SIX LECTURES ON
DYNAMICAL SYSTEMS
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SIX LECTURES ON
DYNAMICAL SYSTEMS
editors
B. Aulbach & F. Colonius Institut fur Mathematik Universitat Augsburg Germany
World Scientific Singapore* New Jersey •London •Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farcer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
SIX LECTURES ON DYNAMICAL SYSTEMS Copyright © 1996 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA.
ISBN 981-02-2548-2
This book is printed on acid-free paper.
Printed in Singapore by Uto-Print
Preface This collection of articles grew out of a tutorial workshop which has been conducted at the Department of Mathematics at the University of Augsburg in June 1994. This workshop, organized by the editors of this volume, was part of a particular graduate student program ("Graduiertenkolleg Mathematik: Analyse, Optimierung und Steuerung komplexer Systeme") financed by the DFG and the Bavarian Ministry of Education, Culture, Science and Art. The six articles in this volume cover different facets of the mathematical theory of dynamical systems. The topics range from the topological foundations through invari ant manifolds, decoupling, linearization, perturbations and computations to control theory. In any case, emphasis is put on motivation and on guiding ideas, and the theoretical explanations are accompanied and illustrated by many examples. Each contribution is self-contained and provides an in-depth insight into some topic of current interest. We hope that this volume will stimulate further research in the field of dynamical systems.
Augsburg, December 1995
Bernd Aulbach and Fritz Colonius
v
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Contents Preface
v
Dynamical Systems: The Topological Foundations by Ethan Akin 1
Introduction: Discrete Time Dynamics and Closed Relations
1
2
Recurrence for Closed Relations
7
3
Invariant Sets and Lyapunov Functions
14
4
Attractors and Basic Sets
24
5
Maps Between Dynamical Systems
33
Bibliography
42
Integral Manifolds for Caratheodory Type Differential Equations in Banach Spaces by Bernd Aulbach and Thomas Wanner 1
Prologue
45
2
Basic Definitions and Results
48
3
Quasibounded Solutions
60
4
The Fundamental Existence Theorem for Integral Manifolds
76
5
Hierarchies of Integral Manifolds
90
6
The Classical Five Integral Manifolds
97
7
The Missing Sixth Integral Manifold
106
8
Epilogue
109
Appendices
110
A
Strong Measurability and the Bochner Integral
110
B
Banach's Fixed Point Theorem and Related Results
114
References
117
vii
Contents
viii
Control Theory and Dynamical Systems by Fritz Colonius and Wolfgang Kliemann 1
Introduction
121
2
Control Systems as Dynamical Systems
125
3
Limit Behavior of Controlled Trajectories
131
4
Two Perturbation Theorems for Ordinary Differential Equations
134
5
Two Examples
139
6
Analysis of Linearized Control Systems
144
7
Spectral Theory of Nonlinear Control Systems
150
References
156
Shadowing in Discrete Dynamical Systems by Brian A. Coomes, Huseyin Kogak and Kenneth J. Palmer 1
Introduction
163
2
The Infinite-time Shadowing Theorem
165
3
Finite-time Shadowing
175
3.1 The Finite-time Shadowing Theorem
175
3.2 Implementation of the Theorem in Dimension One
177
3.3 Example: The Logistic Map
181
3.4 Implementation of the Theorem in Dimension Two
182
3.5 Example: Tinkerbell Map
188
Periodic Shadowing
190
4.1 Periodic Shadowing Theorem
190
4.2 Necessary and Sufficient Condition that L~l Exists
193
4.3 Implementation of the Theorem in Dimension One
194
4.4 Example: The Logistic Map
196
4.5 Implementation of the Theorem in Dimension Two
196
4.6 Example: The Henon Map
201
4.7 Lyapunov Exponents for Periodic Orbits of Maps
201
4.8 Example: The Henon Map
208
4
References
208
Contents
ix
Perturbations of Invariant Manifolds of Ordinary Differential Equations by George Osipenko and Eugene Ershov 1
Introduction
213
2
Basic Notions and Motivational Examples
216
3
Perturbation of Equilibria
223
4
Invariant Manifolds of Pseudo-Hyperbolic Linear Maps
231
5
Perturbation of Invariant Manifolds Through an Equilibrium
245
6
Perturbation of Periodic Orbits and Invariant Manifolds Through Them . 251
7
Sacker-Neimark and Mane Theorems
254
8
Locally Non-unique Invariant Manifolds
258
References
263
The Reduction of Discrete Dynamical and Semidynamical Systems in Metric Spaces by Andrejs Reinfelds 1
Introduction
267
2
Preliminaries
268
3
Auxiliary Lemmas
270
4
Fixed Point
273
5
Invariant Sets
274
6
Conjugacy of Homeomorphisms 1
281
7
Conjugacy of Noninvertible Mappings
290
8
Conjugacy of Homeomorphisms 2
293
9
Application to Stability Theory
304
Shadowing Lemma
305
10
References
308
Dynamical Systems: The Topological Foundations Ethan Akin Mathematics Department The City College 137 Street and Convent Ave. New York, N. Y. 10031 USA
[email protected] Abstract Of importance for the study of smooth dynamical systems, the concepts of Lyapunov function, attractor and chain recurrence are entirely topological. The theory connecting these ideas provides tools with which to begin the study of a wide variety of dynamical systems.
1
Introduction: Discrete Time Dynamics and Closed Relations
A dynamical system is a mathematical attempt to describe the phenomenon of change. The label is applied to a wide variety of models: differential equations in continuous time, iterations in discrete time, and stochastic processes. The first examples which students meet usually occur in calculus and differential equations courses: falling bodies, radioactive decay and the simple harmonic motions of a spring. These are differential equations with which you are probably familiar. Let us use them to introduce the conceptually simpler, but less familiar, iteration processes that will be the focus of our attention. 1
2
Dynamical Systems:
The Topological
Foundations
An initial value problem consists of a vectorfield on some Euclidean space R n , i.e. a function £ : R n -> R \ and an initial value x € R n . The associated solution path, a?*, is an R n valued function on some time interval containing t = 0 which satisfies: x't = t{xt)
and x0 = x.
(1)
Provided the vectorfield function is reasonable, e.g. continuously differentiable, the problem has a unique solution. Governed by the vectorfield we move in the space R n as time t varies, proceeding along a path uniquely determined by our initial position a. While the time t is the variable on the solution path, the initial value .T is a parameter, that is, it is a constant in some ways and a variable in others. The initial position is the unchanging point of origin for the entire solution path. When we replace one initial point by another, we replace the corresponding solution path entirely. We can reverse this perspective, viewing time as the parameter and the initial point as the variable. This defines for each fixed time t, the time t map / ' : R n —> R n , associating to .T the point /*(#), the time t position of the solution path starting at z, i.e. /*(#) = xt using the solution path of (1). Clearly, f° is the identity map since x0 = x. Furthermore, we have the important semigroup condition:
r s = fo/s.
(2)
This equation holds because our vectorfield is time independent (also called autonomous or stationary), that is, the vector £(x) attached to the point x remains unchanged. (Of course, £,{xt) changes as xt moves from point to point by varying t.) We can begin at point x and run along until time £, reaching the point y = xt. Then we can begin the process again with new initial point y and run along until time -s, reaching y3. For a stationary system we arrive at the same place if we start at x and just run until time t + s. The equation xt+s — y9 with y = xt is (2). We get the associated discrete time system by restricting attention to only integer times. In particular, let / denote the time 1 map f1. By induction on n, we see from (2) that the time n map fn for n = 1 , 2 , 3 , . . . is just the composition / o / o / . . . o / of n copies of / . So we get the entire discrete time process by iterating / . Thus, the map / suffices to describe it.
Discrete Time Dynamics and Closed Relations
3
Because the solution paths of (1) extend to negative times, the maps / ' are all homeomorphisms. In fact by (2) the map f~l is the inverse map for /*. We will be iterating maps which are not homeomorphisms and even more general relations. So we do not assume our dynamical systems are reversible in this way. As we are abandoning calculus we no longer need the vector space struc ture on our underlying state space. So we can consider systems on quite general metric spaces. However, compactness will be an essential require ment. All of our spaces are assumed to be nonempty, compact, metric spaces. For such spaces a subset is closed if and only if it is compact. Hence, the continuous image of a closed set is closed. Furthermore, if {Cn} is a decreasing sequence of closed, nonempty subsets of such a space X then the intersection C = C\nCn is nonempty, as well as closed. Furthermore, if U is any neighborhood of C in X then for sufficiently large n, Cn C U. More generally, if {Cn} is any sequence of closed subsets of A", define: limsup{C n } = nnUjt>nCfc.
(3)
The following exercise collects the properties of the lim sup operator. Exercise 1 With {Cn} a sequence of closed subsets of X let C denote the lim sup{Cn}. Prove the following properties: (a) If U is any neighborhood of C then for sufficiently large n, Cn C U. Furthermore, C is the smallest closed subset satisfying this condition. In particular, if the Cn 's are nonempty then C is nonempty. (b) x £ C if and only if there is a sequence of integers {n,} tending to infinity and a sequence {x{} in X with X{ G Cni such that x = Xmii-+00Xi. (c) If {Cn} is a decreasing sequence, then C = nnCn. If {Cn} is an increasing sequence then C = U n C n . In general, UnCn = C U (U n C n ). (d) If f : X —► A"i is a continuous map then f(C) = l i m s u p { / ( C n ) } . □ Our approach to dynamical systems in these notes will largely follow the book The General Topology of Dynamical Systems [1]. We will be able to highlight the more interesting results and arguments, referring back to the book (hereafter denoted GTDS) for the more tedious or elaborate details. We will be considering dynamical systems obtained by iterating not just maps, but more general relations. At first glance this appears to be
4
Dynamical Systems:
The Topological
Foundations
a generalization devoid of interest, a typical pedant's delight. However, we will see later than even were we to begin with a homeomorphism, its study would lead naturally to more general relations. So with that firm promise as motivation we introduce the notation for and basic properties of relations. There are two, equivalent, ways of thinking about a relation between sets A"i and X2. The multiple valued function picture, which we will not use, associates to each point of X\ a subset, possibly empty, of X2. Instead, we go back to set theory, where a function / : X\ —+ X2 is defined as a particular subset of the Cartesian product X\ x X2- Thus, the set theorist identifies the function / with the set {(#, f(x)) : x £ A"i} C X\ x X2. Adopting this language, as we do, we say that, for example, the identity map lx : X —> A" is the diagonal subset {(x,x) : x € A"} of A" x X. We now define a relation f : X\ —»• X2 to be a subset of A'i x A"2, any subset. Associated with a point .T € A'i we denote by f(x) the subset of X2 defined by f(x) = {y.(x,y)ef}cX2.
