m ~ ' + • • • + bm.

(3.27)

Then, as was proved in Sect. 1, the output x ( t ) is related with the input u ( t ) by the equation x ( t ) = Í h i t - s\ 7>(s) d s \

Jo

(3.28)

I Basic Concepts

18 here

h(t\ T) = ((/ - eAT)~1eAty,

c)

(0 < t < T)

(3.29)

and the matrix A and the vectors γ and c are described in Subsect. 1.2. Denote by Η the operator defined by the right-hand side of formula (3.28). The following statements are consequences of (3.29): Theorem 3.3. The operator Ή acts from L\ to cl~m~l

and is continuous.

Theorem 3.4. If ρ > 1 then the operator Ή acts and is completely continuous from each Lp to Cl~m-y. Note that the operator Ή is not completely continuous regarded as an operator from Li to C'-m~l. Later we will need some properties of the operator Ή considered in the space

l2Denote .

(k= 1,2,...). (3.30) The functions (3.30) form an orthonormed basis in L2. Denote by Eq the one-dimensional space of constant functions and by E^ (k = 1 , 2 , . . . ) , the two-dimensional spaces with a basis gk(t), e kit). Denote by Pk the operator of orthogonal projection to E¡,. Since (3.25) is a differential equation with constant coefficients, then the Tperiodic output x(t) corresponding to the Γ-periodic input u(t) e Ek will also be a function from Ek. Therefore each subspace Ek is invariant with respect to the operator Ti. Hence this operator commutes with the projector Pk'. HPk = PkH

(k = 0 , 1 , . . . ) .

(3.31)

Direct computations show that WHuW^ = \w(—)|

M|i2

(ueEk)

where

From this follows Theorem 3.5. The operator Ή : L¿ —•y L2 is completely continuous and \ \ H \ \

L 2

max

4=0,1,2,...

ηψ).

(3.32)

§ 4 The harmonic balance method ΙΙ(Λ> + · · · + Λ ν ) ? % 2 =

max

19

ηψ)

Consider now a system of automatic control with the feedback u — f{t,x) where f{t, x) is a function Γ-periodic in t. The functioning of such a system is described by the differential equation L(^)x(t)

= M{jt)f(t,x(t)).

(3.33)

To find the Γ-periodic solutions of equation (3.33) one has to solve an appropriate operator equation. One of them, by virtue of (3.28), has the form JC

= A(x)

where A(x) = [ h{tJo

s\ T)f(s, x(s)) ds

(3.34)

and the kernel h(t; T) of the integral operator A is defined by formula (3.29). It is often convenient to use another operator equation. Let χ* — x j t ) be a Γ-periodic solution of equation (3.33). Then ** = Hy+

(3.35)

y*(r)=/(f,jc,(r)).

(3.36)

where

Replacing x* in (3.36) by formula (3.35) we have y*(t)=f(t,Hy*).

(3.37)

Hence finding of Γ-periodic solutions of equation (3.33) is reduced to that of solutions y* of the operator equation J - B(y)

(3.38)

Β = f οΗ

(3.39)

where

and f is a superposition operator

f (x)=f(t,x(t)).

(3.40)

If the solution )>* of equation (3.38) is known, then the Γ-periodic solution x* = **(r) of equation (3.33) can be found using formula (3.35).

I Basic Concepts

20

§ 4 The harmonic balance method 4.1 Forced oscillations. Consider the Γ-periodic regimes of a dynamic system described by the equation dx — = / ( f , jc)

(xeRN)

(4.1)

whose right-hand side is Γ-periodic in time. Assume that equation (4.1) has a Tperiodic solution χ*(/)· Harmonic balance method approximants xn(t) of x¥(t) are constructed as follows. In a trigonometric polynomial xn(t) =

+

sm-— +Bkcos— ) (4.2) 1 1 k=ι the unknown vector coefficients A* and Bk are found by using the equation 1

j x n ( f ) = ΡJit, xn(t))

(4.3)

where P„f(t, xn(t)) is the n-th partial sum of Fourier series of the Γ-periodic function/(i, xn(t))· For equality (4.3) of two trigonometric polynomials to be true the coefficients at the same modes must be equal. What we have now is a finite system of ( 2 n + 1 )N equations Rk(a°¡, . . . , a%,...,

al, ..., anN, b\, . . . , blN,..., {k=

b\,...,

bnN) = 0

(4.4)

l,...,(2n+l)N)

with (2η + 1 )N variables. Solutions of this system are substituted into (4.2) and an approximant x*n(t) of the periodic solution χ*(ί) is thus obtained. Some natural questions arise concerning this method: 1. What are the conditions for equations (4.4) to be solvable? 2. What conditions ensure the convergence of approximants (4.2) to the periodic solution χ»? 3. In what spaces does this convergence occur? 4. What is the convergence rate? These questions are studied in Chapter 3 of this book.

4.2 Free oscillations. Let the above dynamical system be autonomous. Then its dynamics is described by the equation dx — =/(*)

(leR")

(4.5)

with its right-hand side not depending on t. As noticed above the problem to find periodic solutions of equation (4.5) is much more complicated than the analogous

§ 4 The harmonic balance method

21

problem for forced oscillations. One difficulty is that the oscillation period is unknown a priori. Therefore in the approximations (4.2) the period Τ is treated as unknown as are the amplitudes A* and B k . Consequently the equation jxn{t)

= Pnf(Xn(t))

(4.6)

gives a system of (2η + 1 )N equations Rk{a\, ...,a0N,...,anl,...,anN,b\,...,blN,...,bnl,...,b"N,T) (k= 1,...,

= 0

(4.7)

(2n + 1)JV)

with ( 2 n + 1 )7V + 1 variables. To make system (4.7) definite, it is necessary to add some auxiliary equation Ma°i, - • •, a°N, . . . , a?, . . . , anN, b\,...,

blN,...,

b\,...,

bnN, T) = 0.

(4.8)

Then the number of variables in system (4.7), (4.8) will coincide with the number of equations in this system. There are no specific ways to choose auxiliary equation (4.8), a successful choice in any particular case depends on the scientist's skill. One way to construct such equations is described in Chapter 3.

4.3 Oscillations in the systems of automatic control. Oscillatory processes in closed-loop single-circuit system of automatic control are described by the differential equation L{j^)x

= M{j^)f{t,x)

(xeR*)

(4.9)

with a nonlinearity f(t, χ) Γ-periodic in time. As in the above cases the approximant to the T-periodic solution is represented as a trigonometric polynomial (4.2) with unknown scalar coefficients A¿ and Bk. These coefficients are found from the equation

1

Φ

Χη({) Μ =

Φ

Ρη/(ί

'

X n ( t ) )

where P n is, as above, the n-th partial sum of Fourier series of the Γ-periodic function/(i, xn(t)). Coefficients at the same modes in both sides of equation (4.9) are equal, so we obtain a system of 2n + 1 equations Rk(Ao, ·. ·, A„, Bi, . . . , B n ) = 0 with 2n + 1 variables Ao, . . . , A„, B\, ..., Bn. Solutions of this system are substituted into (4.2), and the T-periodic solution x, is thus approximated. The utility of the harmonic balance method in this case will be also illustrated in Chapter 3.

( 4 1 0 )

22

I Basic Concepts

4.4 Harmonic balance method and general theory of projection methods. Let E be a real Banach space and A : E ^

E be an operator. Consider the equation

χ = A(x) Approximants of the solution following procedure called the Let Ε ι c E2 C · · · C En c of the space E. Let Pn : E - > of equations

(xeE).

(4.11)

x* of equation (4.11) are often obtained by the Galerkin method. . . . be a sequence of finite-dimensional subspaces En be projection operators. Consider the sequence

xn — PnA(x„ )

(Xn e E„).

(4.12)

Each equation (4.12) can be treated as a finite system of equations with scalar variables. The number of variables of this system is that of equations and is equal to the dimension of the space E„. (To prove this fact it is sufficient to choose in En a basis e¡, ..., ekn and to develop elements in both sides of equation (4.12) in this basis; the coefficients of the same elements of basis must be equal, in this way a finite system of equations equivalent to equation (4.12) is obtained). Equation (4.12) is called the Galerkin equation. It is an approximation of equation (4.11). If the Galerkin equation is solvable, then it is natural to consider its solution χ* as an approximant of the solution x* of equation (4.11). General theory of projection methods studies conditions under which approximating Galerkin equations (4.12) are solvable and investigates how close Galerkin approximations are to solutions of original equation (4.11). We show that the harmonic balance method can be treated as a variety of the Galerkin projection method. The reasoning below will not be absolutely strict, but Chapter 3 will prove its validity. Let E be some Banach space of functions x(i) defined in the segment [0, Γ] which can be developed into Fourier series in functions

. 2?Tt

2πί

1, s i n — , cos — ,

. 2mt

2nnt

. . . , sin — ,

cos - γ -

Denote by E„ a subspace of the space E which is a linear hull of the functions

. 27Γί

2nt

1,' s i n -rrτ- , ' c o s - — , > . . . , ? sin rji

2nnt rwi

," cos

2nnt ryi

and by Pn the operator whose value on the function x(t) is the ra-th partial sum of Fourier series of this function. Operator Pn is obviously a projector from E to En. Consider the Γ-periodic solutions of equation (4.1). As mentioned above to find them is to solve the integrofunctional equation

x(t) = x(T)+

[ Jo

f(s,x(s))ds.

(4.13)

§ 4 The harmonic balance method

23

If for any function χ = x { t ) e E the function A { x ) = x(T) + [ f ( s , x(s)) ds

Jo

(4.14)

belongs to the space E, then equation (4.13) can be treated as the operator equation x = A(x)

( x e E )

(4.15)

in the space E. The Galerkin equations corresponding to the system of subspaces E \ , £2, · · · , E n , . . . and projectors Pi, P 2 , . . . , P n , • • · have the form xn = P n A ( x n )

(xn e En)

(4.16)

where PnA(xn) = xn(T) + P n f f ( s , xn(s)) ds

Jo

(4.17)

and x n is the trigonometric polynomial (4.2). But the operator P n commutes with integration. Hence equation (4.16) can be rearranged into x n ( t ) = x n ( T ) + ί Ρrif (s, xn(5)) ds.

(4.18)

Jo

Solutions of this equation are solutions of equation (4.3) which is equivalent to the system of equations of the harmonic balance method (4.4). In an analogous way one can prove the equivalence of the harmonic balance method and the Galerkin method used for the equation

χ = A(x) where

A(x) =

1

ί ( ; φ ) + s f ( s , x(s))) d s + f f ( s , x(s)) ds Jo Jo

and that these two approaches to the problem of forced oscillations are equivalent. Let us have a closer look to the case when the equation has the form

dx dt

- — = Ax + f ( t , x)

(xeE^)

(4.19)

where f ( t , x ) is a function Γ-periodic in t . Let the eigenvalues of the matrix A be not equal to 0, ± 2 n i / T , . . . , ± 2 n n i / T , Then finding Γ-periodic solutions of equation (4.19) is equivalent to the solution of the integral equation x(t) = [

Jo

H ( t — s\ T ) f ( s , x ( s ) ) d s

(4.20)

with a kernel H(t; T ) = (/ - eAT)~leAt

(0 < t < T ) .

(4.21)

24

I Basic Concepts

The operator Β given by the right-hand side of equation (4.20) is a superposition of the linear integral operator

J

f-T I H(t — s\ T)x(s) Jo

Tix =

ds

and the superposition operator f(x)=f(s,x(s)).

Hence the operator form of equation (4.20) is x = Hof(x)

(4.22)

and the operator form of the Galerkin equation is xn = PnH of(*„).

(4-23)

We show now that in the space E0 c E of Γ-periodic functions the operators Ρn and Tí commute. Let g e E0. Then PnTig is the n-th partial sum xnJt) of the Fourier series of the Γ-periodic solution χ* (ί) of the equation dx — =Ax at

+

g(t).

Multiplying both sides of this equation by the operator Pn and using the fact that Pn commutes with differentiation operator ^ and with the matrix A we have dxn(ñ ^gl=Axl{t)+Png{t).

Hence x"(t) is a Γ-periodic solution of differential equation ^ = A x ( t ) + P

n

g ( t )

and can therefore be written in the form A =

HPng.

Consequently, HPng = PnTig.

Since operators Pn and Tí commute, the Hammerstein equation can be rewritten in the form xn = HPnf(xn).

(4.24)

But solutions of equation (4.24) are those of the equation d —xn(t) dt

= Axn(t) + PJ(t,

xn(t))

(4.25)

25

§ 5 The method of mechanical quadratures

equivalent to the system of equations of the harmonic balance method corresponding to differential equation (4.19). This means that the harmonic balance method of approximate construction of T-periodic solutions of equation (4.19) is equivalent to the Galerkin method of approximate solution of operator equation (4.22). It follows from the above reasoning that the harmonic balance method of finding the oscillatory regimes in the system of automatic control L

Ùx=MÙf(t'x)

(4 26)

·

can become the Galerkin method for approximate solution of the operator equation χ = l·) o f (x)

(4.27)

where T

f) (x) = r/ h(t — s\ T)x(s) ds, Jo f (x)=f(t,x(t)), and h(t; T) is an impulse response of the linear unit with a transfer function W(p) = M(p)/L(p).

§ 5 The method of mechanical quadratures 5.1 The Galerkin method with perturbations. The method described in the previous Section is a variety of the Galerkin method. A modification of the Galerkin method, the Galerkin method with perturbations, is often used in applications. An outline of this method follows. Let Α : E E be some operator. Consider the operator equation x = A(x). To solve equation (5.1) approximately take up a sequence of subspaces £ i c £ and operators An : En

2

(5.1) finite-dimensional

c - c £ „ c . . .

(5.2)

En associated with the approximating equations xn — An{xn).

(5.3)

Solutions x* of equations (5.3) are considered as approximants of the solution x* of equation (5.1).

I Basic Concepts

26

The fact that equations (5.1) and (5.3) are akin signifies that the operators Sn

=

*An

(5.4)

Pn-A

are small (in an appropriate sense) where P \ E - * E is the projection operator from E to En. Equations (5.3) are called the Galerkin perturbed equations and the method using them, the Galerkin method with perturbations. In case when A„ = PnA the Galerkin method with perturbations coincides with the classical Galerkin method. n

n

5.2 Quadrature processes. Let a¡ n , . . . , a„„ be some positive real numbers and 0 < < · • · < snn = Τ be distinct points of the segment [0, T], Consider a quadrature process determined by the sequence of quadrature formulae

f

ds

=

Jo

a

^ 2

jnX(Sjn)

+

(« = 1 , 2 , . . . ) .

Rn(x)

(5.5)

j=i

The numbers ci\n, . . . , ann are called coefficients and the numbers ,îi„, . . . , snn are called nodes of quadrature formula (5.5). The difference pT R

n

( x ) =

" x ( s ) ds

-

ûijnx(sjn)

Jo

(5.6)

,=i

is called the residual term of quadrature formula (5.5). The quadrature process (5.5) is called convergent if for any vector function x(t) continuous in the segment [0, T] lim \Rn(x)\ n—>oo

=

0.

(5.7)

If the quadrature process (5.5) converges, then for some constant C η

sup "

a J=

in < C

(5.8)

1

and lim

max (sj+i „ — Sj n ) — 0.

(5.9)

η->· oo 1 nn = i^n-l.n,

Snn]

I Basic Concepts

28

and let Xj„(t) be the characteristic function of the set T>j„: Xjn(t) =

fi \ o

if t e Vjn, if,

(5.15)

Let /0\ 0

i l0 \ e\

=

\0 /

w

be the standard basis in and En be linear hull of the vector functions ekXjn(t) where k = 1, . . . , TV, j = I, ... ,n. Define the projectors Pn : L^ —> En by the formula: Pnx =

x(sjn)xjn{t)

(x e

(5.16)

;=ι Introduce then approximating operators An '• En —En η η

with

Y , y] OCjnG(Sin, Sj„)f(Sj„,

(0

(5.17)

'V=1 7=1

where η Χ

η = Σ ÇjnXjnit)· j= 1

System (5.13) is equivalent to the operator equation Xn in the subspace En: the vector {£*n, ..., if and only if the function

( -V t; )

(5.18)

ξ*ηη} G M^" is a solution of system (5.13) η

oo n

= 0

(6.18)

where Ρ is the embedding operator of C to La.

§ 7 The method of finite differences 7.1 Formulae of numeric differentiation. Let χ = x(t) (t e I ) be a smooth Revalued function. Consider the formula of numeric differentiation x{k)(t) « h~k ¿

bjx{t + jh).

(7.1)

J = - r

Here bj e M.N, h is a small parameter and positive integers r and s satisfy the inequality r + s > k. Numeric differentiation formula (7.1) is called convergent if for any smooth function x(t) and for any t lim h~k Σ A-»· 0

bjxit + jh) -

x{k\t) = 0.

j=-r

7.2 Discretization of the differential equations. Consider the problem of finding the Γ-periodic regimes of the nonlinear system = f(t, χ,χ',...,

X(m_1))

(x G R w )

(7.2)

with the right-hand side Γ-periodic in t. One of the most widely used methods of solution of this problem is the method of finite differences. Approximants of the Γ-periodic regime x*(t) of system (7.2) are written in the form of Γ-periodic network function xh = (...,

X-I, Xo, Χι, ...)