(4)
In regarding a function as a special kind of relation we use the slightly abusive notation of identifying a singleton set with the point it contains. We will usually write y € f{x) for the equivalent statement (.T,y) € / • So for A C A'i we define the image f(A) C X2: f(A) = {y:ye
f(x) for some x <E A} = [j f(x).
(5)
Because relations are just subsets of X\ x X2 we can speak of inclu sion between two relations and take unions and intersections of families of relations. This kind of language is unfamiliar because, for example, if / , g : A'i —> X2 are functions and / C g then it is easy to check that f = g. Any relation / : A'i —> X2 has an inverse relation f~l : X2 —> X1 defined by
/ _ 1 = {(y,x) ■ (x,y) e / } c l 2 x AY
(6)
As an exercise, the reader should check that the following two definitions of the domain of / : X\ —> A"2 agree: Dom(/) = {x : / ( * ) ^ 0} = rHX2).
(7)
Discrete Time Dynamics and Closed
Relations
5
Despite all these novelties we mimic function notation, writing / : X\ —* X 2 , for example, because we want to enlist the readers' intuition about how ordinary functions work. Crucial for this purpose is the observation that composition of functions extends to relations. If / : X\ —> A~2 and g : X2 —> A'3 are relations then g o f : X\ —* A 3 is the relation defined by: 9° f
=
{(xi z) '■ There exists y E X2 such that y E f(x) and z 6 g(y)}
=
n13((fxX3)n(Xlxg))
(8)
where 7r13 : A'i x A"2 x A"3 —► A"x x A"3 is the coordinate projection map. It easily follows that:
(9 o f)(s) = g(f(z)).
(9)
From this equation the associative law for composition follows as does the related equation {g o f)(A) = g(f(A))
(AcXt).
(10)
Observe also that / o 1Xl = f = ^x2 ° / and {g o / ) _ 1 = / - 1 o g~l. We also have that composition distributes over union. For sequences {/„} and {gm} of relations from A^ to X2 and X2 to A"3, respectively, we see that (U m 9m)0 ( U „ / n ) = U m , n (y m O / n ) .
(11)
When X\ = X2 = X so that / : X —» X we call / a relation on X. Just as with self maps we can iterate such relations. For / : X —> X we define f° to be the identity map 1^ and for n = 1,2,... we define fn to be the composition of / o / . . . o / of n copies of / . Finally, we let f~n denote ( / - 1 ) n - The analogue of (2) follows from the associative law by induction: fn1+m
=
jm Q p7
j n i f J a >0)
(12)
That is, the composition law holds provided that the integers n^ and n 2 have the same sign. As usual, it is important to take note of the rules which fail to hold; / o f~x and / - 1 o / need not equal or even contain the identity map.
6
Dynamical Systems:
The Topological
Foundations
Recall now that our spaces are compact metric spaces. From the metric we get an important class of relations. Given e > 0 define in X x X: Vt
=
{(.Ti,;r 2 ):c?(.r 1 , ; r 2 ) < e} (13)
Vre =
{(x1,x2)
: d(xux2)
0, with analogous conditions for the V^s. Notice that Vt is open and Vc is closed in X x X but V€ may contain more than the closure of Vt. For compact metric spaces X\ and X2 the product is a compact metric space, as well. Here we use the max metric on the product:
di2{{xuy\),(x2,y2)) = max.(dr(xi,x2),d2(yi,y2)).
(14)
We will call a relation / : Xi —> X2 a closed relation when it is a closed subset of Xi x X2Proposition 2 Let f : X\ —► X2 and g : X2 —+ X3 be closed relations. (a) The domain Dom(f) is a closed subset of X\. (b) The inverse / - 1 : X2 —> X\ is closed. (c) The composition g o / : X\ —> X3 is closed. (d) If A C X\ is closed then f{A) C X2 is closed. (e) If B C X2 is closed then {x : f(x) C\ B ^ 0} C Xi is closed. (f) If U C X2 is open then {x : f(x) C U] C X\ is open. Proof: (b) is obvious. The image f{A) is 7r 2 (/ fl (A x X2)) which is closed by compactness, proving (d). Similarly, (c) follows from (8). Now (a) follows from (7), and (e) holds because, generalizing (7), the set in question is f~l(B). Finally, {.r : f(x) C U] is Xx - f~1(X2 - U) and so (f) follows. D
It is a good exercise to fill in the details of the above, somewhat tele graphic, proof. Note in particular how compactness is used. The assumption of closure is a kind of continuity condition. To see this we rephrase part (f).
Recurrence for Closed
Relations
7
Corollary 3 Let f : X\ —> X2 be a closed relation. For every closed subset A of X\ and every e > 0 there exists 6 > 0 such that: f o V6(A) = f(V6(A))
C Vc(f(A)) = Vt o f(A).
Proof: By part (f) above, {x : f(x) C Ve(f(A))} is an open subset of X\. It clearly contains the closed set A and so contains V&(A) for some positive S by compactness. □ In particular, a function / : X\ —► A"2 is continuous if and only if it is a closed relation. Notice, however, that 6 depends on A as well as on e. We do not in general get the analogue of uniform continuity. We conclude with a useful limit result. P r o p o s i t i o n 4 Let {Fn} and {Gn} be decreasing sequences of closed rela tions from A'i to A'2 and from A"2 to A3, respectively (i.e. Fn+i C Fn, etc.). If H n F n = / and nnGn = g, then D n (G n 0 Fn) = g 0 / . Proof: Because the sequences are decreasing, (8) implies that g o f = 7Ti3((/ x A 3 ) fl (A'i x g)) = 7r 13 (n n [(F n x A 3 ) 0 (A x x Gn)]). By part (c) of Exercise 1 the intersection of a decreasing sequence of closed sets is the lim sup of the sequence. So by part (d) of Exercise 1, g 0 f = n„7r 13 ((F n x A 3 ) H (A-! x Gn)) = nn(Gn o Fn). (Actually, we only use the special case of the lim sup result that intersection commutes with taking image by a continuous map for a decreasing sequence of closed sets. Another compactness result). D Exercise 5 Let {Fn} be a decreasing sequence of closed relations from X\ to A~2 and {An} a decreasing sequence of closed subset of X\. With V\nFn = / and nnAn = A, prove that f)n(Fn(An)) = f(A). D
2
Recurrence for Closed Relations
With A" a compact metric space, as usual, we begin with a closed relation / on A . What does it mean to regard / : X —► X as a dynamical system?
8
Dynamical Systems: The Topological
Foundations
Given z E A", think of f{x) as the set of successors of x, the set of possible positions to which ;r can move in the next time period. So fn(x) is the set of possible fates for .T at time n. In dynamics our goal is t o compute the long run fate of a given initial point. Our logical first step is to enlarge the relation / to define the orbit relation:
of=\jr
(15)
n=l
i.e. y E Of(x) if y E fn{x) for some n = 1,2, Observe that we begin with n = 1 and not n = 0 so that it need not be true that x E Of(x). Notice that Of is a transitive relation. If y E fn(x) and z E fm{y) then z E fm+n{x). In general, a relation F on X is called transitive when F o F C F. Unfortunately, Of is usually not a closed relation even for a homeomorphism / . In fact, Of(x), the set of futures of a:, is usually not closed and we want to enlarge our notion of future to include the limit points of this set. First, define for any closed subset A of X the limit set: u>f[A] = \imsup{fn(A)}.
(16)
By Exercise 1, this is the set of points upon which the image sets {fn(A)} accumulate. In particular, for A = x, a single point, we use Exercise 1(c) to define: ujf(x)
= limsup{/ n (rr)} (17)
Kf(x) = Of(x) = Of(x)Uuf(x). These equations define new relations Kf and ujf on X. Notice that square brackets are used in (16) because ujf(A) = UxeAuf(x) can be a proper subset of wf[A] (see GTDS Exercise 1.7). While uf(x) and Kf{x) are closed subsets of X for each point x E X , the relations tof and Kf need not be closed relations, i.e. closed subsets of X x X. So we define flf Aff
= limsup{/n} _ = Of =
(18) OfUQf.
Recurrence for Closed
Relations
9
The relations Aff and Qf are closed but in taking the closure we may lose transitivity. As we will see, our theory develops by using the strong properties of closed, transitive relations. This suggests that we consider Qf = the smallest closed, transitive relation on X which contains / .