(*,· e R w , j = 0, ± 1 , . . . )

with values in R^. Here h = Τ/η and [xh]j = [Xhì]+n

(j =

0,±í,...)

34

I Basic Concepts

([Xhij e IR^ is the j-th component of the network function x/,). 7-periodic network functions χ h form a linear space which will be denoted £/,. The components [xh\j (j — 0, ±1, . . . ) of the network function xh are approximants of the values of the periodic regime x{t) in the points jh (j — 0, ± 1 , . . . ): [xh]j

xjjh).

The derivatives in system (7.2) are replaced by finite differences using some formula of numeric differentiation Sk (k k x \t) « h~ Σ hkrt + jh) (k — 0,1,..., m). (7.3) j=-n Denote by UJh the shift operator given by the formula [UJhxh]i = Xi+j

(i,j = 0,±1,

...),

i.e. the i-th component of the network function UJhx¡, is equal to the (i + j)-th component of the network function χ h- Then the difference operator Z)® associated with the numeric differentiation formula (7.3) takes the form Sk D

V

h k

= ~

Σ bMUh j=-n

(k = 0,l,...,m).

(7.4)

Write now the difference system associated with continuous system (7.2): D{™\h = qhf(t, Dfxh,

D(rX)Xh)

(7.5)

where qh is the operator mapping the function x(t) to the network function ( . . . , jc(-A), x(0), x(h), ...). The coordinate form of the difference system (7.5) is [D™xh]j = f(jh, [Df}xh]j,...,

[D(rl)xh]j)

(j = 0,±l,...).

(7.6)

It is a system of η vector equations where the η A'-dimensional vectors [xh]j are the variables. Consequently system (7.6) is a system of Nn scalar equations with Nn scalar variables. The solutions x*h of this system approximate the Γ-periodic solution x#(t) of system (7.2): [x*h]j~ x*(Jh)

0" = 0, 1

π - 1).

7.3 Reduction of the method of finite differences to the Galerkin method. Consider a linear differential equation dmx — /ìm

h TX = y(t)

(7.7)

§ 7 The method of finite differences

35

where y(t) e £ν and the parameter r is chosen so that the homogeneous equation dmx — + rx = 0 dtm

(7.8)

with the periodic boundary conditions 4 0 ) = x(T),

xim~l)(0)

= x(m~1}(T)

(7.9)

has only a zero solution. Then the Γ-periodic solution of equation (7.7) is unique and can be represented in the form x(t)=

[ Jo

(7.10)

G(t,s)y(s)ds

where G(t, s) is Green's function of the differential operator ^ + r with periodic boundary conditions (7.9). The derivatives of the Γ-periodic function x(t) have the form X(k\t)=

Γ

S)y(s)

ds

(k =

... ,m — I).

(7.11)

Denote by /C and IC(k> the operators given by the right-hand sides of formulae (7.10) and (7.11). Now finding Γ-periodic solutions of equation (7.2) reduces to solution of the operator equation y = B(y)

(7.12)

(yeSN)

where B(y) = fit, Ky, fCmy,IC^y)

+ rKy.

(7.13)

In more precise terms, the solutions χ*(ί) of equation (7.2) and the solutions y jt) of operator equation (7.12) are related by the formula x*(t)=

[ G(t, s)y*(s) ds. Jo Consider now the discrete problem D™xh + rDf>xh=yH.

(7.14)

(7.15)

Here yh is a Γ-periodic network function. Under some natural assumptions the invertibility of the operators (7.8) with boundary conditions (7.9) implies that for sufficiently small h equation (7.15) has a unique solution Xh for each network function y h• The solution xh can be written in the form xh - K.hyh

(7.16)

where /C/¡ is an operator in the space £h of the network functions. We introduce now some subspace En of the space SN of Γ-periodic functions. It consists of piecewise linear functions whose nodes are situated in the points

36

I Basic Concepts

kh (k — 0, ± 1 , . . . ), h = Τ /η. The spaces En and Sh are isomorphic and their isomorphism is made by the operator Φ* : £h

En

(7.17)

such that the image of the network function χ h = ( . . . , χ~ι, xo, χι, . . . ) is a piecewise linear function Φ/,χ/, whose graph goes through the points (kh, Xk) (k = 0, ± 1 , . . . ) . Obviously ΙΙΦΑΙΙ

=

II®*

1

II

=

1 .

( 7 . 1 8 )

Using the isomorphism Φ* define the operators P„ = Φ h q n : £N ->

E„

and /C® = Φ λ Ο ^ ^ Φ , ; 1 : En

En.

(7.20)

The operators Pn are projectors of the space £¿v to the subspaces En. One can easily see that \\Pn\\ = 1.

(7.21)

The operators Kf® approximate the integral operators ÌC 0

Με-,

i™ libilo = IMI*··

h-* 0

Although the systems V = (ph) and Q = (qh) consist of the same operators, the Vconvergence and Q-con vergence are defined using different norms. The sequence Xh is "P-convergent to χ if lim \\xh -phx\\m o

= 0

while the Q-convergence of the sequence y h to y means that lini IIyh - qhy||0 = 0. h->- 0 Define now the approximating operators U(xh)

= qhf(t, Dfxh,...,

D^Xh).

Here £>® is the difference operator of differentiation defined by formula (7.4) and qh is the operator mapping the function x(t) to the network function χ h = (..., x(—h), x(0), x(h), ...). The operator f¿¡ acts from to Sh. The approximating equations are qhf(t,D^Xh,...,D(^xh)

= 0.

(8.19)

0" = 0, ± 1 , . . . ).

(8.20)

The coordinate form of equations (8.19) is f(t, \Dfxh\j,[D^xh]j)

— 0

42

I Basic Concepts

Equation (8.20) is a system of η equations with η variables. As a result, both the method of finite differences and the method of mechanical quadratures can be treated as a variant of the factor method. This version of those methods is much easier and more natural than their Galerkin-type realization described in Sect. 5 and 6.

Chapter II Existence theorems for oscillatory regimes

§ 1 Smooth manifolds and differential forms This Section contains some necessary information about closed smooth manifolds and exterior differential forms.

1.1 Smooth manifolds. Let U c M.N and V c MM be domains (open sets) in spaces and K M . Consider a mapping / : U —> V of domain U to V. The mapping / is given by a set of M scalar functions /,·, defined in U: f ( x ) = (fi(xi,

• •·, *n), • •·, Im(x\,

• • -,xn))·

Functions / , · ( / = 1 , . . . , M) are called components of the mapping / . A mapping / :U V is called smooth if all the partial derivatives dTfjjx

x,...,xN)

of the components > · · · , x n ) (i — I,... ,M) of the m a p p i n g / exist and are continuous in U. If X C i " and Y c R M are arbitrary subsets of spaces MiV and E M , then the mapping / : X Y is called smooth if for any point χ e X there exist a neighborhood U c l " and a smooth mapping F \ U R M coinciding with / in unx. A continuous one-to-one mapping / : X F (X c K", F c is called a homeomorphism of sets Χ and Y if f ( X ) = Y and the inverse mapping/"" 1 : Y -> X is continuous. A mapping / : X ^ F ( X C R W J C R M ) is called a diffeomorphism i f f is a homeomorphism of sets X and Y and both / and f~l are smooth mappings. The sets X and Y are called then diffeomorphic. The set Wl c K w is called a smooth M-dimensional manifold if for any point χ e DJl there exists its neighborhood W c such that the set 9JÍ Π W is diffeomorphic to an open set V C R M . The set 9JÏ Π W is called a neighborhood of the point χ in SDÌ. A set SDÌ C Ma' is called a zero-dimensional manifold if any point JC e 971 has a neighborhood SUI Π W containing only a point x.

44

II Existence theorems for oscillatory regimes

A diffeomorphism g : V ÜJIΠ W is called a parametrization of the neighborhood 9ΗΓΊ W. The inverse diffeomorphism g~l : 9JlfW —>· V is called a coordinate system in 9JÍ Π W. Remark. The above definition describes only smooth manifolds without a boundary, or closed manifolds, and does not describe manifolds with boundaries. Manifolds with boundaries are introduced later.

1.2 Tangent spaces. Before introducing a concept of a tangent space to a smooth manifold we define a derivative of a smooth mapping. Let U be a domain in R^. The derivative f'x of a smooth mapping / : U -> V (U c R M ) is a mapping of the space to the space R defined by the formula W,,, f(x + t h ) - f ( x ) r fx(h) = lim f—»0 t

where χ e U, he M.N. The derivative/^, is a linear mapping. \íf¡{x\,..., 1, . . . ,M) are components of the mapping / , then an M χ N-matrix Sf± dx¡

___

9£l\ dxN

dxi

'''

3Ím j Bxn /

xN) (i =

is the matrix of the f'x. This matrix is often identified with the derivative f'x itself. The properties of the differentiation of smooth mappings are similar to that of the differentiation of scalar functions: (1) If / : U

R M and g : U

MM are smooth mappings, then

(f +

(2) If / : U —>·

g)'x=fx+g'x·

is a smooth mapping and α is a scalar, then (af)'x =

af'x.

(3) (Chain rale) If / : U -»· V and g : V W (U c are smooth mappings and f(x) = y, then (gofyx

=

V C RM, W C

g'yofx.

(4) If / : M.N —> R M is a linear mapping, then for any point χ e R N fx=f-

Define now a tangent space of the smooth manifold OTcE". Let χ € Wl be a point in 9Jt. Choose a parametrization g : V -»• SDÌ

§ 1 Smooth manifolds and differential forms

45

of a neighborhood g(V) of the point χ e 9Jt, where V is an open subset of Let ν = g - 1 ( X ) . Find the derivative g'v : —>• of the mapping g. The image of the space in the mapping g'v is called a tangent space TD.Jlx to the manifold DJl in the point x: r m x = g;(M M ). This definition is sound because the image does not depend on the chosen parametrization g : V —»· SDÌ of a neighborhood of the point χ e ÜJl. Let m C R N and Ol C R M be smooth manifolds and / : 5DÎ - » 9Î be a smooth mapping of the manifold SDÌ to 9Î. Freeze a point χ e M and define a derivative f'x of the mapping / in the point x. This derivative will be a linear operator from TMX to T%, where y=f(x). f is a smooth mapping, so there exists an open set W in containing χ and a smooth mapping F :W ->• MM coinciding with / in the set fflflff. Let

fx{h) = KW

(h e TWlx).

Let us prove a soundness of this definition, i.e. it does not depend on the chosen mapping F. Choose parametrizations

g:U

^WlC *N

ÛÏC pili

h :V

of neighbourhoods g(U) and h(V) of points χ and y. Here U and V are domains in and R M respectively. It is possible to assume that g(U) c W and / o g(U) c h(V). Consider a commutative diagram of smooth mappings of domains U, V, W:

W h(A-'o/og)

U -—-Λ

V

The commutative diagram of derivatives associated with the previous one is

Κ K, (A-'o/og)^ where u = g '(χ), ν = h '(y)· It follows from this diagram that the derivative F'x maps ΤΜΧ = g'u(M.n) to T% = h'v(Rm). Also fx — i.e. f'x does not depend on F.

o (h

I Λ-1 o f o g)'u o (g'u)

46

II Existence theorems for oscillatory regimes

1.3 Orientation. Consider the set of all bases in M^, or the set of all ordered collections of Ν linearly independent vectors. The set of bases is divided into two classes of equivalence: two bases (a\, . . . , aN) and (b\, . . . , b^) belong to the same class if the determinant d e t C of the transition matrix C from basis («ι, . . . , αχ) to basis {b\, ..., bN) is positive. The orientation in is a class of equivalence of ordered bases. The standard orientation in M^ is generated by the basis (1, 0, . . . , 0), (0, 1, . . . , 0), . . . (0, 0, . . . , 1).

(1.1)

A smooth manifold 9JÏ is called oriented if all its tangent spaces T9Jlx are oriented consistently. Consistency of the orientations is defined as follows: if dim 971 — M > 0, then for any point χ e DJl it is assumed that there exist a neighbourhood U c and a diffeomorphism h : U M'w preserving the orientation, or such that for any y GU the isomorphism h'y maps the chosen orientation of the space Τ Tì y to the standard orientation of the space R". The orientation of a zero-dimensional manifold is defined as one of the characters, + 1 or —1.

1.4 Manifolds with boundaries. Let Hm = {x = (xu

. . . , % ) e l

M

: % > 0}.

HM is a closed half-space in R M . The boundary DHM of the half-space HM is defined as a hyperplane 8Hm = {x = (xu

...,

xm)gRm

:xM = 0}.

Consequently, 9HM = χ 0. A set X c is called a smooth M-dimensional manifold with boundary if for any point χ e X there exists its neighbourhood Χ Π U diffeomorphic to a neighbourhood HM Π V (here U and V are open sets in and R M , respectively). The boundary dX of the manifold X is defined as a set of all points of X which are mapped by these diffeomorphisms to the points of dHM. The boundary dX of a smooth M-dimensional manifold Ζ is a smooth (Μ — 1)dimensional manifold. The set X \ dX is called the interior of the smooth manifold X. It is a smooth manifold of dimension M. The tangent space T X x to the manifold with boundary is defined by the same way as for closed manifolds. Note that the tangent space TXx in the point χ e dX has a dimension equal to the dimension of X. One should distinguish this space from the tangent space TdXx of the manifold dX in the point x. There is a natural imbedding TdXx C ΤXx\ and dimT'dXx = dim TXx - 1.

47

§ 1 Smooth manifolds and differential forms

To describe the procedure of orientation of the boundary dX of the iV-dimensional manifold X, note that the tangent space TXx in the point χ e dX contains vectors of three kinds: 1) Vectors tangent to the boundary dX (they make a (TV— 1)-dimensional tangent space TdXx). 2) Vectors directed "outside" the manifold X (they make an open half-space bounded by a subspace TdXx c TXx). 3) Vectors directed "inside" the manifold X (they make a complementary halfspace). If some orientation of the manifold X is chosen (call it positive), then for χ e dX choose in the tangent space TXx a positive oriented basis (x\, ..., xM) such that X2, . . . , xm £ ^Xx and the vector x\ is directed "outside". Then the basis (x2, • • •, xm) determines the orientation of the boundary dX of the manifold X. If the dimension of the manifold Ζ is 1, then any point χ of the boundary dX is given an orientation + 1 or — 1 depending on whether the positive oriented vector of the tangent line to dX is directed "outside" or "inside" X. The term "smooth manifold" will be used hereafter only for smooth manifolds without boundary. If a smooth manifold has a boundary it will be referred to as a "manifold with boundary".

1.5 Exterior forms. The ft-th degree exterior form (or fc-form) on is a function cük(xi, ..., Xk) (xi e R' v , i = 1 , . . . , k) of k vectors which is poly linear (linear on each its argument) and skew-symmetric: a>k{x\

+ μχ] — Xcùk(x\,...

k

(ù {xh,...,

,xi+u...,xk) ,x°¡, xi+1, ..., Xk) + μω!ί(χ i,..., ν

xik) = (-1) ω*(χι,...,

x}, xi+i,...,

χ

xk).

where Γ0 ν=
M.N of a neighbourhood U C M'v+1 of the point xo such that F(x) = G(x) (x e U Γ) HN+l) and G(x) has no critical points in U. By Lemma 2.1 G~l(y) is a smooth one-dimensional manifold. Let Ρ be an orthoprojector from to the coordinate axis x w + ) and ρ be its restriction to the manifold G~l Cy). Then zero is a regular value of the mapping ρ : G~l(y) xN+ iNaturally, the line lx tangent to the one-dimensional manifold G~l(y) in the point χ e p~l(0) coincides with the kernel of the derivative G'x = F'x : RN+l

Rn,

and the assumption that the restriction / : 3H N + l -> R w is regular in the point χ ensures that lx does not stay in dHN+x. It means that zero is a regular value of the mapping p. Lemma 2.2 thus implies that the set G~l (y) Π HN+l = F~l (y) Π U containing χ 6 G - 1 (y) such that ρ (χ) > 0 is a smooth manifold with boundary p~\ 0). •

2.3 The degree of a mapping. Let 371 and 0Í be A'-dimensional oriented manifolds, ÜJI be compact, and 0Í be connected. Consider a smooth mapping / : 9JI -> 9Í.

II Existence theorems for oscillatory regimes

54

Let χ e SDÌ be a regular point of the mapping / i.e. f'x : TWlx - > ΓΟΐ/φ be a linear isomorphism of the oriented vector spaces TWlx and Τ^(χ). Denote , J 1 sign Λ - I

if orientations of TdJlx and Γ9ΐ/· ω coincide, ο^^ί^

For any regular value >> the degree deg ( f , y) of the mapping / : 97Î by the formula deg(f,y)=

Σ

is defined

si§nfx-

Since y is a regular value of the mapping /, the classical implicit function theorem implies that the full preimage f~l(y) of the point y consists of isolated points x,. The number of points in the p r e i m a g e ' ( y ) is finite because 97t is compact: f~\y)

= {x\, ···,

**}·

This implies that k deg(f,

sign/;..

= f=l

It follows from the definition that deg ( f , y) is a locally constant integral function of y defined in an open subset of full measure (by the Sard theorem) in the manifold an. It turns out that deg(/, y) does not depend on the chosen regular value y, or the function deg ( f , y) is constant in the set of regular values. The common value of d e g { f , y) is called the degree of the mapping / : 9Jt ->· 0Î and is denoted as deg/. It is proved in this Subsection that this definition of the degree of a mapping is sound since deg ( f , y) is constant in the set of regular values. Lemma 2.4. Let the manifold 971 be a boundary of a compact oriented manifold X, and the orientation of 97X coincide with that of 8X. If f : 971 VI is the restriction to 971 of a smooth mapping F : X —»· then d e g ( f , y) = 0 for any regular value y. Proof. First let y be a regular value both for / and for F. By Lemma 2.3 the set F~1 (y) is a smooth one-dimensional manifold which boundary dF~] (y) is an intersection of F~l(y) and the boundary 971 of the manifold X. A compact onedimensional manifold is a union of finite number of simple arcs (diffeomorphic images of intervals) and simple closed curves (diffeomorphic images of circles). The boundary points of arcs belong to 971 = dX. Let A c F~1 (y) be one of those arcs, and dA = [a] U {b}. Let us show that sign/; + sign/; = 0.

§ 2 Degree of a mapping

55

The Lemma follows from this equation because preimage f'(y) of the arcs lying in the preimage of

consists of ends

F'(y).

To prove the previous equation, analyze how the orientations of manifolds X and 0Í determine the orientation of the arc A. Let χ € A and {uj, . . . , υ/^+ι} be a positive oriented basis in TXx

such that v\

is tangent to the arc A. Then υ ι makes the tangent line TAX positively oriented if and only if F'x maps {v2,...,

i>w+i} to the positively oriented basis of the space

T%. Let v\ ( x ) be a positively oriented vector of unit length tangent to A in the point x. One of the two vectors ν ι (a) and v¡ (b) is directed inside and the other outside the manifold X. This means that the numbers f'a and f'b have different signs and the sum sign/' a + signf' b is equal to zero. Consider now the case when the point y e Oí is a regular value of the mapping / : X

0Í but is not a regular value of F : X

Oí. Since every small

neighbourhood V C Oí of the point y consists of regular points of the mapping /, the function deg(/, y) is constant in V. The set of regular values of the mapping F : X

0Î is dense in V (see Corollary of the Sard theorem), hence there is a

point yo in V which is a regular value of both / and F. A s proved above,

•

d e g ( f , y ) = deg(/-,yo) = 0. Let T I C

be a smooth manifold. Denote as SUt χ [0, 1] a subset in R N + 1

which consists of all {χ, λ } where χ e SDÌ, λ e [0, 1], The set 9Jlx [0, 1] is a smooth manifold with boundary. Its boundary consists of two copies of the manifold SDÌ, denote them as 3JÌ χ 0 and SDÌ χ 1. If SDÌ is oriented then ÛJI χ [0, 1] can be oriented as a product of two oriented manifolds, SDÌ and [0, 1]. Copies 3JÌ χ 0 and 3JI χ 1 of the manifold SDT are oppositely oriented. T w o mappings /o : SDÌ tfo~/i)

0Í and / ι : StJl

0Í are called smoothly homotope

if there exists a smooth mapping F : 9JÌ χ [0, 1]

F ! (χ) = fi (χ)

Foix) = fo(x),

Oí such that

(χ e art).

The mapping F is called a smooth homotopy connecting /o with f \ . L e m m a 2.5. Let the mappings fo

: 9JÌ —»• 0i and f\

: 9JÌ

0Í be

smoothly

homotope. Then for any common regular value y of the mappings fo and f\ deg(fo,y) Proof

Let F : 9Jt χ [0, 1]

=

deg(fuy).

0Ì denote a smooth homotopy connecting fo with f\.

A restriction / of the mapping F to the boundary of the manifold SDÌ χ [0, 1] is given by the formula fo (χ)

if {χ, λ } e SDÌ χ 0,

/ ι (χ)

if {χ, λ } e an χ 1.

56

II Existence theorems for oscillatory regimes

Since 9Jt χ 0 and 971 χ 1 are oppositely oriented, for a regular value y the degree d e g ( f , y) of the mapping/ : 9(971 χ [0, 1 ]) —^ OT is equal to the difference of degrees deg(/i, y) and deg(/o, y). On the other hand, by Lemma 2.4 deg(f, y) = 0. • The diffeomorphic mappings / o , / i : 9JÍ ^ 0Î of the smooth manifold 971 to the smooth manifold 0Î are called smoothly isotope if there exists a smooth homotopy F : 9ÏÏ χ [0, 1] 0Î connecting/o with f i such that for any λ e [0, 1] the mapping Fx(·) : 971 0T is a diffeomorphism from 971 to 0T. Lemma 2.6. Let yo and yi be arbitrary points of a connected manifold 971. Then there exists a diffeomorphism h : 97Ì — O í mapping yo to y \ which is isotope to the identity mapping. Proof First construct a smooth isotopy F° : RN χ [0, 1] of the space E w to itself which leaves all points outside the unit ball Β in their places and maps the origin to the given point XQ e Β. To do this, consider the differential equation χ = φ(χ)χο

(χ e

1

where φ : —»· E is a smooth function such that φ(χ) > 0 when χ € Β and φ(χ) = 0 when χ e RN \ Β. Let p(t, x) be a solution of this equation with the initial datum p(0, x) = x. Consider a solution pit, 0) of this equation. Evidently there exists such ίο > 0 that p(to, 0) = xq. Denote F SN~1. This mapping is a diffeomorphism. This diffeomorphism preserves orientation if det A > 0 and changes it if det A < 0. By Theorem 2.3 deg A/\A\

= ind(0, A) = sign det A.

•

Since sign det A = (—1)^

(3.17)

where β is the sum of multiplicities of real negative eigenvalues of the matrix A, Theorem 3.12 implies Theorem 3.13. The topological index of the isolated zero of the linear field A is given by the equation ind(0, A) = (—1)^.

(3.18)

8. Direct sum of vector fields Let the space R N be a direct sum of spaces Ε o and E\\ = Eq ® E\. Let PQ :RN ^ Eo and P\ : position (3.19).

(3.19)

E\ be projectors corresponding to the decom-

Theorem 3.14 (Leray-Schauder). Let Ω C be a bounded domain and Ωο = Ω Π £ ο φ 0 . Suppose that the field Φ is defined in 9Ω, nondegenerate and satisfies the condition ρ , φ ( χ ) = ργχ

(χ e 3Ω).

Then the rotation / ( Φ ; 3Ω) of the field Φ in 9Ω coincides with the )/(Φ 0 ; 9Ωο) of the restriction Φο of this field to 3Ωο·

rotation

Proof Consider only the case when the domain Ω is connected. Let first |χ(Φ 0 ; 9Ω 0 )| = Η Φ 0. Let A : EQ ->· EQ be a nondegenerate linear mapping and sign det A = sign γ(Φο\9Ωο). Choose in Ω different points x\, ..., xn and the number r such that the balls BN(r, x¡) (i = 1 , . . . , ή) do not intersect and lie in Ω.

72

II Existence theorems for oscillatory regimes

Define in Ω a continuous mapping χ : Ω EQ in the following way. First take up χ(χ) = ΡΟΦ(χ) when χ e 8Ω and χ(χ) = A(PQX — χ,·) when χ e BN(r, χ,·) (i = 1 , . . . , η). Then extend this mapping to Ωο so that new zeros do not emerge and the image of Ωο lies in EQ. Then extend it to the whole set Ω preserving the continuity with values in EQ (it is possible by the Uryson theorem, see e.g. [191]). Define Ψ(χ) = χ (χ) + P\x. The points x, and they alone are zeros of the field Ψ. By Theorem 3.13 ind(x,·, Ψ) = ind(x ; , Ψο) where Ψο is a restriction of the field Ψ to Ωο- Then Theorem 3.9 on an algebraic number of singular points implies equation / ( Ψ ; 3Ω) = χ(Ψ 0 ; 9Ω 0 ). But Ψ(χ) = Φ (χ) when χ e 8Ω. The two latter equations imply the equation )/(Ψ; 9Ω) = )/(Φο; 3Ωο). If /(Φο; 9Ωο) = 0, then the proof simplifies. In this case the mapping ΡοΦ is to be extended to a mapping with values in EQ nondegenerate in Ωο· Then it should be extended in an arbitrary way to the continuous mapping χ defined in Ω with values in EQ. If the field Ψ is defined by the formula Ψ(χ) = χ(χ) +P\x, then the rotations / ( Ψ ; 9Ω) and ΚΨο; 9Ωο) are equal. Consequently / ( Ψ ; 9Ω) = / ( Φ ; 9Ω) and ΚΨο; 9Ω 0 ) = γ(Φ0·, 9Ω 0 ). • Theorem 3.14 is needed not only to pass to spaces of lower dimensions. It is also the core of theory of rotation of vector fields in infinite-dimensional spaces. The following statement, Theorem about direct sum of vector fields, is also an important tool of computing the rotation. Let Ωο C EQ and Ω] c E¡ be bounded domains in the spaces EQ and E¡. The product Ω = Ωο χ Ω ι of the domains Ωο and Ω ι is a domain in M'v which consists of points χ = xo + xi where xo e Ωο and x\ e Ω], Let Φο and Φι be nondegenerate vector fields defined in Ωο and Ωι with values in EQ and E\, respectively. The direct sum Φ = Φ 0 θ Φι of the vector fields Φο and Φι is a vector field defined in Ω Φ(χ) = Φ ο ( / ν ) + Φ ι ( Ρ ι χ ) ·

(3.20)

Theorem 3.15. If the fields Φο and Φι are nondegenerate in 9Ωο and dQ¡, respectively, then the field Φο Θ Φ ι is nondegenerate in 3Ω and / ( Φ 0 ® Φ ι ; θ Ω ) = ΚΦ0;9Ω0) · Κ Φ ι ; 9 Ω 0 .

(3.21)

Proof. Computing the rotation /(Φο θ Φι; 9Ω), one can redefine the vector fields Φο and Φι in an arbitrary way in internal points of domains Ω 0 and Ω ι . Then without loss of generality the fields Φο and Φι can be assumed to have only finite number of zeros in domains Ωο and Ωχ, respectively. Let x¿, . . . , xg be zeros of the field Φο in the domain Ωο and x\, ..., x'" be zeros of the field in the domain Ωι. Then the points x'0 + x7, (i = 1, ..., n, j = 1 , . . . , m) are the zeros of the direct sum Φ = Φο Θ Φι in the domain Ω = Ωο χ Ω ι. The following equation

§ 4 Completely continuous vector

fields

73

is a corollary of Theorem 3.9 on the algebraic number of singular points: /(Φ; 9Ω) =

Σ

ind(x[) + xÌ, no the iterations A" map Ω to itself and A" (*) φ χ for χ e 9Ω. Then γ(Ι - A\ 3Ω) = 1. Some auxiliary statements are to be given prior to the proof of the Browder theorem. Let Ωι and Ω 2 be bounded domains in and Λ : ^ and Λ : IR'7 ^ w M be continuous operators. Let Fq and F¡ denote sets of zeros of the vector fields φ 0 = 1 - Α ι oAq,

(5.1)

82

II Existence theorems for oscillatory regimes Φ ι = / - Λ ) θ Λ

(5.2)

in Ωο and Ωι respectively. The domains Ωο and Ωι are called domains with the same core with respect to Φο and Φι if 4)(*o)cni,

i,(Fi)Cfi0.

(5.3)

The inclusions (5.3) imply that Φο and Φι are nondegenerate in 3Ω0 and 3Ω! respectively and also Ao(Fo) = Fu

Al(F1)

= F0.

(5.4)

Lemma 5.1. Suppose that the bounded domains Ωο and Ωι have the same core with respect to the fields Φο — / — A\ o Ao and Φι = I — Ao o A\. Then y(Oo;ano) = K ® i ; a n i ) .

(5.5)

Proof. Consider a direct sum R2N = ® of two copies of the space RN. Let Ωο lie in the first copy of the and Ωι in the second. The points of the first R w will be denoted as χ and of the second Mw as y. The points of the direct sum will be denoted as {x, y}. Let Ω = Ωο χ Ωι c Consider in Ω the vector field

This field is nondegenerate in 9Ω. Indeed, if Ψ(^ο,)Ό) = 0> then x0 = jo = Α (*ο) and so xo £ yo £ F\• Therefore

A\(>ό),

{x0, yoì e F0 χ F ι c Ω. As long as the field Ψ is nondegenerate in 9Ω, its rotation γ(Ψ; 9Ω) is defined. Consider in Ω the field x(x, y) - { x - A i o Ao(x), y - A W ) · This field is also nondegenerate in 3Ω and its vector in the point [x, >>} € Ω and the vector of the field Ψ are not oppositely directed because the vectors of these fields can be oppositely directed only in points {x, >>} such that y = AQ (X) but in such points these vectors coincide. Thus the fields Ψ and χ are homotope in 3Ω and their rotations in 3Ω are equal: / ( Ψ ; 9Ω) = / ( χ ; 3Ω).

(5.6)

Let the set .Αο(Ωι) belong to the ball B(r). Denote G = Ω ο χ B(r). Since χ is defined in G and nondegenerate in G \ Ω, Theorem 3.4 implies the equation y(x-dQ)

= Y(.x;dG).

(5.7)

The vectors of the fields χ and 9(x,y)

= {x-Ai

o AW.

y)

§ 5 Fixed point principles and solution of operator equations

83

in the points of 3G are not oppositely directed, and so χ and θ are homotope fields in dG and y(x;9G) = y(0;9G).

(5.8)

But since the field θ is a direct sum of the field Φο and the identity field I, by Theorem 3.15 7 ( 0 ; 9 Ο = ΚΦο;9Ω ο ).

(5.9)

Equations (5.6)-(5.9) imply the equation ΚΦΟ;9Ω 0 ) = 7(Ψ;9Ω).

(5.10)

/(Φ1;9Ω1) = /(Ψ;9Ω)

(5.11)

The equation

can be proved in a similar way. Now (5.5) is a corollary of (5.10) and (5.11).

•

Introduce now a vector field Φρ by the formula φp = I - A

p

where ρ is a positive integer. If the point xo is a zero of the field Φ /; , then the points A(Xo), A2(Xo), ·. ·, AP~X{Xo) are also its zeros. If XQ is an isolated zero, then so are A(XQ), A2(xo), • • •, AP~X(*o).

Lemma 5.2. If XQ is an isolated zero of the field Φρ, then the equation ind(„4(x 0 ), Φ Ρ ) ^ ind(x 0 , Φ Ρ )

(5-12)

holds. Proof. Choose neighbourhoods Ωο and Ωι of XQ and A(XQ) such that XQ is the unique zero of Φρ in Ωο and A(xo) is the unique zero of Φρ in Ω). The domains Ωο and Ωι have the same core with respect to fields / — AoAp~x and I — Ap~x o A. By Lemma 5.1 Κ(Φ Ρ ;Ω 0 ) = χ(Φ Ρ ;Ω,).

(5.13)

But χ(Φ ρ ;Ω 0 ) = ind(x0, Φ ρ ),

γ(Φρ\ίΙ\)

= mà(A(xQ),

Equations (5.13) and (5.15) imply (5.12). The following statement is a corollary of Lemma 5.2:

Φρ).

(5.14) •

84

II Existence theorems for oscillatory regimes

Lemma 5.3. If XQ is an isolated zero of the field Φρ, then ind(x 0 , Φ ρ ) = indM-Oo), Φρ) = • • · = md(Ap-1

(χο),ΦΡ)·

Let now Ω be some bounded domain in M.N. Denote the set of zeros of the field Φ ρ in Ω as F p. Consider also the field Φ — I — A. Lemma 5.4. Let ρ be a prime, A{FP) c Ω, and let the field Φ ρ be nondegenerate in 9Ω. Then γ(Φρ; 9Ω) = γ(Φ; 9Ω)

(mod ρ).

(5.15)

Proof Without loss of generality the operator A is assumed to be smooth, the field Φ to have only a finite number of zeros χι, . . . , Xk in Ω, and 1 not to be an eigenvalue of the derivatives A(x\), • . •, A'(xk)· Choose a real number ρ such that the balls B(p, x¡) (i — 1 , . . . , k) are mutually disjoint and lie in the domain Ω and 1 is not an eigenvalue of the derivative A'(x) when χ belongs to one of those balls. Define in M.N the operators k A W = A(x) + ε Σ Π/2(* i= 1 ίφί

~ xi)(x

"

Xi)

where / is a linear functional such that/(χ,·) φ f ( x j ) when i φ j. If ε is sufficiently small then γ(Φερ, XI) = γ(Φρ· 9Ω),

γ(Φε·, 9Ω) = γ(Φ· 3Ω).