(19)
We can obtain Qf by intersecting over the family of all closed transitive relations containing / (e.g. X x X is such a relation). On the other hand, it is easy to check that a relation F on X is closed and transitive if and only if AfF = F. This suggests that we approach Qfhy successive applications of «A/\ F0 = / , F n + 1 = AfFn, Such a procedure will work but transfinite induction is usually required (see GTDS Exercise 1.18). There is a somewhat larger closed, transitive relation whose construction is more direct and natural, the chaining relation defined by Charles Conley, Cf. To motivate the definition of Cf let us reflect upon the meaning of our earlier constructions. Given x E A", y E 'R-f(x) = Of(x) if for every e > 0 there exist a whole number n and y\ £ /*(#) such that d(y,y\) < e. When / is a function this means we run along the orbit sequence { / ( . T ) , / 2 ( . T ) , . . . } and at some time make an e jump to y. On the other hand, y E Aff(x), i.e. (x,y) E 0 / , if for every e > 0 there exist a whole number n and Xi E A , ?/i E fn{&i) such that d(x,xi) < e and d{y,y\) < e. In the function case this means we make an initial e jump to rri, run along the orbit sequence of this new point {/(•Ti),/ 2 (.r 1 ),...} and at some time make an e jump to y. These little jumps suggest the inevitable measurement errors which occur in real applications. However, if you attempt to compute the orbit of a point on a computer by iterating the map, the little round off errors arise with each iteration, not just at the beginning and the end. From this observation comes the idea of a chain. Given e > 0, an e chain for a closed relation / on A is a sequence {rr 0 ,a- 1 ,...,,T n }, with?? > 1, such that for i = 0,. . . , n - l d(xi+uf(xi)) < e, that is, there exists y, + i E /(&,-) such that d{xi+i, y t + i) < e. We then call it a chain from ;r0, the initial point, to xn, the terminal point. The number n is the length of the chain. We say that x f chains to y, or just chains to y when / is understood, if for every positive e there is an e chain for / from
10
Dynamical Systems:
The Topological
Foundations
x to y. This defines the relation:
Cf = 0 0(V, o /).
(20)
As the intersection of transitive relations, it is clear that Cf is transitive. It is not obvious from the definition that Cf is a closed relation. However, the following identity implies closure. Cf=f]Af(VtofoVl).
(21)
While we will omit the proof of (21), referring to GTDS Proposition 1.8, the reader might want to undertake it as an exercise with the following hints. Choose for ;r 6 X and e > 0 a positive 8 < e/3 such that / o Vs[x) C K/ 3 o/0c},.and show; that 0{V6of oV6)(x)_C 0(V€of)(x). Then show that 6i + 62 0, (B X B) C\ F f\ V€ — 0. By compactness we can choose a finite cover {U0,U\,... ,Un] of B by sets of diameter less than e. Suppose {;r 0 ,;ri,... ,.Tjt} is a 0 chain for F in B with k > n. By the pigeonhole principle there exist .T^J and Xj2 of the chain with j \ < j 2 such that x^ and Xj2 lie in the same element of the cover, and so (XJ^XJ2) E Vt. But .r t+1 € F{xi) for i — 0 , 1 , . . . , fc — 1 because the sequence is an 0 chain. As F is transitive this implies Xj 6 F(xi) for 0 < i < j < k. Thus, (.Tjj, Xj7) G (B x B) n F fl V€. This contradiction proves the lemma. □ P r o p o s i t i o n 7 Let F be a closed, transitive relation on X. (a) F contains the limit relation SIF. In fact, SlF = f)Fn
= uF.
(27)
n
Furthermore, \F\ = \QF\. (b) SIF is closed and transitive. In fact, nFonF
= FottF
= nFoF
= QF.
(28)
(c) y G OF(.r) if and only if there exists z G \F\ such that z G -F(:r) and y G F(z). That is. the limit relation $IF factors through the cyclic set \F\ = \OF\.
Recurrence for Closed
Relations
13
Proof: (a) Since F is transitive, F2 C F and so {Fn} is a decreasing sequence of closed relations. So by Exercise 1(c), O F = l i m s u p { F n } = n n F n . In particular, O F C F. Similarly, OJF(X) = limsup{F n (z)} = nnFn(x). But clearly, ( n n F n ) ( . r ) = n n (F n (.T)). Finally, x G F(x) implies x G Fn(x) for all n and so x G SlF(x). Thus, | F | C | 0 F | . The reverse inclusion is clear since O F C F . We prove (b) and (c) together. First, from (27) it is clear that F o O F and O F o F are contained in O F . Since O F C F , each of these contains OF o OF. Next suppose that z G | F | with y G F ( z ) and 2 G F(x). Then | F | = | 0 F | implies z G OF(z) and so y G F 0 O F 0 F(a?). By the above inclusions F o O F o F C O F 0 F C O F and so y G O F ( z ) . Conversely, assume y G O F ( . T ) . Let B be the closed set F(x) D F~1(y). For any n, y G F n + 2 (a;) and this means 5 contains 0 chains for F of length n. By Lemma 6, we must have J B n | F | ^ 0. Any point z in F ( x ) n F _ 1 ( y ) n | F | will serve as the factorization point we want. Thus, we have proved part (c). Finally, if y G SlF(x) and we choose z in F(x) D F J (y) fl | F | as above, we have, from (a) that z G SlF{z). So z G O F 0 F ( . T ) C O F ( . T ) and y G F o O F ( ^ ) C QF(z). Hence, y G O F o O F ( ^ ) . This inclusion completes the proof of (28) in (b). □ We conclude by gathering some useful basic properties. You can use the hints to work out the, fairly direct, proofs or use the references to the appropriation section of GTDS. Exercise 8 (a) For any relation F on X, -prove that \F\ = | F - 1 | . (b) For f a closed relation on X, prove that A(f~x) — (Af)~x for the relation operators A = O, 0 . M, Q, QQ, C and OC. (Hint: For 0 use Exercise 1(d) and for C use (21), or see GTDS Proposition 1.11(b).) (c) Let X be the interval [0,1] in R and f : X —► X be the homeomorphism defined by f{x) = x2. Prove that \Cf\ = \f\ = {0,1}. Show that with A — % or us, it is not true that A(f~l) = {Af)~l in this case. (d) With f a closed relation on X and F = Of, Qf or Cf, prove the identities: fUfoF = F = fUFof. (29)
14
Dynamical Systems:
The Topological
Foundations
(Hint: With Fi = / U / o F, prove that F^ C F and that Fx is transitive. Conclude Of C Fl and. when F is closed, Qf C Fx. Then show Cf C (Ve o / ) U (Ve o / ) o Af(Ve o / ) for e > 0 and intersect over e > 0 to get F C Fi when F = Cf. For the second identity apply the first to f~x. See also GTDS Proposition 1.11(d).) (e) Prove the identities of (26). (Hint: Use (29) and induction to show for n = 1,2,... F = f U f2 U . . . U fn U fn o F C Of U Fn+1 C F. Intersect over n and use (27). See also GTDS Proposition 2.4 (c).) □
3
Invariant Sets and Lyapunov Functions
If A is a subset of A" we say that A is + invariant for a relation / on X if f(A) C A, i.e. I G A implies f{x) C A. A is called invariant if f(A) = A, i.e. y G A if and only if y (E f(x) for some x £ A. Notice that for two subsets A and B of A" in general f(A) cB&[Ax
{X - B)] n / = 0.
(30)
In particular, A is + invariant for / if and only if the complement X — A is + invariant for / - 1 . L e m m a 9 Let f be a closed relation on X and A be a closed subset + invariant for f. uflA] = nnfn(A) (31) is a closed sttbset of A invariant for f and containing every f invariant subset of A. If, with F — Qf orCf. the subset A is F + invariant then F(A) = f(A) and u>F[A]=u>f[A]. (32) In particular, A is f invariant if and only if it is F
invariant.
Proof: If f(A) C A then fn+l(A) C fn(A) and so {fn(A)} is a de creasing sequence of closed subsets each / + invariant. So the intersection is / + invariant. It equals wf[A] by Exercise 1 (c). If y £ fn+1(A) then n n f-\y) fl f (A) ± 0. So y e uf[A] implies { / ^ ( y ) D f (A)} is a decreasing
Invariant Sets and Lyapunov Functions
15
sequence of closed nonempty sets, and the intersection f~l{y) 0 wf[A) is nonempty. Thus, u>f[A] is / invariant. Clearly, f(B) = B and B C A im ply, by induction, B C fn{A) for all n. So B is contained in the intersection, Now / C F implies / ( A ) C F(A) for any subset A. If F ( A ) C A then the identity F = fUfoFoi (29) implies F ( 4 ) C / ( A ) . Inductively applying this result to the F + invariant set Fn{A), we see that Fn+1(A) = f(Fn(A)) = / n + 1 ( ^ ) - So (32) follows from (31) applied to / and to F. In particular, F{A) = A if and only if F(A) C A and / ( A ) = A. □ Observe that if A is + invariant for / then it is automatically + invariant for Of. If, in addition, A is closed then it is -f invariant for 1Zf as well. However, + invariance with respect to Aff is, in general, a strictly stronger condition. Moving from there to Qf + invariance and on to Cf + invariance are each further strengthenings of the demands upon the set A. As we will now see, each of these conditions has a dynamic interpretation. If / is a map then a single point e is + invariant, or, equivalently, invariant, when it is a fixed point, i.e. e = / ( e ) . A fixed point is called stable if for points x near e the entire orbit Of(x) remains near e. As we will now see this kind of stability is exactly associated with Aff invariance of the fixed point. P r o p o s i t i o n 10 For f a closed relation on X and A a closed subset of X the following conditions are equivalent. (1) A is Aff + invariant. (2) For every neighborhood G of A there exists a neighborhood U of A such that x € U implies Of{x) C G. (3) Every neighborhood G of A contains some f + invariant neighbor hood U of A. Proof: (3) => (2): Obvious. (2) =£► (1): Given any positive e, there exists a positive 8 < e, so that ,r e V8{A){C U) implies Of{x) C Vt{A){= G). Now suppose x <E A and y € Aff(x). There exists a zero chain beginning 6 close to .T and ending 6 close to y. By choice of 0 was arbitrary, y € A.