Therefore it is sufficient to prove Lemma 5.4 for the fields Φ£ and Φ£ with a small ε > 0. Let us show that if ε is small, then xj is the unique zero of the field Φ ε in B(p, xj). xj is the unique zero of the field Φ in B(p, x/) and therefore ΚΦ;5(/>, *,)) = ιηά(χ 7 ;Φ) = ( - i f ^ where ß(xj) is the sum of multiplicities of real eigenvalues of the operator A'(xj) which are greater than 1. Thus if ε is small, then Y(e-S{p,Xj))

=

(-l)^K

But if ε is small, then the number 1 is not an eigenvalue of all linear operators Α'ε (x) (x e B(p,Xj)). This implies that every zero of the field Φ ε in the ball B(p, Xj) is isolated and that all the numbers βε(χ) (χ e B(p, Xj)) are odd or even simultaneously (β ε (χ) is the sum of multiplicities of real eigenvalues of the operator A!g(x) which are greater than 1). This means that the indices of all zeros

§ 5 Fixed point principles and solution of operator equations

85

of the field Φ ε in B(p, x f ) are the same. Theorem 4.9 implies now that the rotation /(Φ ε ;5(/0, Xj)) is divisible by the number of zeros of the field Φ ε in B(p, χ/), and so the field Φ ε has only one zero in B(p, xj), and it is x¡. Consequently if ε is small, then Φ ε has no zeros except x\, jc*. One can easily see that A'e(xj) - A'(Xj) + ajsl

( j = 1 , . . . , k)

where the numbers a¡ are not equal to zero. As a result, with any small nonzero ε the operators Α'ε(χ/) have no eigenvalues in the unit circle. Therefore it would be sufficient to prove Lemma 5.4 assuming that the operators A'(xi), . . . , A'(xk) have no eigenvalues in the unit circle. But then 1 is not an eigenvalue of all the operators (Ap)'(xj) (j = 1 , . . . , k). Hence x\, ..., jc* are isolated zeros of the field Φ ρ . The index of each zero x¡ of the field Φρ depends on the sign of the determinant det(7 — (Ap)'(xj)). Since (Ap)'(xj)

= (A'(xj)Y

(j=l,...,k),

it is true that ηκΙΟ^Φρ) = ηκΙ(χ;;Φ)

(5.16)

ind(^;®p) = ίηά(^;Φ) (modp)

(5.17)

if ρ is odd and

if ρ = 2. _ Denote now as U a neighbourhood of points x\, ..., Xk such that U C Ω and the field Φρ has no zeros in U except x\, . . . , xt- It follows from equations (5.16), (5.17) and Theorem 3.9 on the algebraic number of zeros that γ(Φρ·, dV) = γ(Φ\ dV)

(mod p)

where V = Ω \ U. Therefore it would be sufficient to prove the Lemma assuming that the field Φ has no zeros in Ω. Let Φ(χ) φ 0 for χ € Ω. Perturbing the operator A slightly we can restrict our considerations to the case when the field Φρ has only a finite number of zeros in Ω. Let y e Q b e some zero of the field Φρ. All the points y, ^.(y), ..., Ap~l(y) lie in Ω and are zeros of the field Φρ. Since ρ is a prime then none of these points coincide. Indeed, assume that this is not true, or for some I and m (I < I < m < ρ - 1) the equation Al(y) = Am(y) holds. Then Al+p-m(y)

= Ap(y)=y.

(5.18) k

Let k be a minimal positive integer such that A (y) = y. By virtue of (5.18) 1 < k < p. So for some integer η > 1 the equation ρ = nk holds and ρ is not a prime.

86

II Existence theorems for oscillatory regimes

Consequently none of the points y, A(y), ..., Ap 1 (y) coincide. Therefore all the zeros of the field Φρ can be split into mutually disjoint groups {yu A(yi),

...,

Ap~' (yO),

{y η, A(yn),

...,

Ap-\yn)}.

Theorem 3.9 on the algebraic number of zeros implies now the equation η γ(Φρ; 3Ω) - ^ (indty, Φρ) + md(A(yj), y=i

Φρ) + · · · + indCT"'(}>,), Φ ρ )),

and by Lemma 5.3 η γ(Φρ·, 9Ω) = ρ Σ ind(yj, Φρ) = 0

(mod ρ).

(5.19)

;=ι Since the field Φ = / — A is nondegenerate in Ω, γ(φ· an) = 0. Equations (5.19) and (5.20) imply now (5.15).

(5.20) •

We proceed now to the proof of the Browder theorem. Proof. Corollary 5.1 implies that if η is sufficiently large then γ(Ι — An; 9Ω) = 1. It follows then from Lemma 5.4 that the number y(/ — Α; 9Ω) — 1 is divisible by all sufficiently large primes. It is impossible unless γ(Ι — A; 9Ω) = 1. • Theorem 5.2 implies the fixed point principle of F. E. Browder: Theorem 5.3. If the assumptions of Theorem 5.2 are true, then the operator A has at least one fixed point in Ω.

5.3 The Schauder theorem. The fixed point principle of J. Schauder is an analog of the Brouwer theorem for completely continuous operators in infinite dimensions. Theorem 5.3 (J. Schauder). If Ω c E is a bounded convex domain in the Banach space E and completely continuous operator, the operator A has at least one fixed point in Ω.

§ 5 Fixed point principles and solution of operator equations

87

Proof. As in the proof of the Brouwer theorem one can assume that 0 € Ω and A(x) φ χ when χ e 9Ω. Then a completely continuous family of vector fields φ λ ( χ ) = X - (1 - λ)Λ(χ)

(χ G 9Ω; 0 < λ < 1)

is a homotopy between the fields Φ 0 (χ) =x-

A(x)

and Φι(χ) = χ. Since /(Φι;3Ω) = 1, it is true that γ(Φο\ 9Ω) = 1. Theorem 4.4 completes the proof. • Corollary 5.2. Under the assumptions of Theorem 5.3 / ( / — Α; 3Ω) = 1. The Schauder principle is often represented in the following form: Corollary 5.3. Let /C be a convex compact in the Banach space E and A: K, —» K, be a continuous operator. Then A has at least one fixed point in IC.

5.4 The Leray-Schauder principle. Consider an equation χ = A(x)

(5.21)

in a Banach space E where A : E E is a completely continuous operator. The solutions of equation (5.21) are said to have an a priori estimate if there is po > 0 such that equation (5.21) has no solutions outside the ball B(po). The existence of an a priori estimate by no means ensures the existence of a solution. Consider a family of equations χ = λΑ{χ)

(0 < λ < 1)

(5.22)

along with equation (5.21). Theorem 5.4 (The Leray-Schauder principle). Assume that the solutions of all equations (5.22) have a common a priori estimate. Then equation (5.21) is solvable. Proof. Consider a completely continuous family of vector fields Φ λ (χ) = χ - λ^Ι(χ)

II Existence theorems for oscillatory regimes

88

in the sphere S(po) of the big radius po· Since Φ λ (χ) φ 0 when ||jc|| = po, 0 < λ < 1, it is true that the fields Φο and Φι are homotope in S(po)· Then Κ(Φι;5(ρ 0 )) = y(Oo;5(po)) = K 0 are such that the norm ||(7||* of the matrix C corresponding to the norm |·|* in R^ is less than 1. Lemma 5.5 shows that this is possible. Introduce now the norm ||·|| £ in the space E by the equation Me

= LPWI*

(5.42)

where X = {Xl, . . . , XAf}, p(x) = {He,,

..·,

Ilxwbj·

Then (5.38) takes the form p(A(x)-A(y)) CN is also completely continuous. Moreover, by virtue of the estimate (6.3) with any xeCN l l £ ° f l l C w < m\

C.

Hence the operator A maps the ball ||x||Cn < ||/C|| C to itself and by the Schauder theorem it has a fixed point x*. The function χ* = x*(i) is a Γ-periodic solution of equation (6.1). •

6.2 Systems of automatic control. Consider a system whose dynamics is described by the differential equations Li (£t)xi

= Mi (£t)f

....• d

ι ( t , x x

) ,

N

-

Ln{¿¡)xN=

(6.7) Χι, . . . , ΧΝ)·

Mf/(j¡)fN(t,

Here Ljip)

= Plj + dlPl'~l

Mjip)

= y(ipm'

+ · · · + dh

+ yiPm~l

( j = l , . . . , N ) ,

+ --- + bim.

( j = l , . . . , N )

where the degrees lj and m¡ of the polynomials L¡ and M¡ satisfy the conditions lj>mj

1

+

(/'^.•••.AO-

Assume also that the nonlinearities f ¡ are continuous in all their variables and Γ-periodic in time and the numbers 0, ±βψ, . . . are not roots of the polynomials Lj (j = 1 , . . . , N). Finding the Γ-periodic regimes of system (6.7) is equivalent to solving the system of integral equations χ

ι ( 0 = Io

*ι 0 -

s ; Ό f i ( s , Xi (s),....

XN(S))

ds,

f

XN(t)=

f0

hN(t

- s\ T)fN(s,

Xi (s),...,

xN(s))

(6.8)

ds

where hj(t; Τ) =

((/

- eAiT)-leA''Yj,

/ 0 0

1 0

0 1

0

0

0

ν

- d

1

-dn-2

c)

(0 < t < T, j = •

° \ 0

• •.

- d j

1

l, . . . , Ν),

(6.9)

96

II Existence theorems for oscillatory regimes (Á À

/1\ 0 c

Yj =

—

\

¿-χ

0

Vo/

\yi

)

and yi =

= ··· =

=

0

Aj-m, = Κ Aj-mj+ì + Àj-m/ì yj, + Aj-Â

= Vx

+ · · · + y'lj-mjOL = Κ

Denote then Wj 1 =

max t=o,±i,...

=

/ 2nki \ T

max *=o,±i,...

(6.10)

where Wj(p) =

Mjip) Lj(p)

(6.11)

Theorem 6.2. Suppose that the nonlinearities f j (j — 1,..., side of system (6.7) satisfy the estimates \f\(t,xi,...,xN)\

An analog of Theorem 1.1 for equation (1.1) in the Hilbert space is

(1.36)

116

III Convergence of numerical procedures

Theorem 1.6. Suppose that the completely continuous operator Α : Ω — Η is continuously Fréchet dijferentiable in Ω, equation (1.1) has a solution χ* € Ω, and the linear operator I is continuously invertible in H. Then there exist «0 and po such that for η > no equation (1.2) has a unique solution x* e H„ in the ball B(po, x«) and lim H** — χ*\\η = 0. «->•00

(1.37)

Proof. We show first that for sufficiently large η the operators I — P„A'(:c*) are invertible in Η and the norms of their inverse operators with any η are smaller than one constant. Assuming that this is not true, for some sequence hk e Η ( ||Α*||Η = 1, k = 1 , 2 , . . . ) and for some sequence of subscripts nk lim \\hk-PntA\x,)hk\\H k—>oo

= 0.

(1.38)

Since the operator A!(x*) is completely continuous, the sequence yk = A'(x*)hk is compact; without loss of generality one can assume it to be convergent to some 3>o as k —> oo. Then IIAt - yollH < IIAt - Pntyk\\H + II¿V* - Pnkyo\\H + \\Pnjo - yoll« ·

(1.39)

Since the projectors P„k converge strongly to the identity operator and \\Pn\\c{H) = 1, (1.38) and (1.39) imply the convergence lim IIhk - y 0 | | Ä = 0. k-* oo Using (1.38) we have yo = A'(x*)yo and Hyollff = 1 which is in conflict with the fact that the operator I t/4. invertible. The proved statement about invertibility of operators I — PnA'(x#) allows us to replace equations (1.2) by equivalent equations x = Bn(x)

(xeH)

IS

(n > no) (1.40)

where Bn(x) = * - ( / - ΡηΑ(χ*))-χ (X PnA{x)). For sufficiently small po and sufficiently large η the operators B n are contraction mappings of the ball Β(ρα,χ^). Hence equation (1.40) (and simultaneously equation (1.2)) is solvable. The convergence of solutions x* of equations (1.2) to the solution λ:* of equation (1.1) follows from the inclusions x* e B(p0, xt) in which the radius po can be taken arbitrarily small for sufficiently large n. •

117

§ 1 Projection methods

Here is an analog of Theorem 1.2 for Hilbert space. Let Ω be a bounded domain in Η and 9ÏÏ be the set of solutions of equation (1.1) lying in Ω C H. Theorem 1.7. Suppose that equation (1.1) has no solutions in the boundary 9Ω of the domain Ω and / ( / - Λ; 9Ω) ^ 0.

(1.41)

Then for sufficiently large η the sets ΐϋίη of solutions of equations (1.2) are nonempty and the following convergence occurs: lim sup inf \\x* - x,\\H = 0. n—»00 ^«ggjj x,eWl

(1.42)

Proof To prove Theorem 1.7 it is sufficient to note that under the assumptions of the Theorem equation (1.18) holds. Assume that it is not true. Then for some sequence of indices nk and for some sequence of points xk e Ω there exist ε > 0 such that \\A(xk) - PnkA(xk)\\H

> ε.

(1.43)

Since the sequence A{xk) is compact, without loss of generality assume that for some point xq lim \\A(xk) — xoll// = 0· k-KX> Since the projectors P„t converge strongly to the identity operator,

lim \\A(xk) -P„kA(xk)\\H

k^xx

< lim \\A{xk) - x0\\H

κ—oo + lim ||xo -Pnkxo\\H+ lim ||*o - ·Α(**)||Η = 0. oo k—>-oo

This is inconsistent with the inequalities (1.43).

•

1.6 Estimates of the convergence rate. Under the assumptions of Theorems 1.1, 1.4, and 1.6 the convergence rate of approximants x*n to the solution of equation (1.1) can be estimated. Only equations in Hilbert spaces will be considered in detail in this Subsection. The estimation procedure in the general case is the same except some technical details. Theorem 1.8. Under the assumptions of Theorem 1.6, for some constants c, C > 0 the following estimates of convergence rate of approximants x* to the solution x* of equation (1.1) hold: c ||x* -Pnx*\\H

< 11^ ~x*\\h

< C II** - P „ x * | | # .

(1.44)

118

III Convergence of numerical procedures

Proof. As proved (see the proof of Theorem 1.6) for sufficiently large η the operators I — PnA'(x*) are invertible and there exist constants Co and Cι such that for some no the following inequalities hold: 11/ - P„A\X,)\\C(H)

||(/ - ΡηΑ'(χ*)Γι

no),

(1-45)

(η > no).

(1.46)

Since (7 - PnA'(x*))(x*

- χ*) = (7 - Pn)x* - Pn(A(x*„) - A(x*) - A'(x*)(x*n -

x,)),

it is true that PnA'(x*)rl\\c(H)

< II (I -

χ ( ||(7 - Pn)xJH < CK ||(7 - Pn)xJH

+ \\A(x*n) - A(x.) + \\A(x*n) - A(xJ

- A'(x*)(x*n -

x.)\\H)

- A'(x*)(x*n - x*)||„) (1.47)

and I I * * - * ; i i h > HI -

PnA'(x*Mc¡H}

X ( ||(7 - Pn)x*\\H + \\A( En and qn : F -> Fn are connecting operators if lim \\p x\\ = ΐ—>oo n En lim l l ^ l l ^ =

ME,

(2-1)

M F -

(2.2)

The families of Connecting operators pn and qn(n = 1 , 2 , . . . ) will be denoted as V and Q, respectively, The sequence xn G En is called ^-convergent to the point χ e E if lim \\x - p x \ \ E = 0. η—>·οο n n

(2.3)

The Q-convergence is defined in a similar way. The sequence of operators A„ : En -> Fn is called Pô-convergent to the operator A : E —>• F if for any sequence x„ ^-convergent to the point χ the sequence A„(xn) is Q-convergent to the point A(x). The concept of "PQ-convergence was introduced in Sect. 8 of Chapter 1 where the approximation of nonlinear equations by equations in factor spaces was considered. The regular convergence and compace convergence are important concepts to be introduced in addition to the concept of Ρ Q-convergence. Their definitions follow. An operator sequence An '• En —Fn is regularly convergent to the operator A : E -> F if An is "PQ-convergent to A and the following regularity condition is met: if \\XN\ÌE„

< const·,

then the sequence {A„(x n )} is Q-compact.

2.2 The a posteriori estimate lemma. Let the operator A : E F be Fréchet differentiable. Assume that an approximate solution xo of the equation A(x) = y

(2.4)

is somehow found. In applications it is important to obtain information about solvability of equation (2.4) using the approximant XQ and to estimate how close the solution of this equaiton is to its approximant xo- Statements providing such information are called a posteriori error estimates, such as

120

ΠΙ Convergence of numerical procedures

Lemma 2.1. Let the operator A'(xo) : E —F have the bounded inverse operator {A{x0))~x : / —> £ and let for some constants τ, κ, S, e [0, 1) the following inequalities hold: Μ^ο)Ιΐ£(*.ί} α > 0. •tedQ

(2.28)

We claim that lim inf n—rOQ

||ΦΠ(*„)||£„ > 0.

(2.29)

If it is not so, then for some sequence xn e 9Ω„ lim ||*„-A(*„)II £ „ = 0 .

η—too

(2.30)

Since the sequence {A„(xn)} is P-compact, it can be assumed without loss of generality to be "P-convergent to some point xq. Therefore by virtue of (2.30) the sequence {xn} is "P-convergent to the same point. The consistency condition d) implies then that xo € dQ. But the sequence [A„(xn)} is "P-convergent to A(xo)· Therefore Jto = A{x{)), which is inconsistent with the condition that the field Φ is nondegenerate in 3Ω. This proves inequality (2.29). Reducing the constant a, if necessary, assume that lim inf η—>oo

||Φ„(χ„)||£π > α > 0.