16
Dynamical Systems:
The Topological
Foundations
(1) => (3): Because Aff (A) C A and G is a neighborhood of the closed set A, Corollary 3 applied to Aff implies that for some positive b~ Aff{Vs(A)) C G and we can choose 6 so that V6(A) C G as well. Thus, GDUX=
VS(A)
UAff(V6(A))
DU2 =
VS(A)
U
Of(V6(A)),
and U2 is clearly an / + invariant neighborhood of A. Remark: If / is also transitive then Aff = Of and so U2 = U\ is closed as well as + invariant. If / is a continuous map then it can still be shown Ui is / + invariant and so U can be chosen closed in either case. See GTDS Remark after Proposition 2.7. However, the result stated there that U can be chosen open for a map is false. Q Exercise 11 Define the subsets o / R x R ; A'0 =
{(0,0)}U{(l/77,0):n = l , 2 , . . . }
A'i
=
{ ( l / ( ? i + l ) , l / ( n + m ) ) : n , m = 1,2,...}
X
=
A'QUA'I.
Prove that X is closed and so is compact. Define the function f\ : X\ —> A" by \-TTI v
n+l'n + m
n+l'
■
1)
m > 1
n+m—1'
(M)
(n = l , 2 , . . . ) m = 1
and f : X —* X by f = lx0 U f\. Prove that f is a continuous map and that with e = (0,0) Aff{e) = e. (Construct a base for the neighborhood system of e in X consisting of closed f invariant sets.) Prove that X is the only open f + invariant set containing e. D In order to describe the meaning of Qf + invariance we introduce the important concept of a Lyapunov function. A Lyapunov function L for a closed relation / on X is a continuous real valued fimction on X such that y G f{x) implies L(y) > L{x), i.e. fC c] is closed and is / + invariant because L is a Lyapunov function. Because L is automatically a Qf Lyapunov function the set is, in fact, Qf + invariant. In particular, {x : L(x) = sup.L} is Qf + invariant. We get the reverse direction from the following lemma applied to F = Qf. □ L e m m a 15 Let F be a closed, transitive relation on X and A be a closed F + invariant subset of X. There exists a Lyapunov function L : X —► [0,1] for F such that A = L~1{1). Proof: We mimic the beautiful proof of the Urysohn Lemma. Let {0, l , r i , r 2 , . . . } be a counting of the rationals in [0,1]. Define UQ = X, U\ = A and Ur = 0 for all rationals bigger than 1. Now apply Proposition 10, and the Remark thereafter, to define, inductively, UTn a closed, F = MF + invariant subset so that r n < r m implies Interior UTn D UTrn. We make our choices so that n^=2UTn = A. Define L(x) to be the value of the associated Dedekind cut: L{x) = sup{r„ : .T £ Ur„} = inf{r : x 0 Ur}. As usual, L is continuous because {x : L{x) < t} = U ^ ^ A ' - UTn and {x : L(x) > t] = U rn> flnt Urn. Because each Ur is F + invariant, x G Ur implies F(x) C Ur and so L is a Lyapunov function. Finally, A = n^=2Urn = L~1{1). □
Invariant Sets and Lyapunov Functions
19
P r o p o s i t i o n 16 Let F be a closed, transitive relation on X. There exists a continuous real valued function L on X such that y £ F(x) implies L(x) < L(y) with equality only if, in addition, x E F(y). In particular, L is a Lyapunov function for F with \L\ = \F\. Proof: Notice first that if a; G F(y) and y 6 F(x), i.e. (x,y) € FnF'1 then L(x) = L{y) for any Lyapunov function L. Next construct for any fixed pair (a?,y) in F — {F fl F~l) a Lyapunov function L(XtV) such that L(Xiy)(x) < jL(Xiy)(y). To do so, apply Lemma 15 to the F + invariant set A = {y} U F(y) and observe that x $ A since y € F(x) but x £ F(y). For each continuous function L, the set 1, let B = f{Ua). As Ua is inward B C Int Ua and, •by assumption, fr~l{B) C Int Ua+T. By the continuity result, Corollary 1.3, there exists 8 > 0 such that with Ua+i = Vs(B), Ua+X C Int Ua and fr~l{Ua+\) C Int Ua+r. Ua+i is inward because f(Ua+i) C / ( U a ) = B. The gap from ?7a+i to Ua+r is filled in by using the inductive hypothesis.
24
Dynamical Systems:
The Topological
Foundations
This completes the proof of the lemma and so allows us to fill in the gaps in the sequence between r?0 + . . .rc*.and no + ••• + «* + rik+i to define the complete sequence {Un} of neighborhoods of A satisfying (36), (37) and n (38).
4
Attractors and Basic Sets
The theme of this section is the relation between the attractors for a closed relation / on A" and the chain recurrence set |C/|, i.e. the set of chain recurrent points for / . For any set A we call A D \Cf\ the trace of A on \Cf\. By Lemma 19 an attractor is determined by its trace. This follows from equation (41) which we rewrite here: Cf(An\Cf\) = A
(44)
for any attractor A for / . For any preattractor £?, e.g. any inward set, the limit set A = u>f[B] is an attractor by part (b) of Proposition 20. Since B is Cf + invariant, Lemma 9 implies that equation (32) applies to B and so A = nn[Cf)n(B) = QCf(B). (Notice the apparent ambiguity in the middle expression and use Exercise 5 to eliminate it.) So if z E B fl |C/|, z E SlCf(B) and so z e A. Thus, for any preattractor B n \Cf\ = A D |C/|, i.e. the preattractor B and the associated attractor A = ujf[B] have the same trace. P r o p o s i t i o n 22 For a closed relation f. the class of attractors is finite or countably infinite. Proof: By Proposition 20 the trace of an attractor is an open-andclosed subset of the compact metric space \Cf\. A compact open set can be written as a finite union of members of any base for the topology. Since \Cf\ has a countable base, it has at most countably many open-and-closed sets and so countably many attractor traces. By (44) distinct attractors have distinct traces. Thus, there are countably many attractors. D An attractor R for the inverse relation / _ 1 is called a repellor for / . So for a repellor equation (44) becomes Cr\Rn\Cf\) = R.
(45)
Attractors
and Basic Sets
Recall from Exercise 8 that (Cf)
25 l
= C(f *), so we omit the parentheses.
1
Also \Cf\ = IC/- !. An attractor-repellor pair consists of an attractor A and a repellor R such that A D i? = 0 and | C / | C A U i?. P r o p o s i t i o n 2 3 Let A be an attractor for a closed relation f on X. There is a unique repellor R for f such that A, R form an attractor-repellor pair. Called the repellor dual to A, R is characterized by: R = Cf~\\Cf\
-(AH
\Cf\)) = X - {x : QCf(x) C A}.
(46)
In particular, if x G X - (A U i?) then £lCf(x) C A and Q,Cf~l(x) C R. Proof: Because attractor traces are open in | C / | , \Cf\ — (An \Cf\) is a closed set and so R is closed, and C / - 1 + invariant by transitivity of Cf-1. Notice that x e R implies x E Cf~\z) for z G \Cf\ -(An \Cf\). l Since z 6 Cf~ (z) we see that z £ R. R contains \Cf\ - (An \Cf\) and x G Cf~l(R). Thus, R is Cf'1 invariant. If .T were also in A then z G Cf{x) would imply z e An \Cf\ which is not true. Thus, A n i? = 0 and | C / | C AU i?. In particular, .Rn \Cf\ = \Cf\ - (An \Cf\) which is open-and-closed in \Cf\ = \Cf~x\. So by part (a)(6) of Proposition 20, R is a preattractor for / _ 1 and by part (b)(3) it is an attractor for / _ 1 , i.e. a repellor for / . If a; E R then, x G Cf~l(z) for some z in Rn\Cf\. Thus, z € RnQCf(x). In particular, QCf(x) meets R. If, on the other hand, x £ R and y € QCf(x) then by Proposition 7(c), y G Cf(z) for some z G Cf(x) D | C / | . Because JF 0 i?, 2 G A fl | C / | and so by Cf + invariance of A, y G A. Hence, O C / ( . T ) C A. Thus, R is the complement of {x : ftC/(ar) C A}. Applying this result to the attractor R for / _ 1 we see that {.T : QCf~1(x) C R] is the complement of A. So x e X - (AU R) implies both QCf(x) C A and fiC/-H-T) C R. a Picking up now our announced theme, we characterize the chaining relation by using attractors. P r o p o s i t i o n 24 Let f be a closed relation on X and x,y G X. (a) If x G \Cf\ then y G Cf(x) if and only if every attractor for f which contains x also contains y. (b) If x 0 \Cf\ then there exists an attractor- repellor pair A, R such that x 4 A U R.
26
Dynamical Systems:
The Topological
Foundations
Proof: If y g Cf(x) then by part (c) of Proposition 20 applied to the Cf + invariant set C / ( . T ) , there exists an inward set U containing Cf{x) with y &U. Let A be the associated attractor for U, namely wf[U}. If .T G \Cf\, then .T G Cf(x) and so x G U n | C / | , the trace of *7, which is contained in A. So A is an attractor containing x but not y. The reverse implication in (a) is obvious from the Cf + invariance of attractors. If .T ^ \Cf\ and we choose y = .T in the first paragraph then x g U and so x £ A, but by Proposition 7(c), QCf(x) C Cf(Cf(x) D \Cf\) and so flC/(s) C Cf(U n |C/|) = C/(i4 n \Cf\) = A. Therefore, by (46) x is not in the repellor dual to A either, i.e. x £ AU R, proving (b). D By using attractors we can construct Lyapunov functions in a way which bypasses the Urysohn Lemma argument of Lemma 15. L e m m a 25 Let A, R be an attractor-repellor pair for a closed relation f on X. There exists a Lyapunov function L : X —► [0,1] for Cf such that A = L - ^ l ) , R = L-^O) and \L\ = A\J R. Proof: If U is an inward neighborhood of some closed set B and L : X —> [0,1] is continuous with L(x) = 1 for x G B U f(U) and L(x) = 0 for # G X — Int U then X is a C / Lyapunov function. For if y G Cf(x) then either ;r G U in which case y G Cf(U) = f(U) (cf. Lemma 9) in which case L(y) = 1 > -^(.T), or x E X — Int ?7 in which case L(x) = 0 < L(y). Now if x ^ A U i? then we can choose an inward neighborhood of A U Cf{x) which does not contain x and apply the above construction to get a Cf Lyapunov function which is 1 on A U Cf{x) and O o n i . Similarly, we construct a Cf'1 Lyapunov function which is 1 on R U Cf~l(x) and 0 on .T. Taking the difference and adjusting linearly we get a Cf Lyapunov function Lx : X -* [0,1] with R C £ _ 1 ( 0 ) , A C L _ 1 ( l ) and x e Gx = X — \LX\. Choose a sequence {.xn} in X — (A U R) so that {GXn} covers X — (A U R). With Se n a positive series summing to 1, T,enLXn yields the required Lyapunov function. (For details see GTDS Proposition 3.10.) □ On the recurrence set \Cf\ the relation Cf is reflexive as well as tran sitive. Thus, Cf fl Cf~l is a closed, symmetric, transitive relation with domain \Cf\. On \Cf\ it is a closed equivalence relation. The associated
At tractors and Basic Sets
27
equivalence classes are called the basic sets for / . So we see that for x G X:
c/wncr I w = (f1, , .
4
...
x
*fJt\
(47)
[ the basic set containing x x G |C /1 By Proposition 24, two points of \Cf\ lie in the same basic set exactly when they are contained in exactly the same attractors. Notice, too, that if L is any Cf Lyapunov function then it is constant on each basic set. P r o p o s i t i o n 26 Let f be a closed relation on X. There is a Cf Lyapunov function L on X such that \L\ = \Cf\ and L takes distinct basic sets to distinct values, i.e. L{x) = L{y) for x,y G \Cf\ if and only if x G Cf(y) and y G Cf(x). Proof: By Proposition 22, the class of attractors is countable. Let {An} be an infinitive sequence of attractors (repetitions are allowed) which includes all of them. For each ??, let Rn be the repellor dual to An and Ln : X —> [0,1] a Cf Lyapunov function with An = Z/~ 1 (l), Rn = L'1^) and \Ln\ = An U Rn. Define L(.T) = ^ = 1 2 . 3 - n L n ( ; r ) .
(48)
As the series converges uniformly, Exercise 13(b) implies that L is a Cf Lyapunov function with \Cf\ C \L\ C f| \Ln\ = f)(An U Rn) C \Cf\ n
using Proposition 24(b) for the last inclusion. Thus, \Cf\ = \L\. If x G \Cf\ then the sequence {Ln(x)} consists of 0's and l's with the l's indicating which attractors in the sequence {An} contain x. Equation (48) then associates to .r the point in the classical Cantor set with ternary (i.e. base 3) expansion given by the sequence {2L n (.r)}. If L{x) = L(y) for x,y G \Cf\ then uniqueness of the ternary expansion on the Cantor set implies x and y lie in exactly the same attractors. So x and y are in the same basic set. D A Lyapunov function satisfying the conditions of this Proposition is called a complete Cf Lyapunov function. The restriction of such a function
28
Dynamical Systems:
The Topological
Foundations
L to \Cf\ yields an embedding of the quotient space of basic sets, i.e. the space of equivalence classes under CfC\Cf~l on \Cf\, onto the set of critical values i(|Zr|) in R. In the special case constructed in the proof above, .L(|£|) is a closed subset of the Cantor set and so is zero- dimensional. Hence, for any complete Cf Lyapunov function L the critical value set is zero-dimensional and hence nowhere dense in R. Thus, we can regard the quotient space of \Cf\ by the equivalence relation Cf D Cf-1 as the space of basic sets of / , a compact zero-dimensional metric space which is, via any complete Cf Lyapunov function, homeomorphic to a compact nowhere dense subset of R. Cf induces a partial order on the basic sets: a closed, reflexive, antisymmetric and transitive relation on \Cf\/Cf D Cf-1. Notice that the space of basic sets for / and for f~l agree. Any Cf -f- invariant subset B of X contains any basic set it meets and so the trace B 0 \Cf\ is the union of these basic sets. In particular, we have the following dual results: Proposition 27 For a closed relation f, two attractors are equal if they contain exactly the same basic sets. Two basic sets agree if they are con tained in exactly the same attractors. Proof: If A\ and A2 are attractors which contain exactly the same basic sets then the traces Ax D\Cf\ and A2H \Cf\ agree as these are each the union of this common family of basic sets. So AY = A2 by (44). If B\ and B2 are basic sets contained in exactly the same set of attractors then with X\ £ B\ and x2 E B2, xux2 E \Cf\ and so by Proposition 24(a), Xi G Cf(x2) and x2 £ Cf(xi). So ;ri and x2 lie in a common basic set which is Bi = B2. □ On the other hand, while a closed relation admits only countably many attractors there may be uncountably many basic sets. Exercise 28 Let C be the Cantor set in the unit interval [0,1] and let k : [0,1] —> [0,1] be a continuous, nonnegative function which vanishes exactly on the Cantor, e.g. k{x) = inf{|.r - .Ti| : xx E C). Let X = [0,1] x [0,1], a unit square in R 2 . Define f : A" —> A' by: f(x,y)
= (x,y + -2/(1 - y)[(2y - l ) 2 + k(x)]).
(49)
Attractors
and Basic Sets
29
Prove that f is a homeomorphism with \f\ = \Cf\ = ([0,1] x {0,1}) U (C X {|}). (Hint: Show that L(x,y) = y is a Cf Lyapunov function, though not a complete one.) Show that [0,1] x {0},, [0,1] x {1} and each point {(ar, | ) } in C x {2} o,re basic sets. Describe the attractors and dual repellors. □ Finiteness for the class of basic sets coincides with finiteness for the class of attractors. P r o p o s i t i o n 29 For a closed relation f there are finitely many basic sets if and only if there are finitely many attractors. To be precise, if f has k distinct basic sets, it has at least k distinct nonempty attractors. If f has at most k distinct basic sets it has at most 2k distinct attractors. In particular, f has no basic sets if and only if 0 is the only attractor for / , and f has exactly one basic set if and only if it has exactly one nonempty attractor. Proof: Each attractor is characterized by the set of basic sets it con tains. If there are k basic sets then there are 2k sets of basic sets and so at most 2k attractors. For example, if A" is a set of k points and / = lx then the k points of A' are the basic sets and every subset is an attractor. So there are 2k attractors. Suppose now that / has at least k distinct basic sets. Choose L a complete Cf Lyapunov function. L is constant on each basic set and takes different values on distinct basic sets. Hence, we can number the k basic sets B\,B2i.. •, B^ so that the corresponding values for L satisfy 61 < 62 < . . . < bk- As the critical values for L form a nowhere dense set we can choose regular values C\,.. .Ck so that C\ < 6j < c2 < b2 < c 3 < 6 3 . . . 6jt_i < c^ < 6fc. By Exercise 18(b) the set U{ = {L > q } is an inward set with associated attractor A{ — wf[lh] containing J B J , B , + I , .. . , B * but not the remaining basic sets on the list, for i = 1 , . . . , k. So the attractors Ai, A2,... ,Ak are nonempty and distinct. The entire space A' is an inward set (/(A') C Int X = X) and the associated attractor is u>/[A~] = D n / n (A"). If for some n = 1,2,... the set fn(X) is empty, or equivalently the relation fn is 0, then 0 is the only attractor, in fact the only / invariant subset. For sufficiently small positive e, (Ve o f)n is 0 as well. (Hint: Use Proposition 4 and induction.) Hence, {Cf)n = 0, \Cf\ = 0 and there are no basic sets. These peculiar,
30
Dynamical Systems:
The Topological
Foundations
nilpotent, closed relations are described in GTDS Exercise 2.16. If, on the other hand, fn(X) / 0 for n = 1,2,... then the intersection uf[X] is a nonempty attractor, A. As 0 is an attractor with empty trace, A must have a nonempty trace, by (44) again. So \Cf\ / 0 and there are some basic sets. If there is exactly k = 1 basic set then there is at least 1 and at most 1 2 — 1 = 1 nonempty attractor. Finally, more than one basic set implies more than one nonempty attractor. E The unique basic set case of this proposition brings up an alternative use of the word "transitive". A group acting on a space, a group of rotations of the sphere, for example, is said to act transitively if any point of the space can be moved to any other point by the action of some member of the group. With this in mind we call a closed relation / on A chain transitive if any point of A" chains to any other, that is, if Cf — X x A". Proposition 30 For a closed relation f on X the following conditions are equivalent: (1) f is chain transitive, i.e. Cf = X x A'. (2) X is a basic set for f. ('V \Cf\ — X and f has a unique basic set. (4) Dom(f) = X and X is the only nonempty attractor for f. (5) f{X) = X = f~\X) and f has a unique nonempty attractor. Proof: Since \Cf\ is the union of the basic sets, the equivalences be tween (1), (2), and (3) are clear. (2) => (4): If A' is a basic set for / and A is a nonempty attractor then A meets A and so A contains A'. Thus, A = X. X is also a basic set for f"1 and so A" is also a repellor. In particular, A" is / _ 1 invariant. That is, A = /- 1 (A") = D o m ( / ) . (4) =*► (5): The unique nonempty attractor is A". In particular, / invariance of A implies A = / ( A ) . Finally, A = D o m ( / ) = / _ 1 ( X ) . (5) =^ (3): There is a unique basic set by Proposition 29. If .T £ A" then {/"(•?)} is a sequence of nonempty closed sets since / _ 1 ( X ) = D o m ( / ) = A . Hence uf(x) C Qf{x) C QCf(x) is nonempty. So by Proposition 7(c) there exists zx € \Cf\ such that zx G Cf{x). Similarly, /(A") = A" implies there exists z2 6 \Cf\ such that z2 £ Cf~1(x). But there is a unique basic set which must therefore contain both z\ and z2. In particular, z2 € Cf{z\). By
Attractors
31
and Basic Sets
transitivity, zx G Cf{x). Then z2 G Cf{zi) and x G Cf(z2) imply x G Cf{x), i.e. rrG | C / | . Thus, \Cf\ = X. R e m a r k : As C ( / _ 1 ) = ( C / ) ~ \ condition (1) is symmetric in / . So / chain transitive implies / _ 1 is chain transitive. D For any closed subset B of A" we define the restriction of the closed relation / to £?, or the subsystem induced by B , to be the closed relation IB on B: fB = fn (B x B). (50) For example, if / is a function then the restriction fg is a function if and only if B is an / -f- invariant subset. We call B a chain transitive subset for / on A", if the restriction fg is a chain transitive relation on the compact metric space B. A basic set B is a chain transitive subset but the proof is not as easy as it first appears to be. Given .T, y G B we know that x chains to y, that is, for any e > 0 there is an e chain {;ro,.Ti,... , x n } with n > 1, XQ — $ and ;rn = y. However, to show B is chain transitive we must construct such e chains remaining entirely inside B. By contrast if x and y lie in different basic sets but y G Cf(x), we usually cannot construct such e chains lying entirely in \Cf\. The key result requires another concept. If for a closed relation F a subset B is F 4- invariant then any chain for F (i.e. 0 chain) beginning in B remains in B. If B is F~l + invariant then any chain for F ending in B lies in B. We call B F semi-invariant if any chain which begins and ends in B lies entirely in B. We can express this as the inclusion G{F){B)
D C{F)-\B)
C B.