(2.31)

126

III Convergence of numerical procedures

3. Let Κ = {yi, ...,

yr} be a finite f-net of the compact set Y, or sup min yeγ 1 ^ + ^

+8

(k = I,...

,r)

which is in conflict with (2.32). Hence for large η (ra > no) the inequality sup min II A„(x„) - y^\\En x„en„ 1 «o)

where if

^ - W y - y A s

l l y - ^ b < |>

μ* (y) =

a

»

if Ib-Jirllfi > 2 '

Í Ä

«

)

=

Il^-^lk

if I I ? » - A ,
n o ) 2-

By virtue of (2.32) and (2.34) the operators β and Qn (n > n 0 ) are defined and continuous in F and A n (Ω„) respectively. Their definition implies the inequalities sup

\ \ A { x ) - Q A { x ) \ \

E

< ^

sup

(2.37)

t

1

xeQ I I A „ ( x n)

QnAn

x„eQ„

(Χ/; ) l k

< 2'

(2-38)

and the ^-convergence of the operator sequence Qn to the operator Q, i.e. Um

„

l|£„=o.

\\Qn(yn)-PnQ(y)

(2.39)

6. The estimate (2.31) implies that the fields Φ„ are nondegenerate in 9Ω„ and therefore their rotations χ(Φ„; 9Ω„) are defined. It follows from (2.31) and (2.38) that for large η the fields I — Q„An are nondegenerate in 3Ω„ and homotope to the fields Φ„. Hence γ(Φη;

But the operator I rotation imply that

— Q„A„

r(I

-

3Ω„)

=

γ(Ι

-

acts from Ω„ to

QnAn,

9Ω„) -

γ(Ι

QnAn; EQn.

-

9Ω„).

(2.40)

Therefore the properties of the

QnAn,

9Ω°)

· (2.41)

{]

where = Ω„ Π E n. Using the invariance of the rotation with respect to the isomorphism Ψ„ : EQ — E ° N we have Y(I

-

QnAn,

3Ω°) =

Y(I

-

Ψ " 1 QnAn^n,

90^°)).

(2.42)

7. For sufficiently large η the operators Ψ,7' δηΑιΨ« will be shown to have no fixed points in the set \ (Ψ~'Ω° Γ)Ωο). Assuming that this is not true there exists a sequence e Ψ ^ Ω ^ c EQ, xk Ωο such that Xk = ^ Q n k A n k ^ n k X k .

(2.43)

Equations (2.35) and (2.36) imply the compactness of the sequence {xk} in E. Therefore it can be regarded as convergent to the point xq e Eq. Hence xq φ. Ω

III Convergence of numerical procedures

128

and the P-convergence of the sequence Ψ ^ χ * e xo £ Ω· Therefore xo € 9Ω. Passing to the limit for k have

c Cìnk to xo implies that oo in equation (2.43) we

xo = QA(XQ). Then by virtue of (2.38) inf ||x 9Ω

A ( X ) \ \

E


oo in equation (2.47) we have xo = QA (xo). Then as in the previous stage of the proof a contradiction with the inequality (2.44) occurs. 9. Using the reasoning such as that used in the seventh stage of the proof one can prove that the operator QA has no fixed points in the set Ωο \ (Ψ~'Ω° Π Ωο) for large η. Hence γ(Ι - QA; 3(ψ- 1 Ω^ η Ω 0 )) = y(I - QA; 9Ω 0 ).

(2.48)

Considering the operator QA : Ω EQ c E as an operator taking values in E and using the properties of the rotation implies that γ(Ι - QA, 9Ωο) = γ(Ι - QA; 3Ω).

(2.49)

§ 2 Factor methods

129

It follows from (2.31) and (2.37) that the fields I — QA and Φ are homotope in ΘΩ. Therefore / ( / - QΑ; 9Ω) = /(Φ; 9Ω).

(2.50)

10. We obtained a chain of equations (2.40), (2.41), (2.45), (2.46), (2.48), (2.49), (2.50). The beginning and the end of this chain lead to equation (2.26). • We proceed now to the proof of Theorem 2.2. Proof.

By Lemma 2.4 for sufficiently large η the sets 97t„ are nonempty. Now it's necessary to prove that equation (2.24) holds. If it is not so, then for some ε > 0 and some sequence x*k e Ti„k (k — 1 , 2 , . . . ) the inequalities in

t \\4-Pnkx*\\En X* 6 VJC

*

>ε

(2.51)

hold. Since xl=Ank{x*k),

(2.52)

by the second condition of the Theorem the sequence x*k is P-compact. So without loss of generality it can be assumed P-convergent to some point XQ e Ω. Passing to the limit as k —> oo in (2.52) we have χ*ϋ =

Α(χΙ),

or XQ € SDÌ. Then by (2.51) 114 -PnA\\E„k

>

ε,

which is inconsistent with the P-convergence of the sequence {x^} to x*y

•

2.5 Convergence of the factor method for equations with a linear main part. Assuming that equations (2.4) and (2.13) have the form Cx + B(x)=y C„xn

+ Bn(xn)

= yn

(χξΩ),

(2.53)

(x„ e Ω„)

(2.54)

where the linear bounded operators C : E -»· F and Cn : En these equations are equivalent to X = A(x)

Fn are invertible,

(x G Ω)

(2.55)

(x„ e

(2.56)

and Xn = An(x„)

Ω„)

130

III Convergence of numerical procedures

in which A(x) =

C-1(y-B(x)),

A„(xn) = C~l(jn -

(2.57) (2.58)

Bn(Xn))·

Theorem 2.2 implies Theorem 2.3. Suppose that the following conditions are met: (1) the bounded domains Ω C E and Ω„ c En satisfy the consistency conditions a), b), c), and d); (2) the operator sequence {Cn} converges regularly to the operator C; (3) the operator Β : Ω —> F is completely continuous and the finite-dimensional operators B„ : Ω„ —> Fn are continuous; (4) the operator sequence Bn converges compactly to the operator B; (5) equation (2.53) has no solutions in the boundary 9Ω of the domain Ω and y(I - Α; 9Ω) φ 0; (6) the sequence {yn} is Q-convergent to the point y; (7) the set Q of limit points of any sequence Bn(xn) (xn e Ω„; η = 1 , 2 , . . . ) is compact in F. Then for sufficiently large η the sets SJl and 9Jtn of solutions of equations (2.53) and (2.54) are nonempty and the following convergence occurs: lim sup

inf \\x*n-pnx*\\E

=0.

2.6 Additional remarks. 1. Theorems 2.2 and 2.3 assume that the sets of limit points of the sequences A„(xn) and Bn(xn) are compact. These assumptions can be left out if the spaces E and F are separable. In separable spaces the set of "P-limit points of any "P-compact sequence is compact. In nonseparable spaces this is generally not true.

2. In connection with condition (2) of Theorem 2.2 and condition (4) of Theorem 2.3 of compact convergence of the operators A„ to A and Bn to Β a natural question arises, when can a completely continuous operator be compactly approximated by a sequence of operators in factor spaces. If the space E where the operator acts is separable, then a compact approximation always exists. In nonseparable spaces there exist completely continuous operators without compact approximations. Let us describe the compact approximation of completely continuous operators. Let E be real Banach space, En be finite-dimensional approximations of E, pn E —> En be connecting operators possessing the property (2.1) and A : Β —> E be a completely continuous operator defined in the unit ball B c E.

§ 2 Factor methods

131

Without loss of generality one can assume that the operator A is extended to the whole space E preserving complete continuity and so that A(E) c coA(B). This is possible by the Dugundji theorem [60]. Let rn : En -> E be some sequence of continuous operators. Lemma 2.5. With the operator sequence rnpn converging strongly to the identity operator in the ball B: lim IIr p x - x\\E = 0 n—>oo n n

(x e B),

(2.59)

the operators An(xn) = pnA(rnxn)

(xn € Bn = pnB)

converge to the operator A compactly in B. Proof. Since the operators pn satisfy the condition (2.1) then according to the Banach-Steinhaus theorem (see e.g. [97]) for some constant a \\Pn II C(E,E„) < a•

(2-60)

The inequality \\pnA(x) - pnA(rnpnx)

II£n < α ||Λ(χ) - A(rnpnx) || £

(x e Β)

and the condition (2.59) imply the P-convergence of the operators A„ to the operator A. Let now xn e B„ (n = 1 , 2 , . . . ) . Since the sequence [A(rnxn)} is compact in E, the sequence {pnA(rnx„)} is P-compact. • Theorem 2.4. Let the space E be separable. Then the completely continuous operator A : Β —> E has a compact approximation. Proof. By virtue of Lemma 2.5 it would be sufficient to have a sequence of operators rn : En —> E satisfying the condition (2.59). Let x(l\ x{2\ ..., x(n>, . . . be countable and everywhere dense in B. Denote (k> E the linear hull of the points x(V\ x{2), ..., x(k). By virtue of (2.1) for any k there exists a number N(k) such that for η > N(k) the restrictions pik) of the operators p„ to are one-to-one mappings of the spaces E(k> to E(nk) = pnE® and the norms of the inverse operators (p'f'T 1 are bounded by one constant: Il (Pn}rlï\c(EÎ\E^)^C

(n>N(k))·

Without loss of generality an assumption can be made that N(l) < Ν(2) < The maximal k for which N(k) < η will be denoted as k(n). The function k(n) is defined for η > Ν (I). Denote =£,

ρη=ρΐ{η)).

132

III Convergence of numerical procedures

The operators pn

1

are defined in E(n) and IIÄ'1 II£(£("),£«»))) -

C

(n>N(D).

Introduce the sets Bn=pn(BDEw). Obviously for each χ e Β lim inf \\ρ η χ-Ζη\\ Ε „ = 0.

(2.61)

n œ

^ z„eB„

Since the sets B„ are closed and convex then (see [120]) there exist retractions Rn : En Bn such that II*» - RniXn)llfi. < 2 inf ||*„ -

Zn\\E.

(*« e

Ea).

(2.62)

z„\\E„)

-Pnx\\En

| | £ + 2 C inf \\Pnx - zn\\E„ · Z„€B„

These estimates and equation (2.61) imply the convergence lim II* - rnpnx\\E = 0. n-*oo

•

The compact approximation is not, however, always possible in nonseparable spaces. For example let E = m be the space of bounded sequences * = £2, . . . } with norm ||*|| m = sup 1&1

§ 2 Factor methods

133

and En = mn be the spaces of «-dimensional vectors xn = {£1, §2, · · ·. norm

with

\\xn\\m„ = max l&l. Let pn : m —> mn be the operator mapping the sequence χ e m to the vector composed of its η first components. Evidently the operators (p„) are connecting operators satisfying the condition (2.1). The set of such elements of the space m that finitely many of their components are equal to 1 and the rest are equal to zero will be denoted as M\, and the. set of such elements of the space m that infinitely many of their components are equal to 1 and the rest are equal to zero will be denoted as M2· M\ and M2 are closed disjoint sets in the unit ball Β c m. By virtue of the Uryson theorem (see e.g. [60]) there exists a real continuous function a{x) defined in m such that α (JE) = 0

(χ e Μι),

a(x) ξ 1 (je e M2)

and 0 < a(χ) < 1

(JE

€ m).

Define a completely continuous operator Λ : m A(x) = {α0*0, a{x),

m by the equation ...}.

The operator A will be shown to have no compact approximations in B. If this is not true, there exists a sequence of continuous operators An : Bn ->• mn (Bn = pnB) such that for each χ e Β lim \\p A{x) - AniPnx) \Ln = 0. n—>o0 n Take the sequence of elements

X(2) = {1, V ..., V «1 x(3) = {1, N ..., ν «1

1, 0, 0, . . . } ' 1, 0, . . . , 0, 1, 1, . . . } ' ^^ ν ' «2

such that \\PniA(xw)

- Ani(pnixm)\\miti

< i

WPm+mAix^) — yl„1+„2(p„1+„2x(2))||mni+(i2 < - ,

(2.63)

134

III Convergence of numerical procedures

The definition of the operator A gives , ti\ A(x(k))=

ι {0, 0, . . . } ' ' ' {1,1,...}

if k is even, ' if k is odd.

(2.65)

Consider the element ** = {1, . . . ,

1, 0 , . . . , 0 , 1, . . . ,

Π1

—' V

Π2

' N-

1,

«3

Since x* e M2, for any η \\PnA(x,)\\mn

= 1.

(2.66)

Formula (2.63) implies now that there exists a number Ν such that for η > Ν \\pnA(x*)


\

(n>N).

(2.68)

Denote k n(k)

= ^ ^

m.

i= 1

It follows from (2.68) that for large k the inequality > 43

\\An(k)iPn{k)X*)\\mnW

(2.69)

holds. But since Pn(k)X*

= Pn(k)X

(k)

,

it is true that IIA®(p„W*W)lkw >

(2.70)

On the other hand, it follows from (2.64) and (2.65) that for even k \\An(k)(Pn(k)X{k))\\m„m


ης, the restrictions p^ of the connecting operators pn to Εε define a one-to-one mapping of Εε to E(¿:> and W ^ r ' h ^ K E ^

2

(">«o)·

(2-82)

Proof. If this is assumed to be false, then there exists a sequence of elements Xk 6 Εε (k — 1, 2 , . . . ) of unit norm such that \\PnkXk\\Enk
·οο * k—>oo k—>oo < a lim

lUo - χ Λ ε + \

k—>-oo

What we have is a contradiction.

¿

=

ζ •

Lemma 2.7. When the elements x(>\ ..., x(k) form an ε-net of the set T, the elements p n x . . . , pnx(k> form an αε-net of the set Tn. Proof The proof follows from estimate (2.60).

•

§ 2 Factor methods

137

Freeze now the numerical sequence ^ (m = 1 , 2 , . . . ) . For every m take a ¿-net of the set Τ composed of elements x^, . . . , x(^m> e T. Let E(m> and E{™] be linear hulls of elements x(£m> and p„Xm\ · • • , PnXmm\ respectively. For any m find a number n{m) such that for η > n(m) the restrictions p^1 of the operators pn to the subspaces E(m> are one-to-one mappings of E(m) to with Il

II£(£ n{m) the operators S(nm) : E^m) φ)(χιύ

=

.... E(nm> by the equation

E ^ j M P ^

(285)

/ jj— 1 ßjn (Xn) where

ßjn(Xn) =

— - \\Xn-PnXm\\En m

if ll*n -PnXm\\E„
-· m

Lemma 2.7 implies that the denominators in (2.85) are not zero and the operators £(m) a r c thug correctly defined. It follows from the definition of the operators and from Lemma 2.7 that \\xn - S^\xn)||£)1

< m

{xn € Tn, η > n(m)).

This estimate implies the inequality IIPnA(r n (x n )) - Slm)p„A(rn(xn))||En

a < m

(xn e Ω„, η > Mm)).

(2.86)

The inclusion S^pnAtrniCln)) c E^>

(n > Mm))

(2.87)

follows from the definition of the operators S ^ . The greatest m such that the inequality n(m) < η holds will be denoted as m(n). The function m(n) is defined for large η, η > n( 1) being sufficient. Denote S„ = S n( 1)), («>«(!)).

III Convergence of numerical procedures

138

It follows f r o m L e m m a 2.6 that every operator pn has a bounded inverse operator defined in the subspace E„ = E(nm(n>> with II PnX

\\c(E'"^,E)

7l(l)).

Lemma 2.8. With x(n>, χ* e Τ and lim | | χ < β ) - * . | | £ = 0, η—>oo it is true that lim \\p-lSnPnx(n)

-X*\\E

=

(2.88)

V·

IIp-'SnPnX^ - Χ Λ ε =

\\Ρ-%ΡΗΧ(Η)

-P~NPNY{N)

< 2 IISnPnx^ < 2 II SNPNX^

(n)

Pny

\\En

-PNX(N)\\E„

< 2 | | S n P n x w -pnx{n)\\En +

+ Y

+ \\yw

W

-

-

X*\\E

xJE

+ 2 II PnX (n) ~ P t f ™ I k + II/"* " + 2 II P n x { n )

2\\pnx*-pnyM\\En

-PnX*\\E„

+

\\yM-ΧΛΕ·

F r o m this w e get the f o r m u l a IIp-XSnpnx^ - x j

£

< 2 f l ( — — + ||* (n) - x J

E

+ ||yW -

X

JE).

This estimate and the equation lim m(n) = oo

•

imply equation (2.88).

(2.89)

lim inf n—>-00

||37„ — xn\\En — 0.

(2.90)

§ 2 Factor methods

139

Proof. Note first that equation (2.89) implies the equation \\yn-Xn\\En

liminf

=0

(2.91)

where TN=PN{TR\E(M(N))).