(51)
Note that if F is transitive, 0(F) = F. L e m m a 31 Let f be a closed relation on X and B be a closed subset of X. We have the inclusion C{fg) C {Cf)g with equality when B is Cf semi-invariant, i.e. when Cf{B)f\Cf~1(B) C B. Proof: The inclusion is an easy exercise. The equality in the semiinvariant case is proved in GTDS as Theorem 4.5. D
32
Dynamical Systems:
The Topological
Foundations
Corollary 32 Any basic set for a closed relation f on X is a chain tran sitive subset for f. Proof: If B is a basic set then Cf(B) = Cf(x) and Cf~l{B) = Cf~l(x) for .T in B. So Cf(B) nCf'^B) = Cf(x)nCf-1(x) = B. Thus, B is Cf semi- invariant. Also, B x B cCf and so B x B = (Cf)B. By the lemma,
C(fB) = BxB.
n
What does it mean when each basic set is as simple as possible, i.e. a single point? The classical definition of a Lyapunov function L for a homeomorphism / on X is a continuous real- valued function which is strictly increasing on nonequilibrium orbits, i.e. {L(fnx)} is increasing in n unless f(x) = x. Motivated by this idea we call L a strict Lyapunov function for a closed relation / if L is a Lyapunov function with \L\ = \f\. Proposition 33 Let f be a closed relation on X. (a) f admits a strict Lyapunov function if and only if \f\ = \Qf\, i.e. every generalized nonwandering point is a fixed point, f admits a strict Cf Lyapunov function if and only if \f\ = \Cf\, i.e. every chain recurrent point is a fixed point. (b) If every basic set for f is a single point, i.e. y G Cf(x) and x G Cf(y) imply x = y, then any complete Cf Lyapunov function is strict and injective on \Cf\ = \f\. Conversely, if f admits any Cf Lyapunov function L which is injective on \Cf\ (C \L\) then every basic set is a single point. Proof: (a) If L is a strict Lyapunov function for / then | / | C \Qf\ C |L| = | / | . So \Qf\ = | / | . Similarly, if L is a strict Cf Lyapunov function, |£/l = l/l- Conversely, if | / | = \Qf\, or | / | = | C / | , we can apply Proposition 16 to F = Qf or Cf to get a strict Lyapunov function. (b) If B is a basic set then by Corollary 32, B is a chain transitive subset. If B = {;r} then by Proposition 30 applied to f&, we have x 6 f(x), i.e. x G l/l- Thus, if every basic set is a singleton, then \Cf\ consists entirely of fixed points, and a complete Cf Lyapunov function is injective on \Cf\. As \L\ = \Cf\ = | / | , L is strict. Any Cf Lyapunov function L is constant on each basic set B. So, if L is injective on \Cf\ then B is a singleton, for each basic set B. D
Maps Between Dynamical Systems
5
33
Maps Between Dynamical Systems
Given a continuous map h : X —> Y we write h x h : X x X —> Y x Y for the product map with (h x h)(xi,x2) = (h(xi),h(x2))< If / is a relation on A", i.e. a subset of A" x A', then {h x h)(f) is a subset of Y x Y and so is a relation on Y. Similarly, if g is a relation on F , then the preimage (h x h)~l{g) is a relation on A'. It is easy to check that we can describe these constructions using composition of relations: (hxh)(f)
=
hofoh-1 (52)
{hxhr\g)
=
hTlogoh.
We say that h maps f to g (written h : f —> g) if for all x\,x-2 E A", •t'2 £ f(xi) implies /?(.T 2 ) € g{h(xi)). This condition is easily seen to be equivalent to each of the following inclusions: (hxh)(f)Cg fcihxhr'ig)
(53)
h o / c g o h. For example, if / is a relation on X and B is a closed subset of A', then the restriction of / to B was defined in the previous section by f& = ff)BxB. The inclusion map from B to A" clearly maps / # to / . P r o p o s i t i o n 34 Let h : X —► Y map the relation f on X to the relation g on Y. Then h maps f to g, / _ 1 to g~l, fn to gn for n = 0 , 1 , 2 , . . . and (when f and g are closed) Af to Ag for each of the relation operators
A = o,u,n,ii,Af,G,ng,CMC. Proof: We use the conditions of (53). (/? xft) _ 1 (#) is closed and contains / . So it contains / . Clearly, (/ix/?)(/ _ 1 ) = {(hxh^f))'1 which is contained l n n in g~ . ho f C g o h is clear for n = 0, assumed for n = 1 and follows by induction for higher n. If {fa} a n d {#«} a r e families of relations with h mapping fa to gQ for each a then since (h x h)~l preserves union and intersection, h also maps
34
Dynamical Systems:
The Topological
Foundations
UafQ to Ua. For * G A" fixed &("/(*)) = h{hmsup{fn{x)}) C n limsup{gf (/?(;r))} = w#(/?.(#)) by Exercise 1(d). So h maps u>f to u> 0 there exists 6 > 0 so that h o Vs C Vt o h. Hence, h oVso f cV€o g oh. That is, h maps V$ o f to Vto g. So h maps C / C 0(Vs o / ) to C7(Ve o ^). Intersecting over e > 0 we see that (/? x h){Cf) is contained in Cg. □ Recall that for functions, as opposed to more general relations, inclusion implies equality. So if / and g are functions and h maps / to g, then we have ho f — goh. (54) For general relations / on X and g on Y we call the continuous map h : X —► I" a semi-conjugacy from f to g when equation (54) holds. Thus, ft is a semi- conjugacy if h maps f to g and, in addition, the lifting property illustrated by the following diagram holds:
(55)
Given yuy2 with y2 G g{yi) and a-j such that h(xi) = y ^ there exists x2 such that /?(x2) = y2 and .T2 G f(x\).