(2.92)

This follows from the obvious inequality 2 m f \\yn-xn\\E y„eTn


c0

(* = 1 , 2 , . . . )

(2.94)

where Co > 0 is some constant. It follows from (2.93) that there exists a sequence Zk e Τηι, such that lim II £-»00

— χ* II ρ = 0. *

Denote

Since x(k> e Τ, the sequence x(k) is compact; without loss of generality it can be assumed convergent to some element χ* e T. We show that χ* φ. Ω. Indeed, if x* € Ω then x* e Tq and pnx* e T°N. Hence inf 11^ - xk\\E„.k < I\Pnkx* - Xk\\Enik < I\xk - Zk\\Enik +a | | x w - x* y^T°„k and therefore lim

inf

k^OOyk€T0

||;y*-Xill£

=0

which is inconsistent with (2.94). Consequently, x* £ Ω. Then by virtue of (2.77) for sufficiently large k the following inequality holds: inf

\\yk -Pnkx*\\Enk

>ci

(2.95)

140

III Convergence of numerical procedures

where ci > 0 is some constant. On the other hand, inf \\yk-pnkX*\ÌE„k < \\Xk -PnkX*\ÌEnk k (1Λ < IIXk - Zk\\E„k + IlPnkX( -PnkX*\ÌE„k

ykEÌÌn

< Wxk-ZÚE^ "k +a\\x{k)

-χ*\\ε·

Hence lim

inf

\\yk-PnkxAE„=Q·

This is inconsistent with (2.95).

•

Lemma 2.10. Suppose x-oo n Proof. Obviously I\rn(pnx{n)) - **||£ < \\rn(pnx(n)) -

\\rn(pnx*) ~

rn(pnx*)\\E+

X*\\E

•

The Lemma follows now from this estimate and conditions 1 and 2 of the Theorem. • We proceed now to the proof of Theorem 2.5 itself. This proof is rather cumbersome and will be made in several stages. In the first six stages equation (2.78) is proved and the last stage contains the proof of equation (2.81). Proof. 1. We prove the formula: lim inf η—too

\\xn-pnA{rn{xn))\\En>Q.

(2.96)

Assume that this is not true. Then there exists a sequence Xk e 9Ω„, such that lim \\xk -PnkA(rnk(xk))\\E k-yoo

k

= 0.

(2.97)

Equation (2.97) implies the equation lim inf \\yk-xk\\Eii=0. k^ooyteT„k *

(2.98)

Now it follows from (2.98) and Lemma 2.9 that lim inf \\yk-xk\\En,. k—too yk£T° "k

= 0.

(2.99)

§ 2 Factor methods

141

From (2.97) and (2.99) we have lim

\\yk -pntA(rni:(xk))\\E

inf

k-+ooyker>k

= 0.

(2.100)

Therefore the equation lim \\rnt(xk)

- rnk(pnkA(rnk(xk)))\\E

k—

= 0

(2.101)

follows from (2.97), (2.99), (2.100) and property 1 of the operators r„. Since the operator A is completely continuous, the sequence {A(rnk(xk))} compact. Therefore it can be assumed convergent to some point x* e Τ: lim \\x*-A(rnk(xk))\\E

= 0.

k—>oo

is

(2.102)

Let us show that the inclusion JE* e Ω

(2.103)

is true. Indeed, otherwise the condition (2.77) would imply the inequality lim

inj

k^-ooykeQ„t

Wyk-Pnkx*\\E„k

> 0.

(2.104)

On the other hand, \\PnkX* ~ xk\ÌE„k

-

II PnkX* - Ρ nk Air

< a

nk(xk))

- A(r„k(xk))\\E

+ p„kA(r„k + ||pnkA(rnt(xk))

(xk))

- Xk\\E„k - xk\\Enk .

This estimate and formulae (2.97) and (2.102) imply the equation lim \\pnkx* — Xk\ÌE„k — k—>oo *

0·

(2.105)

Formula (2.104) contradicts formula (2.105). Consequently, the inclusion (2.103) holds. But then x* e To and by virtue of Lemma 2.10 lim \\rnk(pnkA(rnk(xk))) - x * | | £ = 0. k—>-oo

(2.106)

Comparison of (2.101) and (2.106) gives lim I M * * ) - * * b = 0.

k-*· oo

(2.107)

This convergence and formula (2.102) imply the equation x, = A(x,).

(2.108)

Since the operator A has no fixed points in 3Ω, x* 6 Ω. Formula (2.76) gives then lim

inf

k^ooyk€dCi"k

\\yk -Pnkx*\\E„t

'

> 0.

(2.109)

III Convergence of numerical procedures

142

On the other hand, II** ~PnkX*\\E„k = II** - PnkA(r„k(xk)) + P„hA(rnk(xk)) -pnkX*\\E„k < II** -PnkA(rnt(x*))||£„t

+a \\A(rnk(xk)) - x*\\E .

This estimate and formulae (2.97) and (2.102) imply the equation lim IIp x„ ~ xk\\E = 0, k-KX nt * which is in conflict with (2.109). Hence formula (2.96) is proved. 2. It follows from formula (2.96) that for sufficiently large η inf \\xn-pnA(rn(xn))\\En> x„edn„

a

(2.110)

where a > 0 is some constant. We can assume that for those η the inequality a a m{n) 2 is true. Then for those η the inequality (2.86) implies the estimate sup \\pnA{rn(xn)) - SnpnA(rn(xn))WEn


η ( 1)).

The inclusion (2.87) and property 10 of the rotation of completely continuous vector fields (Theorem 4.10) imply the equation y(7 - SnpnArn- 9Ω„) = γ(Ι - SnpnArn·, 9Ω„).

(2.113)

(m(n))

Since the restriction pn of the operator pn to the subspace E has a bounded x (min)) inverse operator p~ : En E , it follows from property 11 of the rotation (Theorem 4.11) that for large η γ(Ι - SnPnArn, 9Ω„) = γ(Ι - p-lSnpnArnpn\p-xmn).

(2.114)

4. We denote ΩΦ)=Ω磫"« and show that for large η the equation Yd - Ρη 1 SnPnArnpn \ p~19Ω„) = γ(Ι - p~1 Snp„Arnp„ ; dCim(n) )

(2.115)

§ 2 Factor methods

143

holds. Both vector fields in the latter formula are in the same space E(m(n>). Since the inclusion Qm(n) C p n Ώ « holds, to prove equation (2.115) it is sufficient to show that the operators p~SnpnArnpn for large η have no fixed points in the sets Rn = Ρ η " η \ ^m(n) • Assume that this is not true. Then there exists such sequence x{k) e R„k that — Pnk SnkPnk-ArnkpnkX^ ^

(k — 1 , 2 , . . . ) .

The sequence y{k) =

A(rnk(pnkx^))

is compact. Without loss of generality it can be considered convergent to some point y* e Τ: lim | | y , - y ( t ) | | £ = 0. 00

Then Lemma 2.8 implies the equation lim k—>00

=

We claim that the inclusion y,en

(2.116)

holds. Indeed, if (2.116) does not hold, then the condition (2.77) gives lim

inf

\\xk ~Pnky*\\Enk > 0·

(2.117)

Since Λ

c^ ñPn- ' 1ñ¿n k k

then pnkx(k) e ñnk c n„k. And since I\Pnky* -PnkxW\\E„k

< a ||y» -

x(k)\\E

then lim \\p„ky*-pnkx(k}\\En 00

so formulae (2.117) and (2.118) are inconsistent.

*

=0,

(2.118)

144

III Convergence of numerical procedures

Thus inclusion (2.116) takes place. Then y* e TQ and Lemma 2.10 implies the equation lim \\rnt(pnkx(k)) κ->·οο

- y*\\E = 0.

Moreover, lim \\A(rnk(pntx(k)))-y*\\E

k-nX)

= 0.

So >>* = Aiy*). Since x(k) £ Ω, then y(k) φ Ω also. But y* e Ω which means that >>* € 9Ω which contradicts to the assumption that the field I — A is nondegenerate in 9Ω. Equation (2.115) is thus proved. 5. It follows from property 10 of the rotation (Theorem 4.10) that γ(Ι - p~1 SnpnArnpn ; 3Ω,η(η) ) = γ(Ι - p~1 SnpnArnpn ; 9Ω).

(2.119)

We prove that for large η the equation γ(Ι - p~1 SnpnArnp„ ; 9Ω) = γ(Ι - Α; 3Ω)

(2.120)

holds. This equation will be proved if we show that for large η the completely continuous vector fields I - Xp-lSnPnArnPn

- ( I - λ )A

(0 < λ < 1)

have no zeros in 9Ω. If this is not true, then there exist sequences x(k> e 3Ω and Xk € [0, 1] such that = Xkp-lSnkpnkA(rnk(pnk(x(k))))

+ (1 - Xk)A(x(k)).

.

(2.121)

The sequence y(k> = A(r„k (p„k (xa>)) is compact, and so can be assumed convergent to some element y* e T. Then Lemma 2.8 implies the equation lim \\ρ-%ρη^}

k-i-oo

-y*\\E

= 0·

Without loss of generality, the sequences A(x{ky) and Xk can be assumed convergent to some element x* e Τ and some number λ» e [0, 1], respectively. Then (2.121) suggests that the sequence x(k> converges to the element ζ* = λ*^* + (1 — λ*)χ#. But the set Τ is convex, so z* e Τ, and since x(k> e 9Ω, it is true that z* e 9Ω. But then ζ* € Γ 0 and by Lemma 2.10 lim \\rnk(pnkx{k))-z*\\E

k—too

= V·

Therefore lim

k-*oo

— .A(z*)||e = 0

§ 2 Factor methods

145

and by Lemma 2.8 lim IIp-ßn k Pn k A(.r nk (Pn k {x (k) ))) - A{Z*)\\E = 0. k-too Passing to the limit as k oo in (2.118) we have the equation z* = ·4(ζ*). This is a contradiction and equation (2.120) is therefore proved. 6. We obtained the chain of equations: (2.112), (2.113), (2.114), (2.115), (2.119), (2.120). Having compared the starting and the final terms of this chain we have equation (2.78). 7. We prove now formula (2.81) with γ(Ι - Α; 9Ω) φ 0. Let 9ÏÏ be the set of zeros of the field I — A lying in Ω. Take some small ε > 0 and denote the ε-neighbourhood of the set QJl as Us: υ ε = { χ € Ω : inf ||jt — x*\\E < ε}. Since for χ € Ω \ Ue χ - A(x) Φ 0, for large η and for xn e ρ„(Ω. \ Ue) the inequality Xn - PnA(rn(xn)) φ 0 holds. Its proof is analogous to that of formula (2.96). Then for large η M n c Ω„ \ pn (Ω \ Ue). This leads to the inclusion 9J΄ C pnUE because Ωη\ρη(Ω\υε)

CPnUs.

For any point xn e pnUe inf IIx„ ~P„X*\\E„


En possessing properties 1 and 2.

146

III Convergence of numerical procedures

Proof. As in the proof of Theorem 2.5 freeze the numerical sequence ^ (m — 1', 2 ,' . . . )' and take for each m a finite --net of the set Τ made of the elements m ..., e T. Let E(m) and E(nm) be linear hulls of the elements x^, . . . , and Pn*m\ • • • ,p„x%m\ respectively. By Lemma 2.6 for any m there is a number n(m) such that for η > n(m) the operators p{™] : E{m) are one-to-one mappings with ||(p^ m) ) _1 || < 2 (here p(™> is a restriction of the operator pn to E(m>). Without loss of generality one can assume that n( 1) < n(2)
«(1). Consider the sets Tn=pn{Tr\E(m(n)))

(n>n(

1)).

These sets are convex and closed. Therefore (see [120]) there exist retractions Rn : En —> Τn such that for xn e En \\xn - *„(*„)|| £ „ < 2 inf \\yn - x„\\En.

(2.122)

yneT„

This estimate implies the inequality ||Λ„(*„) - R„(y„)\\En

< \\xn - yn\\E„

+ 2 inf IIZn - Xn\\En + 2 inf ||z„ - y„|| £ „. z,eT„

z„eT„

(2.123)

Define now the operators rn by the equation rn(Xn) = PnÌRn(Xn)

Un G En, Π > « ( 1 ) )

(2.124)

where λγ1 -

( p ^ r

1

·

The operators rn are certainly continuous and will be shown to possess properties 1 and 2. Let for the sequences xn, yn e En the conditions (2.72)-(2.74) be met. Then I M * « ) - rn(yn)\\E

< 2 IIR n (x n ) -

Rn(yn)\\E

and by virtue of (2.123) I\r n {Xn) - rn(yn)\\E

< 2 \\xn - yn\\En + 4 inf | | Ζ „ - λ „ | | £ ι ι + 4 inf

zner„ Since

z„er„

\\Zn -

yn\\En.

(2.125)

§ 2 Factor methods

inf IIZn z„€Tn inf

Xn\\E„

IIz„ - yn\\E

147

< inf

Un ~

*J£„ H

< inf

||z„ - yn\\E

z„eTn

a 77, m{n)

H— m(n)

then equation (2.75) follows from (2.125). Prove now property 2. Let xq e To. Take a sequence xn e Τ Π Eoo

- Pnf {HyJHx^

= 0.

(3.56)

It follows from the estimate (3.55) and equation (3.56) that lim \\A'n(y*)xn - A'(y*)xo\\L2 n—>oo

= 0.

To complete the proof it is necessary to show that the regularity condition is met: if the sequence {yn} is bounded in L2 and the sequence A'(y*)yn is compact in L2» then {yn} is also compact in L2. Indeed, An(y*)y« = y* - Pnf 00

(3.59)

Let us show that lim \\An(yn)-A(y)\\L2 n-i-00

= Ü·

(3.60)

Indeed, \\An(yn) - Ami*

= \\PnB(yn) -

B(y)\\Ll

< \\PnB(yn) - PnB(y)\ÏL2 + IIP n B(y) - B(y)||¿2 (3.61) By the estimate (6.17) from Sect. 6 of Chapter 1 \\PnB(yn) - PnB(y)\\L2

< l l ^ l l c ^ \\B(yn) - B(y)\\c oo

= 0.

(3.63)

But since B(y) e C, formula (6.18) from Sect. 6 of Chapter 1 leads to lim \\PnB(y) — B(y)\\¿2 = 0.

(3.64)

164

III Convergence of numerical procedures

Equations (3.63) and (3.64) imply equation (3.60). Consequently, the operators Λ η are "P-convergent to the operator A. Now, we show that the compactness condition is met. Let yn g Ω. Consider the sequence An(yn) = Pnf ('Hyn). The operator Η \ Li ^ C is completely continuous, and so the sequence 7iyn is compact in C. But then because of the continuity of the operator f the sequence f (7iyn) is also compact in the space C. Without loss of generality this sequence can be assumed convergent in C to some point y e C. Then by virtue of (6.17) and (6.18) (Sect. 6 of Chapter 1) lim \\y-An(yn)\\L2= lim \\y - Pnf (Hyn)\\L2 n-t- oo η-κχ) < lim \\y-Pnyh2+ Hm \\Pny - Pnf (Hyn)\\Ll n—too

n—>oo

< lim \\y-Pny\\L2+ «-»•oo

lim | | P „ | | C ^ ||y - f Qiyn)\\c n^-oo

< lim \\y - Pny\\L2+Vf n—>oo

lim ||y - f (Hyn)\\c η—>-oo

= 0.

This means that the operator sequence A„ is a compact approximation of the operator A. By virtue of Theorem 2.2 the sets 9Î and 9ΐ„ of solutions of the equations y = B(y)

(y e L2)

y = PnB(y)

(y e Li)

and

are nonempty and the convergence lim sup inf 1 1 ^ - ^ 1 1 ^ = 0

(3.65)

occurs. Since the sets Dî, 9΄ and Wl, Wln are related by the equations m = nm,

mn = nmn

and the operator Η : Li -»· cl~m~l equation (3.65).

(η = 1 , 2 , . . . )

is continuous, formula (3.58) follows from •

3.7 The harmonic balance and the collocation methods for periodic oscillations of multi-circuit systems of automatic control. There exist analogs of Theorems 3.4-3.7 for multi-circuit systems of automatic control described by systems

§ 3 Harmonic balance method and collocation method

165

of equations of the form /d χ

Íd\ (3.66)

l

N{^)xN

- MN(^)fN(t,

χι

xN).

The harmonic balance method can here be employed in the following way. The components of the Γ-periodic regime x*(t) = WW,

....

*3v(0}

are sought for in the form xj(t) = I

+ ¿ ( 4 cos 2-ψ- + b) sin *=i

=

N).

(3.67)

Then the equations Lì{j^)x1

x1,...,xnN),

= Mì{j^)Pnfx{t,

(3.68) M ^ K

= MN(jt)PJN(t,

xn{,...,

xnN).

are written where Pn is an operator mapping the Γ-periodic function x = x(t) to the segment of its Fourier series containing modes up to the rc-th order. Assuming that the coefficients in both sides of equation (3.68) are equal, we have a system of (2η + 1 )N equations Rk(a°y, ..., a°N,...,

al ..., anN, b\,...,

blN,...,

bnu ..., bnN)

(3.69)

(k= 1, ..., (2n + 1)A0 with (2n+ l)N scalar variables. The solutions of this system substituted into (3.67) provide an approximant to the desired solution x*(t) of system (3.66). Here are formulations of the analogs of Theorems 3.4 and 3.5 for multi-circuit systems. Theorem 3.8. Suppose that the following conditions are met: (1) the degrees lj and mj of the polynomials Ljip) = Plj + 4Pij-x + · · · + 4 m

Mjip) = ti{p > + ti]Pmrl + --- + Vm.