Maps Between Dynamical Systems
35
The phrase "semi-conjugacy" is used because the term "conjugacy" is the isomorphism concept for dynamical systems. P r o p o s i t i o n 35 Let f and g be relations on X and Y respectively. A homeomorphism h from X onto Y is called a conjugacy from, f to g when it satisfies the following equivalent conditions: (1) h maps f to g and h~l maps g to f. (2) h is a semi-conjugacy from f to g and h~l is a semi-conjugacy from
9 to f. (3) h is a semi-conjugacy from f to g. The relations f and g are called conjugate if there exists a conjugacy mapping f to g. Conjugacy is an equivalence relation between relations on spaces. Proof: Clearly, (2) implies (1) and (3). Given (1) we have ho f C goh and h~l o g C / o h~l. Compose with h on the left and right in the latter to get, because h o h~~l = \y and h"1 oh = lx,gohchof. So equation l (54) is true and (3) follows. Given (3) we compose with h~ on the left and right in (54) to get h~l o g — f o /? _1 which implies (2). The identity on X maps / to / and so conjugacy is reflexive. Symmetry is clear and transitivity follows because the composition of conjugacies is a conjugacy. □ We would like to prove the analogue of Proposition 34, that is, if h is a semi-conjugacy from / to g then it is a semi- conjugacy of Af to Ag for A — 0,11, A/\ etc. In general, this requires the additional assumption that h be an open map. Recall that for a continuous map hi : A" —> Y the preimage of an open subset of Y is open in X. h is called open if the image under h of an open subset of X is open in I', i.e. U C X open implies h{U) is open in Y. The projections of a product onto a factor is an open map as is a map which is locally a projection, e.g. a fibre bundle map or a differential submersion. Exercise 36 Let h : X —> 1* be a continuous map. (a) Prove that h is open if and only if for every e > 0 there exists 6 > 0 such that h o V€ D V$ o h. (Hint: If h is open then lx x h : X x X -* X X Y
36
Dynamical Systems:
The Topological
Foundations
is open and so h o Vt = (lx x h)(Ve) is an open set containing the compact set h C X x Y). (b) If h is open, h(x) = y and {yn} is a sequence in Y converging to y then there exists N and a sequence {xn} in X converging to x with h(Zn) = Vn for n> N. E P r o p o s i t i o n 37 Let / , g be relations on X and Y respectively with h : A' —> 1" a semi-conjugacy from f to g. (a) h is a semi-conjugacy from fn to gn for n = 0 , 1 , 2 , . . . and from Of to Oy. If f and g are closed then h is a semi-conjugacy from Af to Ag with A = 0,u> and IZ. (b) Assume, in addition, that h is an open map. h is a semi-conjugacy from f to g. If f and g are closed then h is a semi-conjugacy from Af to Ag with A = 0,u,Sl,K,Af,g,QG,C and SIC. Proof: (a) h o fn = gn o h is clear for n = 0, assumed for n = 1 and follows by induction for higher n. If {fa} and {ga} are families of relations with h a semi-conjugacy from fa to ga for each a then h is a semi-conjugacy of Uafa to UQga by the distributive law, (11). In particular, h is a semi-conjugacy of Of to Og. Exercise 1(d) again shows, as in the proof of Proposition 34, that h(u)f(x)) = ujg{h(x)). Thus, h is a semi-conjugacy from u>f tog and from Uf = Of U u>f to Kg. (b) By Proposition 34, h maps / to g and Af to Ag in each case. So it suffices to prove the lifting property, (55), for each of these. Assume y\ = h{x\) and (yi,t/2) £ 9- There is a sequence {(yi,y%)} in g converging to (yi,ya). By Exercise 36(b) we can lift the sequence {yj} to a sequence {.T"} in X converging to X\ such that h(x1l) = yj1 for all n (throwing away some initial terms if necessary). For each n, we use the lifting property (55) for / and g to choose .rj G /(tf?) such that /i(xj) = yjBy going to a subsequence we can assume that {.TJ} converges in X to some pointjr 2 - Thus, {(.rj,.?;)} is a sequence in / converging to (xi,x2), i.e. x2 € /(.Ti). By continuity of h< h{x2) = lim n {y£} = y 2 . Thus, the lifting property holds for / and g. If {/n} and {gn} are decreasing, closed sequences of relations with in tersections / and g respectively, and h is a semi-conjugacy from fn to gn
Maps Between Dynamical Systems
37
for each n then h is a semi-conjugacy from / to g, i.e. ho f = C\n(ho/n) = nn(9n ° h) = g o h by Proposition 4. It now follows that h o ft/ = /io (limsup n {/ n }) = (\ims\xpn{gn}) o h = ft Y map f to g. Assume A and B are closed subsets of X and Y respectively such that A C h~1(B). Let h& : A —> B be the restriction of h. Prove h^ maps f& to gg. Assume now that h is a semiconjugacy from f to g and that A = h~1(B). Prove h^ is a semi-conjugacy from fA to gB. (b) Let X = [0,1] and f : X —> X be the homeomorphism defined by f(x) = x2 (recall Exercise 8(c)). Let Y be the quotient space of X obtained by identifying 0 and 1 to get a single point e ofY, and let h : X -4 Y be the quotient map. Prove that f factors throxigh h to define a homeomorphism g : Y —¥ Y with h mapping f to g and so h is a semi-conjugacy from f to g. Prove that h(0) = e. Mg{e) = Y and Cf(0) = {0}. Show that h is not a semi-conjugacy from Af to Ag for A = Q,Af,Q or C. Notice that h is not an open map. D We will now consider how a mapping h g relates basic sets and attractors of / to those of g. Notice first that if h maps F to G then x G F(x) implies h(x) G G{h(x)). So if / and g are closed relations and h
38
Dynamical Systems:
The Topological
Foundations
maps / to g then from Proposition 34 we get h(\Af\)c\Ag\
(A = 0,K1Af,gX)>
.
(™)
L e m m a 39 Let F and G be the closed transitive relations on X and Y, respectively. Assume h : A' —* Y is a semi-conjugacy from F to G. If xx G X and yuy2 E Y with h(xi) = yx and y2 G \G\ D G(y\) then there exists x2 G A" with h{x2) G y2 and x2 G \F\ n F(xi). That is, in diagram (55) a cyclic point y2 can be lifted to a cyclic point x2. In particular, if h : X —» Y is surjective then h(\F\) = \G\.
(57)
Proof: Define the closed subset of A", A = h~1(y2) 0 F(xi). The lifting property (55) implies there exists 0*2 such that h(x2) = y2 and x2 E F(xi), i.e. x2 G A. Thus, A is a nonempty compact set. If £\ E A then we can apply (55) to h(x\) = y2 and y2 G G{y2) to get x2 G X such that h(x2) = y2 and x2 G F{xi). Because F is transitive and x\ G A C -FX-Ti) we have ;r2 G F(,Ti), i.e. ;f2 G A. Thus, the restriction FA satisfies Dom(F,4) = A. So inductively Dom{{FA)n) = A and, a fortiori (FA)n / 0. By Lemma 6 it follows that A H | F | ^ 0. The points of this set are the cyclic point lifts of y2 that we want. Now if yi G \G\ and /? is surjective, then there exists xi G A" with h(xi) = y\. Let y2 = y\ G G'(yi) H |G'| and apply the previous result to get .T2 G \F\ with /?(;r2) = y2 = y\. Thus, h maps | F | onto \G\. D Recall that the basic sets for a closed relation / on X are the equivalence classes of the closed equivalence relation Cf H C / _ 1 on the chain recurrence set \Cf\. On the quotient space of basic sets Cf induces a partial ordering. For any collection of basic sets an element of the collection which is maximal (or minimal) with respect to the partial order Cf will be called a terminal (or initial) element of the collection. We use these terms to avoid confusion with maximal or minimal elements with respect to set inclusion. As an exercise the reader should prove that an attractor which is chain transitive is a single basic set which corresponds to a point which is terminal and topologically isolated in the collection of all basic sets (cf. Proposition 30 and Corollary 32).
Maps Between Dynamical Systems
39
P r o p o s i t i o n 40 Let f and g be closed relations on X and Y respectively. If h : A" —► Y maps f to g then for every basic set B for f in X there is a unique basic set B\ for g in Y such that h(B) C B\. Thus, h induces a continuous, partial order preserving mapping from the space of basic sets on X to the space of basic sets on Y. If h is a surjective semi-conjugacy from f to g then for every basic set Bi for g in Y there exists a basic set B for f in X siLch that h(B) C B\. Thus, a surjective semi-conjugacy induces a surjective map from the space of basic sets on X to the space of basic sets on Y. If h is an open, surjective semi-conjugacy from f to g then for every basic set Bi for g. in Y there exists a basic set B for f in X such that h{B) = B\. In fact, {B : h(B) C B\] is a compact, nonempty subset of the space of basic sets for f. Terminal elements of this collection exist and h{B) = B\ for any such terminal element. Proof: By Proposition 34, h maps Cf to Cg and so all of the points of h(B) are Cg fl Cg-1 equivalent. So they are all contained in a single, clearly unique, basic set. Thus, the continuous map h : \Cf\ —► \Cg\ factors through the quotient map associated with the closed equivalence relations CfnCf"1 and CgC\Cg~l. By definition of the quotient topology the induced map is continuous. Since h maps Cf to Cg, two / basic sets which are Cf related map to g basic sets which are Cg related. Now assume that h is a surjective semi-conjugacy from / to g. We cannot apply Lemma 39 directly since h need not be in semi-conjugacy of Cf with Cg (we are not yet assuming h is open). We adapt its proof defining A = h~1{Bi) where B\ is a g basic set. As h is surjective, A is a nonempty compact set. By Exercise 38(a) h restricts to a semi-conjugacy HA from f& to g[Ui] and A1 H\Cg\ = U\ H\Cg\. Y - Int U\ = U\ is an inward set for g~l and R\ = uj(g_1 )[Ui] is the dual repellor for A\, see Proposition 23. Because h maps / to g and f~l to ^f-1 Lemma41(a) implies U = /i -1 (?7i) and U = h~l(m ) are inward sets for / and f"1 respectively. Let A = tof[U] and R = uf(f~l)[U], attractors for / and / _ 1 respectively by Proposition 20(b). By (58), we have h(A) C i i and h{R) C Rv If B is an / basic set then h(B) C B\ for a unique g basic set B\. If B\ C A\ then B C h~\Bx) C U and s o ^ C ^ f l \Cf\ = A n \Cf\. Otherwise, Bx C i?i and so Bx C U fl \Cf\ = R n \Cf\. Thus, (60) follows and A, R is an attractorrepellor pair for / because (A U R) D \Cf\ = \Cf\ (see Proposition 23). On the other hand, if A, R is an attractor-repellor pair for / with h(A) C A\ and h(R) C R\ then since h(B) is contained in A\ or Rx for any basic set B, (60) holds with A, R replaced by A, R. Hence, A = A, R = R because attractors are determined by their traces. (b) If U is inward for / with uf[U] = A then by Lemma 41(b), h(U) is inward with associated attractor ujg[h(U)] = Ax satisfying h(A) = Ax by (59). By (56) we clearly have h(A n \Cf\) C h(A) fl \Cg\. If y = fcfai) with .ri 6 A and y G \Cg\ then because h is a semi-conjugacy of Cf with Cg by Proposition 37, we can apply Lemma 39 with y2 = yx = y to find •^2 € \Cf\ C\Cf(xi) such that h{x2) = V2 = V- Since A is an attractor it is Cf invariant. So xt € A implies x2 G A. Thus, y 6 /i(A fl |C/|). D
42
Dynamical Systems:
The Topological
Foundations
Exercise 43 (a) Let h : f -> g by the map of Exercise 38(h). Prove that Y is the single basic set for g, i.e. g is chain transitive. Prove that {0}, {1} are an attractor- repellor pair for f in X and are the two basic sets for f. Observe that {e} = {h{\)} = {h{0)} is neither an attractor, a repellor nor a basic set for g. Observe that {e} = h{\Cf\) is a proper subset of \Cg\ = Y. (b) Let f be the homeomorphism of the unit square defined in Exercise 28. Let g be the identity map on the unit interval Y. Prove that h : X —> Y, projection on the first coordinate, is an open surjection mapping f to g and so is an open semi-conjugacy from f to g. Prove that Y is the single basic set for g. Observe that the f basic sets {(x, | ) } for x £ C do not map onto basic sets for g. (c) Assume h maps the closed relation f on X to the closed relation g on Y and that L\ : Y —► R is a Lyapunov function for g. Prove that L — L\ o h is a Lyapunov function for f with
\L\ c h-WLtl). Prove that if h is a semi-conjugacy equality holds in (62).
of f with g and of f~l
(62) with g~l then D
Bibliography In the following short list of related works, Conley's CBMS lectures [3] described chain recurrence, its relation to attractor theory and applied these ideas to a variety of topics. In a classic paper, reprinted in [6], Smale intro duced basic sets for a special class of smooth dynamical systems. Shub's book [5] develops this smooth theory using many of the topological tools we have described above. The relation between smooth and topological dynamics is also elaborated in [2], the first volume on dynamical systems in the Russian Encyclopedia of Mathematical Sciences. Finally, the study of permanence in [4] applies attractor theory to biology. 1. E. Akin, T h e General Topology of D y n a m i c a l S y s t e m s , Grad uate Studies in Mathematics v. 1 (American Mathematical Society, Provi dence, 1993) ( = GTDS).