(j=l,...,N), (j =

l,...,N)

166

III Convergence of numerical procedures

satisfy the inequality q = min (/,·1 — m¡ 1 — 1) > 0; 1 0 \Rnk(xk)\ > ε. can be assumed to converge in CN to

Without loss of generality the sequence some function xq = xo(t): lim

k-too

\\xk — xollc„

=

(4.11)

0.

Then limi /

k^OO \J0

Xk{s)ds—

/ J0

xo(5) ds I = 0. )

On the other hand, by (4.3) Τ ρΤ

li

Xjc(s)ds—

/

Jo

xo (s)

ds

ηk Σ ajnk(Xk(Sjnk) - Xo(Sj„k)) + R„k(Xk) - Rnk(xθ) ;'= 1

(4.12)

174

III Convergence of numerical procedures

ηk \Rn I--'if; t(Xk)\ -

^

ι

-

\Rnk

(*θ)Ι - X ]

(*jnk

I Xk(Sjnk)

~

*o(Wl

Ik ε -

>

|Ä„t(*o)| - \\xk - *ollc* Σ 0 1 * « j=ι

> ε -

|Λ„4(*0)Ι - e \\xk - x 0 llc N ·

This estimate, equations (4.11), and the formula

lim \Rnk(x0)\ = 0 k—>-oo

imply the equation lim ( /rTXk(s) ds — rT xo(s) ds J > ε k-* oo\J 0 JO ) k-*rtX>

which does not agree with (4.12).

•

4.2 Local convergence of the method of mechanical quadratures. Consider a multi-circuit system of automatic control Li(á)*i ^

{ j t )

N

X N

=M1(4)/i(f, *!,·..,**), ~

Μ Ν {-¿ì) f l i t , X\, . . • ,

(4.13)

Xn)·

The assumptions of the previous section on the polynomials L¡ and M, are made. The nonlinearities /,· (i = I,... ,N) will be assumed continuously differentiable in all their variables and T-periodic in t. As mentioned above finding the Γ-periodic solutions of system (4.13) is equivalent to finding solutions of the system of integral equations

xi(t)=

/ Jo

hi(t-s\T)fi(s,xi(s),...,xN(s))ds (4.14)

XN(t) = /

Jo

hN(t - s\ T)fN(s,

x\is),...,

x N (s)) ds

where h¡(t; Τ) is a unit impulse response of the linear unit with a transfer function Wiip) — Mi(p)/Li(p). We use the method of mechanical quadratures for approximate solutions of a system of integral equations (4.14). Denote

x(t) =

:

,

H(r; T) =

( h\ (ί; Τ) 0 0 h2(f,T)

...

V

...

0

0

...

0

0

\

hN(f,T)J

§ 4 Convergence of the method of mechanical quadratures

/ fi(t,

/(*,*)=[

175

x)

;

\ f N ( t , X) , and rewrite system (4.14) in vector form: x(t)=

[ H(t - s; T)f(s, x(s)) ds. Jo

(4.15)

Using quadrature formulae (4.1) replaces the vector integral equation (4.15) by an approximating system η Xin = Σ aj"H(Si" - SJn> T)f(SJn> xjn) 0' = 1, · · · , «)· (4.16) ;=1 System (4.16) is a system of η vector equations with η vector variables xin. It can be considered also as a system of Nn scalar equations with Nn scalar variables. Let E be the space CV of functions continuous in the segment [0, Γ] and En be the space of the network functions xn = {x\n, ..., xnnj with a norm \\xn\\En = l· En is the operator defined by the right-hand side of equation (4.16), y — 0, and yn = 0. The assumptions of Theorem 2.1 will be shown to be true for these spaces and operators.

Since A!(x*) = I — B'{x*) and

Jo the solution of the equation

A'(x*)h = 0 is that of

h = &{xjh, or a Γ-periodic solution of system (4.18). Consequently, ker A (x*) — 0 and assumption 1 of Theorem 2.1 is therefore met. We prove that assumption 2 is also met. Let ωό (t) (t > 0) be a nonnegative increasing function such that

\f'x(t,x)-fx{t,y)\ =

[

/

y (s) d s +

À

y ( s ) d s +

y ( s )

V

a

d s

j n ¡

x*{s

j n ¡

) +

R&n¡(x*).

Ja[+)

(4.44)

On the other hand, by formula (4.1) rT Γ

/

. "i y (s) d s

=

^

Y I

a

j n ¡

y(s

j n i

) +

/?„,.(}>)

7=1 =

Σ

w

f

e

)

+

+

a

Σ

m

x

* i

s

m )

Σ

+^00·

(4.45)

Subtracting equation (4.44) from equation (4.45) we have 0

=

/

d s +

ft

/

y ( s ) d s -

A

Y ]

ot

j n i

y(s

j n i

)

r. -

+ oo h " To prove this Theorem we need some auxiliary statements.

5.5 Auxiliary statements Lemma 5.2. If the formula of numeric differentiation (5.1) is convergent, the corresponding difference differentiation operator D1^' satisfies the estimate \\Dfxh\\Sh

< c \\&hxh\\£h

(c = const., xh e Eh)

(5.17)

§ 5 Convergence of the method of finite differences

193

and the Q-convergence of the sequence {d^x/,} to the element y implies the Q-convergence of the sequence {D^Xh} to y. Proof By Lemma 5.1 the characteristic function χ(ζ) of formula (5.1) can be written in the form (5.3) with the condition (5.4) fulfilled. Moreover, Df

= h-k ¿

bjUi = h-\Uh

- Ihf

j—~r

g

ßjUi

j=-r

Since dh =

hr\uh-ih),

we have nf

= dlvh = vhdkh

(5.18)

where s-k

J=~r

and s-k

ΣΑ

=1

·

i=-r

The operator Uh is isometric, and so s—k

\Wk\\mh) E 4 to meet some additional conditions. Assume that 9Jto is an open smooth ¿-dimensional manifold given by the parameterization: if xo e SOÎo, then for some neighbourhood V of the point xç> in 9Jl0 there is a representation χ = h{u)

(6.17)

where h : intß V is a diffeomorphism of the unit ball intß C R ' to V with h(0) = xo· Then assume that the mapping φ : MQ Rk is a restriction to 97t0 of the smooth mapping Φ : V(3Ho) ^ defined in some neighbourhood V(SDîo) of the manifold 9Λ0 in E. The Fréchet differentiable mapping / : £ - > · R^ is called transversal to the manifold WIQ in the point xo e 9Jto if the kernel ker/'(xo) of its Fréchet derivative Z'(xo) has only one common point with the tangent space Τ CüJt0)X(¡ to the 9Ji0 in the point xq.

203

§ 6 Autonomous systems

Let /i'(0) : R* —»· E be the Fréchet derivative of the mapping h in the point u — 0. Let e\, . . . , e^ be some basis of R.k and denote gi = U{0)ei

(i=l,...,*).

Lemma 6.2. The mapping I : E —> R¿ is transversal to the manifold 9JI in the point X() if and only if h(gi)

···

high)

h(g\)

· · ·

h(gk)

det[/,(£/)] =

where /,- (i = 1,..., mapping I.

#0

(6.18)

K) are components of the Fréchet derivative 1'(XQ) of the

Proof The parameterization of the tangent space Τ (9Άο)Χ(] to the 5Q?o in the point Xo is x = h'(0)u

( u e Rk).

Hence the set of zeros of the linear mapping l'(xο) : E —> T(ÎÏÏIQ)Xq fits the set of solutions of the system of equations li(h'(0)u) = 0 Let u =

-I

+

(6.19) lying in the space

(i=l,...,jfe).

(6.20)

Then system (6.20) takes the form h(gi)$i + --- + h(gk)ïk = 0, (6.21)

, higi)^

+ · · · + higk)Hk = 0.

System (6.21) has a unique solution if and only if det[/,(^·)] Φ 0.

•

Let the Fréchet differentiable mapping Β : E χ -»· R* be defined in some neighbourhood of the pair (XQ, λο) where λο = Φ Oto)· The additional constraint Β(χ·, λ) = 0

(6.22)

will be called admissible in the point (*ο,λο) if B(xo\ λ 0 ) = 0 and the mapping Β(χ;Φ(χ)) is transversal to 9Jt in the point XQ. Theorem 6.1. If the additional constraint (6.22) is admissible in the point xo, the pair (XQ, λο) is a solution of system (6.14), (6.15), isolated in E χ R*. Proof By Lemma 6.1 it would be sufficient to prove that the solution xo of equation (6.16) is isolated. If not, then there exists a sequence xn e Tl0 convergent to xo and such that xn φ xo (η = 1, 2 , . . . ) and the equations B{xn-Φ(χ„))

= 0

(η = 1 , 2 , . . . )

(6.23)

204

III Convergence of numerical procedures

hold. Equation (6.17) shows that xn — h(un) where un -»· 0 as η -»• oo and un φ 0. Then equations (6.23) can be represented in the form |η„Γ^(Α(Μ»);Φ(Α(«„))) = Ο or in an equivalent form: \un\~x B'x(x0-,X0)h'(0)un

+ Iun\~l Β'λ(χ0;λ0)Φ'(χ0)Η'(0)ιιη

+ \un\~l ω(μη) = 0 (6.24)

where lim —;—— = 0. n^oo \un\

(6.25)

The sequence |i/„|—1 un without loss of generality can be assumed convergent to some element uo with |z*ol = 1. Then passing to the limit in (6.24) we have the equation B'x{x0; ko)h'(0)uQ + Β'λ(χ0; λ0)Φ'(χ0)Ιι'(0)ιι0

= 0.

(6.26)

Since Β'(χ;Φ(χ))ν

= Β'χ(χ·,Φ(χ))ν +

Β[(χ·,Φ(χ))Φ'(χ)ν,

it follows from equation (6.26) that the point vo = h'(0)uo belongs to the kernel of Fréchet derivative of the operator Β(χ\Φ(χ)). On the other hand, υ0 e T(Xftο)*0 which is in conflict with the fact that the mapping Β(χ\Φ(χ)) is transversal to the manifold 9Jto in the point XQ. • If the additional constraint has the form λ - λ(*) - 0 where λ(χ) is some operator from E to the form

then the system (6.14), (6.15) takes

A(x; λ) — 0, λ - λ(χ) = 0. This system is equivalent to the operator equation =

(6.27)

This means that the method of the functional parameter is a version of the general method of additional constraints. If xq € λ(*ο) = Φ(*ο) and the mapping λ(χ) — Φ (χ) is transversal to the manifold 9JIq, then by Theorem 6.1 the solution xo of equation (6.27) is isolated and the above numerical procedures can be used for its approximation.

§ 6 Autonomous systems

205

6.4 The degeneracy dimension. Taking up again equation (6.2) assume that the operator A : Ε χ -»· F is Fréchet differentiable. Freeze some point XQ e SDTo with its neighbourhood V C M parameterized by the mapping (6.17) and denote λο = φ(χο)· Consider the equation A'x(x 0 ; λ 0 )h + Α'λ(χ0·, λ)Φ'(*ο)Α = 0.

(6.28)

Lemma 6.3. Each element of the tangent space Τ (ΐΰ\ο)Χο = {h e E : h = h'(0)u} is a solution of equation (6.28). Proof To prove the Lemma it is sufficient to differentiate the identity ^;Φ(ι))ξΟ

(χ e Mo)

with respect to χ in the point χο·

•

Since dim T(xo) = k, the dimension of the set of solutions of equation (6.28) is at least k. Now assume that the following condition is fulfilled: if for some h e E and μ e Rk the equation Άχ(χο;λο)Η + Α'κ(χο·,λο)μ = 0 holds, then h e Τ (ÛJIQ)Xo and μ = Φ'(χο)^· In this case the degeneracy of equation (6.2) in the solution XQ will be called weak. In the case of the weak degeneracy the dimension of the set of solutions of equation (6.28) is equal to k.

6.5 The convergence theorem. An analog of Theorem 2.1 on the convergence of the factor method for the approximation of solutions of equation with parameters will be proved in this Subsection. This Theorem will be used later in analyzing the convergence of various approximate procedures of construction of the periodic regimes in autonomous systems. Let χ* e 9Jt be a solution of equation (6.2) for the parameter value λ = Consider system (6.14), (6.15) in which equation (6.15) is an additional constraint admissible in the point (χ*,λ*). Then by Theorem 6.1 the point (χ*,λ*) is a solution of system (6.14), (6.15) isolated in £ χ M*. Consider the sequence of finite-dimensional subspaces En and Fn (dimf,, — dim Fn) and denote the connecting mappings as p n : E -»· E n and qn : F Fn. To obtain the approximants (χ*,λ*) to the solution (χ*, λ») of system (6.14), (6.15) consider the sequence of approximating systems Α„(χ·,λ) = 0,

(6.29)

Βη(χ·,λ) = 0

(6.30)

III Convergence of numerical procedures

206

in which the operators A„ : E n χ R k differentiable. The following statement holds:

Fn and Bn : En χ Rk

Fn are Fréchet

Theorem 6.2. Assume the following conditions hold: (1) the degeneracy of equation (6.2) in the solution x* is weak; (2) the following equations hold: Hm

\\Α' η (χ η ·,Κ) - Λ(Ρη χ *; λ *)Ιΐ£(£„χΚ*,ί·„) =

(6-31)

\\Xn-PnX»\\E„^>·® K^-K Λ™ \\Κ( Χ η'Λη) -B'„(PnX*;^)\\ C ( E r i X R k : F n ) =0, ΙΙ*ι·-ΡιλΙΙΕ„-»·0 λπ—»λ*

(6.32)

lim | | Λ ( ρ » ^ ; λ « ) | | ^ = 0, «->•00 lim \\Bn(p„xM\\P¡¡ = 0; π-> οο

(6.33)

(6.34)

(3) the operator sequence (A„(pnx*', λ*))'Χη : En —> F„ converges regularly to the operator Α'χ(χ*',λ*) : E F; (4) the operator sequence (An(pnx*', : Κ* -» Fn is IQ-convergent to the operator Α!χ{χ*·,λ*) : Rk —> F; (5) the operator sequence {Bn(pnx*\λ*))* : En —> R* is VI-convergent to the k operator Β'χ(χ„\λ«) : E —lSL ; (6) the operator sequence (Bn(pnx*, λ*))'λ : M.k —> is II-convergent to the operator Β'χ(χ*·,λ*) : R* R*. Then there exist ¿o > 0 and no such that for η > the approximating systems (6.29), (6.30) have unique solutions (χ*, λ*) in the balls B(Sq, (pnx*, λ*)) C En χ R* and the following convergence occurs: lim IIpnx* - x*\\E = 0, n-* oo lim | λ * - λ * | κ * = 0 . ti—>oo Proof The proof follows from Theorem 2.1. Denote Έ = Ε χ Rk, F = En = En χ Rk,

(6.35) (6.36)

Fn = Fn χ R*. pn =

(Ρη,Ι), q„ = ( q n , I ) , X = {x, λ), A(x) = (Α(χ·,λ),Β(χ·,λ)),^χη^= (χ„,λ), An(x n ) = (Λ„(χ„;λ), B n (x n \ λ)) and show that the spaces E, F, En, Fn, the connecting mappings /?„ and qn and the operators A and An satisfy the conditions of Theorem 2.1. This will prove Theorem 6.2. We show first that condition 1 holds, i.e. that the equation

A'(x*)g = 0

(6.37)

207

§ 6 Autonomous systems

where χ* = (χ*, λ*) has only a zero solution in E. Let g = (h, μ ) where /ζ e Ε, μ e M*. Then equation (6.37) is equivalent to the system of equations A'x(x*;

λ*)1τ + Α'λ(χ*',λ*)μ

Β'χ(χ*;λ*)Η

+ Β'χ(χ*;λ*)μ

= 0,

(6.38)

= 0.

(6.39)

Since the degeneracy of equation (6.2) in the solution χ* is weak, formula (6.38) implies that h e Τ(ΰΆ0)χ^ and μ = Φ'(χ*)/ϊ. Therefore equation (6.39) can be written as Β'χ(χ*; λ«)Α + Β'λ(χ*; λ,)Φ'(χ,)Α = 0.

(6.40)

The additional constraint (6.15) is admissible in the point (χ*, λ*), hence (6.40) implies the equation h = 0. But then μ — Φ'(χ*)/ι = 0. Therefore equation (6.37) has only a zero solution. Now we prove that condition 2 holds. By formulae (6.31) and (6.32) „ Ä

\\Λη(Χη)-Α'η(ρηΧ*)\\£(Έηΐπ)

II·*«-Α.*. II—>0 En

-

Ä

\\Λ'η(χη,Κ)

+

A™

\\K(Xn\K)-B'n(PnX*\K)\\aE„x»,Fn)

~ Än(PnX*\>-*)\\c(E„Y.W•00 Then conditions 3 - 6 of the Theorem give lim

oo

\\A'n(pnX*)Xn-qnA'(x*)x\\~ r

"


00 lim λ* = Γ*.

isolated

(6.61) (6.62)

η—> σο

Proof. Let Εη be the space of 27r-periodic trigonometric polynomials whose degree does not exceed n. Denote E = F = Η1, En — Fn, pn = qn = Pn and take the norm in En induced by the norm of Hl. We denote also Λ(χ;λ)

=x(t)-

/ Jo

B(x\ λ)

= x(0) - a,

Λη(χη·, λ) = xn(t) -

Jo

hit — s; k)f{x{s))

ds,

(6.63)

(6.64) h{t-

s; X)Pnfixnis))

ds,

BniXn-,λ) = j c „ ( 0 ) - a

(6.65)

(6.66)

and show that these spaces, connecting mappings, operators and additional constraints satisfy the conditions of Theorem 6.2. The operator A(x\ λ) can be easily proved to act from Η 1 χ E to H l and be continuously Fréchet differentiable. We have A'xix0; X0)h = hit) -

r-Ίπ h(t - .y; X0)f (xo(s))h(s) ds Jo

and Α(χο;

λο)μ = -μ

ρ ¿π / h[it - s\ k0)f(xo(s)) Jo

ds.

Then formulae (6.59) and (6.60) imply that the additional constraint (6.53) is admissible. The degeneracy of equation (6.52) in the solution x*(¿) is weak. Indeed, if for some h e Hl and μ e M the equation hit) -

¡>2π hitJ0

ρ2π s\ Τ *)fixais))

Ks) ds — μ

J0

h'x(t - s; T*)f(x*(s))

ds = 0

holds, then condition 3 implies that μ = 0. But then condition 2 leads to h = vx'^t). This means that the degeneracy is weak. Among all the conditions of Theorem 6.2 only condition 3 is to be proved; the rest is obvious.

212

III Convergence of numerical procedures

We prove regular convergence of the operator sequence ( A n ( P n x * \ T * ) ) ' X n :

En

E n to the operator A ' J x * : , T * ) : H Let h n e E n and

l

Hl.

lim \ \ h n - P n h \ \ E n = 0 . We prove that

lim II ( Α ( Λ λ ; T j y j i n - P n A ' x ( x * , T * ) h \ \ E n = 0.

(6.67)

We have

II ( A n ( P n x * \ T * ) ) ' x h n - P n A ' x ( x T n ) h \ \ E ñ < \\hn-Pnh\\En ρι·2π ¿π + / h ( t - 5; T , ) P n ( f ' ( P n x , ( s ) ) h n ( s ) - f ' ( x , ( s ) ) h ( s ) ) d s Jo E„ < \\hn-Pnh\\En fρ2π in / h ( t - s; T , ) p n ( f ' ( p n x j s y ) ( h n ( s ) - h ( s ) ) ) d s +" Jo

+

[ Jo

h ( t - S-, T m ) P a ( ( f ( P n x * ( s ) ) - f ( x * ( s ) ) ) h ( s ) ) d s E„

All the three terms vanish as η —> 0. Hence equation (6.67) holds. Now we prove that the convergence is regular. Consider the sequence h n e E n such that

sup \\hn\\En η

< 00

(6.68)

and denote

8n = - ( A n ( P n x * t T * y ) x h n .

(6.69)

We show that "P-compactness of the sequence g „ implies "P-compactness of the sequence h n . We have λ 2π

gn = gn(t) = h n ( t ) - /

h ( t ~ S' T J P n f ' i P n X t i s ^ h n i s ) d s .

Jo Consider the sequence

ρ 2π u n Ξ U„(t) =

Jo

h i t - s; T * ) f ' ( P n x J s ) ) h n ( s ) d s .

213

§ 6 Autonomous systems

This sequence is compact in H l , this follows from the fact that the sequence hn is /^-bounded and the operator p2n Vh=

Jo

h(t - s\ T*)f'(x*(s))h(s)

ds

is completely continuous. Consider then the sequence ι·2π vn = v„(t) =

h{t-

s\ Tt)Pllf'(Pnx.Js))hn(s)

ds.

J0 It is easily seen that lim \\u„-v„IIH: η—*·οο

= 0.

(6.70)

The compactness of the sequence un and equation (6.70) imply that the sequence v„ is compact in H]. It follows from this fact that the sequence vn is P-compact. Therefore the 'P-compactness of the sequences vn and hn implies the P-compactness of the sequence gn = hn — vn. Hence all the conditions of Theorem 6.2 are met. •

6.7 The functional characteristic. Let y j t ) a be T*-periodic solution of equation (6.44). Then the function .

(6.71)

is a 27T-periodic solution of equation (6.49) for the parameter value λ = Τ* or, which is the same, a solution of the integral equation x(t)

=

ι·2π h(t-

s\ λ ) / ( ; φ ) ) ds

(6.72)

Jo

with λ = Γ*. Let x*(0) = α

(6.73)

< ( 0 ) φ 0.

(6.74)

x(0) = or

(6.75)

and

Add the constraint

to equation (6.72). Lemma 6.4. Suppose that some neighbourhood B(p, x*) C Hl of the point x* 6 Hl contains no 2n-periodic solutions of equation (6.49) with the parameter value

III Convergence of numerical procedures

214

λ = Γ* different from the functions x*(t + h). Then the additional constraint (6.75) is admissible in the point Proof In this case the operator B(x; λ) defining the additional constraint has the form B{x\ λ) = jc(O)

-

a.

Therefore &(x,\(x*))h = h(P) and so ker#(*„.;o, Φ 3 ) .

(7.86)

For this purpose consider a completely continuous family of vector fields (z{t) - eAWtU(X, z(0)) - J

eAW('-s)Y(X)f((c,

(c, zo(0)) - a )

z(s))) ds,

if 0 < t < r,

=

z(f) - eMX)'U(X, z(0)) - j f eMX)U-s)Y{X)f((c, (c, ζ ο ( 0 ) ) - α )

z{s))) ds, ϊϊτ λ„) and of numbers τη 6 [0, 1] such that the following equations hold: lim ||u>„ — uj*||e = 0, n—>00 z„(t) = eA(k"n (x„U(λ„, z„m χ [ Jo

(7.91)

+ (1 - τ„)(/ -

eA^2n-s)Y(Xn)f((c,

+ f eAM('-s)Y(Xn)f((c,Zn(s)))ds, Jo (c, z„(0)) - α = 0 (n= 1,2,...)

β

2πΑ(λ„)\ —1 2πΑ(λ ">)

zn(s))) ds) (7.92) (7.93)

§ 7 Affinity theory

231

It follows from equations (7.92) that zn(t) is a solution of system (7.58) with the parameter values λ = λη. Thus U{Xn,

z„(0)) = ζη(2π)

(η = 1 , 2 , . . . ) .

(7.94)

These equations and equations (7.92) imply the equations τηε2πΑ(λ"))(ζη(0)

(/ -

-

ζη(2π))

= 0

(«=1,2,...).

The matrices I — τηβ2πΜλη> are invertible because the polynomial L(p) Equations (7.95) are therefore equivalent to Ζη(0)

= Ζη(2π)

(n=

(7.95) is Hurwitz.

1,2,...).

Therefore z„(i) are 2jr-periodic solutions of system (7.58) with the parameter values λ = λ„. It was assumed that the cycle Γ is isolated, and so it follows from this assumption and formula (7.91) that for sufficiently large η λη = Γ*

and

zn(t) = z,(f + hn)

(7.96)

with hn φ 0 and hn —»• 0 as η — o o . Hence equations (7.93) can be represented as x*(h v n) — x*(0) ' — = 0.

(7.97)

Κ

Passing to the limit as η

oo in (7.97) we have the equation 0. Consequently, the zero u>* of the fields of the family Ψ^ is isolated in E uniformly in τ € [0, 1], Therefore indiu;*, ψ ^ ) = ind(w*, ψ ? ) .

(7.99)

But Ψ} = Φ4. For this reason (7.99) implies the equation ind(u;*, Φ 5 ) = ind(io», Φ 4 )

(7.100)

where φ5 =

Ψ3 =

_ eA(X)t(I _ β 2πΛ(λ) Γ 1

ρ2π

Χ

/ Jo

pt

e

AW(2

s)

"- Ya)f((c,

z(s)ï)

ds -

eAW-s)Y(X)f((c,z(s)))ds,

/ Jo

{c,z(0))

6. Denote y — (x(t), the form

λ). In the coordinates y,

Φ 5 (ιι;) = (Φ(y), Z2(t) - Az(y),

Zi(t),

-

α).

. . . , zi(t) the field Φ5 takes

· · · , ziit)

-

A¡(y))

232

III Convergence of numerical procedures

where Φ 0 0 = (jc(0 - [ Jo

hit- s\ k)f(x(s)) ds, x(0) - a)

and h(t — s; λ) is a unit impulse response of the linear unit with a transfer function Μ [ ψ ρ ) j L ( y p ) and 27r-periodic boundary conditions. Consider a completely continuous family of vector fields Ψί(ιι;) = (Φ(γ), z2(t) - xA2(y),

zi(t) -

rMy)).

Let y* = (x-t(t), Γ*). Then for any r € [0, 1] the point wx = (>>*, TA2(y*), ...,

xAi(y*))

is a zero of the fields Ψ^ isolated uniformly in r. Therefore ind(w;0^) =ίηά(ωι,Ψί). But w 1 = ω«,

= Φ 5 . This means that indOo, Φβ) = i n d Φ

5

)

(7.101)

where Φ 6 (κ;) = Ψο(μ^) = (ΦΟΟ, Z2(t), .. -, Ziti)).

(7.102)

The field Φβ is a direct sum of the field Φ and the identity field. Therefore by Theorem 4.10 (Sect. 4 of Chapter 2) on the rotation of the direct sum of vector fields ind(u;o, Φ 6 ) = inclfo, Φ)·

(7.103)

ίηά^,Φ) = μ(^).

(7.104)

But

7. A chain of equations is obtained: (7.64), (7.65), (7.69), (7.86), (7.100), (7.101), (7.103), and (7.102). The beginning and the end of the chain lead to equation (7.61). •

7.7 Additional remarks. Affinity Theorem 7.3 shows the relation between the topological index of the zero ζ* of field (7.29) generated by the Poincaré-Andronoν operator and the topological index of the zero of the completely continuous vector field (7.26) generated by the operator given by the right-hand side of equation (7.25). The latter field Φ is taken up in the space C of functions x(t) continuous in the segment [0, T\. The field (7.26) can be considered also in other functional spaces, e.g. in the space £ of the functions periodic in the segment [0, T] or in the space H[. If the zero x* of field (7.26) is isolated in one of those spaces, then it will be isolated in all of them. Therefore the topological index ind (je*, Φ) of the

§ 7 Affinity theory

233

zero ** of the field Φ is defined both in C and in £ and H 1 . There is an analog of Theorem 7.3 for the field Φ in the spaces £ and H l . This remark remains true also for Theorem 7.4. There exist analogs of this theorem relating the topological index of the cycle and the functional characteristic of the associated periodic solution for the case when the functional characteristic is defined using the spaces £ or H 1 . Theorem 7.1 relates rotations of the field Φο = I —Uγ and the field Φι = / — A where A { x ) = x ( T ) - [ f ( s , x(s)) ds. Jo As mentioned in Sect. 3 of Chapter 1, there exist integrofunctional equations other than (7.2) but equivalent to it. These are some examples of such equations: x ( t ) = jc(0) - £

f ( s , x(s)) ds,

(7.105)

*(0 = j c ( 0 ) - f f ( s , x ( s ) ) d s + [ f ( s , x ( s ) ) d s , Jo Jo

(7.106)

x(t) - ψ ί (x(s) + sf(s, x(s))) d s + f f ( s , x(s)) ds. 1 J0 Jo

(Ί Λ ΟΊΛ

Let A i , A i , A j be integrofunctional operators given by the right sides of equation (7.105)- (7.107) respectively. Denote x¡ - I - A ( i = 1, 2, 3). In a way similar to Theorem 7.1 one can prove Theorem 7.5. I f the b o u n d e d d o m a i n s Ωο C and Ω) c C¡\¡ have c o r e , the r o t a t i o n s γ(Φο', 9Ωο) and γ ( χ ΰ 9Ωι) ( i — 1, 2, 3) are e q u a l : 7(Φ 0 ; 9Ω0) = γ ( χ ϊ , a n , )

the s a m e

(/ = 1, 2, 3).

Let now equation (7.1) have the form dx — = Ax + f ( t , x ) ( x e Rn) (7.108) at where f { t + T , x ) = f ( t , x ) and the matrix A has no eigenvalues of the form 0, ±2πϊ/Τ, ±4πί/Τ, Then to find the Γ-periodic solutions of equation (7.108) is to find the solutions of the integral equation x(t)= f Jo

H(t-s-,T)f(s,x(s))ds

where H { t \ Τ ) = (I — e A T ) ~ x e A '

(0 < t < T ) .

(7.109)

234

III Convergence of numerical procedures

Let Φο = I — Ut where U j is the shift operator during the time Τ along the trajectories of differential equation (7.108) and the field Ψι be defined by the equation Ψι (x) = x(t)~

f H(t — s; T)f(s, x(s)) ds. Jo

Theorem 7.6. If the bounded domains Ωο C core, then

(7.110)

and Ωι c C^ have the same

Κ(Φο;9Ωο) = Κ Ψ ι ; 3 Ω , ) .

(7.111)

The field Ψ! can be considered not only in CN but also in SN and HlN. Formula (7.111) holds in those cases, too. Let the nonlinearity fit, x) in equation (7.108) be continuously differentiable. Then the field Ψι is completely continuous in HlN. Theorem 7.7. If the bounded domains Ωο C core, then

and Ωι c HlN have the same

7(Φ0;9Ω0)-/(Ψ1;9Ω1).

(7.112)

Consider now a single-circuit system of automatic control described by equation (7.24). Make the assumptions of Subsection 7.4 and assume that the nonlinearity f(t, x) is T-periodic in t and continuously differentiable with respect to all its variables. Let then U j be the shift operator during the time Τ along the trajectories of system (7.27) and Λ be the operator defined by the right-hand side of equation (7.76). This operator acts in H l and is completely continuous. As mentioned above, an analog of Theorem 7.3 is true for the case when the field Φ = I — Λ is considered in the space H l . This theorem can be extended also to the case when the fields Φ and Ψ are not only in the neighbourhoods of isolated zeros but also in the domains with the same core. This generalization is formulated as follows. The bounded domains Ωο C Ηλ and Ωι c M.N will be referred to as domains with the same core if the fields Φ and Ψ are nondegenerate in 3Ωο and ΘΩι, respectively, and the first components of the zeros of the field Ψ lying in Ωι coincide with the values at t = 0 of the zeros of the field Φ lying in Ωο. Theorem 7.8. If the bounded domains Ωο C / / ' and Ωι C and the polynomial L(p) is Hurwitz, then 7(Φ;3Ω 0 ) = Κ Ψ ; 9 Ω 0 .

have the same core

(7.113)

235

§ 8 Effective convergence criteria for numerical procedures

§ 8 Effective convergence criteria for numerical procedures 8.1 Use of affinity theory for proving the convergence of numerical procedures. The topological criteria of convergence of various numerical procedures in approximating oscillatory regimes in nonlinear systems proved in Sect. 3-6 do not work because the direct evaluation or estimation of the rotation of the appropriate completely continuous vector field (or the topological index of the desired oscillation) appears to be a very difficult problem. Affinity theory of the previous Section reduces evaluation of the rotation of completely continuous vector fields in infinite dimensions to the same problem for finite-dimensional fields. The latter can often be solved using , some special methods (such as the directing functions method). With numerical procedure the affinity theory and the directing functions method provide effective convergence criteria formulated immediately in terms of the properties of the right-hand sides of the differential equations describing the system dynamics. The next Subsection describes some of those criteria.

8.2 The directing functions method and convergence of numerical procedures. Consider the equation at

=Ax+f(t,x)

(xeRN)

(8.1)

where the matrix A has no eigenvalues of the form 0, . . . and the nonlinearity fit, χ) is continuously differentiable with respect to all its variables and Γ-periodic in t. Recall that the continuously differentiable function V(x) {x e R^) is called directing for equation (8.1) if for some po the inequality (W(jc), Ax + f(t, jc)) > 0

(0 < / < T, \x\ > p0)

(8.2)

holds. The directing function V(x) is called growing if lim V(jc) = oo. 1*1-»· oo

(8.3)

Theorem 8.1. If a growing directing function exists for equation (8.1), then: (1) the set 9JÏ of Τ-periodic solutions of equation (8.1) is nonempty; (2) for sufficiently large η the sets Wln of approximants to the Τ-periodic solutions χ* it) € 9JÍ obtained by means of the harmonic balance method is nonempty; (3) the following convergence occurs: lim sup

inf ||x* — χ*||«ι = 0 .

(8.4)

236

III Convergence of numerical procedures

Proof. Consider first the case when all the solutions of equation (8.1) are extendible to the whole segment [0, Γ], The existence of a growing directing function ensures that the set 9JÌ is nonempty and the common a priori estimate for all Γ-periodic solutions x*(t) e OJt holds: sup max |**(0Ι < RQ < oo x,cm

(8.5)

op0,0 0 there exists 0 such that for any χ G satisfying the inequality —

0 that for any χ satisfying the inequality |λ:-Χ*|oo

(8.16)

holds. An asymptotically stable solution x(t) of equation (8.12) is called uniformly asymptotically stable if the limit (8.16) is uniform with respect to the solutions x(t) with the initial data satisfying inequality (8.15). Let x*(t) be a uniformly asymptotically stable 7-periodic solution of equation (8.12). Then the shift operator Ut during time Τ along the trajectories of equation (8.12) is defined in a small neighbourhood B(p, x*) c Consider the vector field