Bibliography
43
2. D. V. Anosov and V. I. Arnold (eds.), Dynamical S y s t e m s I, En cyclopedia of Mathematical Sciences v. 1 (Springer-Verlag, Berlin, 1988). 3. C. Conley, Isolated Invariant Sets and t h e Morse Index, CBMS conference series N. 38 (American Mathematical Society, Providence, 1978). 4. J. Hofbauer and K. Sigmund, T h e T h e o r y of Evolution and D y n a m i c a l S y s t e m s , London Mathematical Society Student Text N. 7 (Cambridge University Press, Cambridge, 1988). 5. M. Shub, Global Stability of Dynamical S y s t e m s , (SpringerVerlag, Berlin, 1987). 6. S. Smale, T h e M a t h e m a t i c s of Time, (Springer-Verlag, Berlin, 1980).
Integral Manifolds for Caratheodory Type Differential Equations in Banach Spaces Bernd Aulbach h Thomas Wanner Department of Mathematics, University of Augsburg D-86135 Augsburg, Germany
Abstract. This lecture is designed to meet two seemingly contrary purposes. On one hand it is believed to enable a novice to approach the theory of invariant and integral manifolds through a completely selfcontained presentation, and on the other hand it is thought that even the experts may gain some new insight into the theory of integral manifolds, mainly due to a new approach allowing a generalization of the local invariant manifold theory to nonautonomous dif ferential equations x = f(t,x) in Banach spaces with possibly discontinuous t-dependence. More detailed information is given in Section 1.
1
Prologue
This report is addressed to a fundamental problem in the theory of ordinary differ ential equations which in its simplest form reads as follows. Consider an autonomous system of differential equations of the form x = Ax + /(x) where A is a linear operator and / a nonlinear mapping with /(0) = 0. Under the assumption that some decomposition of the x-space into subspaces is known to be invariant with respect to the linear system x = Ax the question arises if there exists a related decomposition of the phase space which is invariant with respect to the given nonlinear system. Since the times of Poincare and Lyapunov this fundamental question of the qualitative theory of differential equa tions has gained much attention and — at least in the context of finite-dimensional ordinary differential equations — the terminal answers have been given in the local theory of invariant manifolds outlined by the following key words: stable and unstable manifolds, center, center-stable and center-unstable manifolds, reduction principle, 45
Bernd Aulb&ch k
46
Thomas Wanner
linearization and decoupling (cf. Grobman [10, 11], Hartman [12, 13], Kelley [17], Pliss [22], Sositaisvili [23]). In this article we aim at the development of a generalization of the local theory of invariant manifolds to nonautonomous differential equations of the form
x = A{t)x + f(t,x) where x lies in some Banach space and t enters the equation in a measurable (i.e. possibly discontinuous) fashion. As a matter of fact, it turns out that in connection with a possibly infinite-dimensional x-space one has to invoke to a stronger concept of measurability as the usual one. The reason for choosing this general setting (and therefore having to deal with some extra technicalities) is the perspective of broad ening the range of applications to a large extent such as e.g. in the stochastic theory of dynamical systems and in control theory. Most texts on ordinary differential equations in Banach spaces consider only equa tions with continuous ^-dependence (cf. Amann [1], Deimling [8], Dieudonne [9]). In the general infinite-dimensional context, however, there are only few results on mea surable t-dependence and those concern mostly linear equations (cf. Daleckii & Krein [7]). Since we do not know of any systematic treatment of nonlinear equations in this general framework, in this paper we state and prove all our results in full detail. Therefore this text is quite lengthy, even though we cannot present the complete theory, that is to say, that we present only the fundamentals but not the various applications. More to this subject will be said in the epilogue, the last section of this lecture. At this point we want to emphasize that this article is not only written for those who are familiar (or want to get acquainted) with the use of measurability in the context of infinite-dimensional differential equations. Those readers who are not familiar with (or are not interested in, or not willing to learn something about) this general concept may regard all of the functions encountered as continuous and/or may restrict the view to finite dimensions. Even then this paper provides a new and interesting approach to the local theory of invariant manifolds and its generalization to nonautonomous systems. The organization of this article is as follows: • Section 2 contains the basic definitions and results on nonautonomous differen tial equations in Banach spaces with measurable t-dependence. First, we prove a global existence and uniqueness theorem for the class of equations being con sidered in this paper. This theorem also provides the continuous dependence of solutions on initial values and parameters. Furthermore, linear differential equa tions, evolution operators, the variation of constants formula and an appropriate version of GronwalTs Lemma will be discussed. • In Section 3 we introduce the notion of quasibounded solutions which turns out to be essential for our approach. We prove several results on the existence and
1. Prologue
47
uniqueness of quasibounded solutions of linear and nonlinear differential equa tions. • Having these results at hand, we present in Section 4 our main theorem on the existence of integral manifolds. In view of later applications (which are not con tained in this lecture, however) this result will be proved for parameter dependent differential equations. • Section 5 contains existence theorems for certain families, so-called hierarchies of integral manifolds. • The existence of integral manifolds generalizing the stable, center-stable, center, center-unstable and unstable manifolds to our general setting is established in Section 6. • Finally, in Section 7 we demonstrate by means of a counterexample that a sixth integral manifold which canonically appears in linear systems does not exist in general for nonlinear systems. • In order to keep the presentation of this lecture as selfcontained as possible, in two appendices we collect (partly with proofs) the background material on measurable functions and fixed point theorems needed for the analysis in these notes. Concluding this introductory section we briefly outline the notation and mention a few conventions which are used throughout this paper without further notice. The Banach spaces we consider may be real or complex, of course of the same type if they appear in a common context. The corresponding norm will always be denoted by || • ||. If X is a Banach space, by L(X) we mean the Banach space of bounded linear operators on X equipped with the operator norm ||A|| := sup{||Ax|| G R : x G X, \\x\\ < 1}, and GL(X) denotes the group of invertible operators in L(X). For a matrix A, the spectrum (i.e. the set of eigenvalues) is denoted by cr(A) and if A is an eigenvalue of a matrix, Re A denotes its real part. For any subset S c R b y x s ^ R — ► { O j l } we denote the characteristic function, i.e. XsM = 1 if £ G S and Xs(0 = 0 if t e R \ 5 . Furthermore, we set R j := {x G R : x > 0} and R+ := {x G R : x > 0}. Unless otherwise stated, intervals may be of any type, bounded or unbounded, with or without endpoints. What measurability is concerned, in statements containing the phrase almost all we usually omit to indicate that the respective exceptional set is of measure zero in the Lebesgue sense. Instead of the derivative ^ we write x and in differential equations x = f(t, x) we often interpret — as in the theory of dynamical systems — x as state and t as time. Finally, for autonomous, i.e. time invariant equations x = f(x) we adopt the term invariant manifold in the usual sense as an invariant subset of the phase space with a manifold structure. 1 The corresponding generalization to nonautonomous equations (i.e. subsets of the extended phase space made up of integral curves) will be called integral manifolds. 1
In fact, the manifolds appearing in this paper are always graphs of suitable functions.
Bernd Aulbach & Thomas Wanner
48
2
Basic Definitions and Results
In this section we present the fundamental facts of the theory of ordinary differential equations of the form x = f(t, x) where x is an element of some Banach space and the dependence of the right-hand side on t is only measurable. It turns out that in this general setting one has to use the concept of strong measurability rather than the usual measurability introduced in standard calculus. For the sake of completeness we collect the basic facts concerning strong measurability in Appendix A. In the present section we use those results freely without further mention. For the theory of differential equations with measurable i-dependence the so-called Caratheodory property of the right-hand side turns out to be essential. In our setting this property reads as follows. Definition 2.1 Let I be a nonempty interval and X a Banach space. A mapping f : I x X —*■ X is then said to have the Caratheodory property if it satisfies the following two conditions: (a) For every t £ I the mapping /(£, •) : X —* X is
continuous.
(b) For every x 6 X the mapping f(-,x) : I —* X is strongly measurable2 respect to the Borel a-algebras on I and X).
(with
In order to develop a solution concept for differential equations whose right-hand sides have the Caratheodory property we need the following result. L e m m a 2.2 Suppose I is a nonempty interval and X a Banach space. Furthermore, let f : I x X —> X have the Caratheodory property and let \i : I —*■ X be strongly measurable. Then also the composed mapping t H-> /ft,//(£)] from I to X is strongly measurable. P r o o f : According to the strong measurability of // there exists a sequence of simple mappings fin : / —► X, n £ N , converging to \i pointwise, i.e. M * ) = £ * » . • ' • X/n,.W -» /*(*)
as n -* oo ,
where kn € N and xnti 6 X. The I„tl,... ,In,kn axe disjoint measurable sets whose union is / . Then the Definition 2.1 ^6^ and the relation
readily imply that the mappings *•-+/(*,//„(*)), 2
!->*,
n€N
For the definition and basic properties of strong measurability we refer to Appendix A.
2.
Basic Definitions and Results
49
are strongly measurable. Because of Definition 2.1 (a) the corresponding sequence of mappings converges pointwise to the mapping / ( • , /x(-)). Hence the latter mapping is strongly measurable. O W i t h Lemma 2.2 at hand we are now able to introduce the type of differential equations (with the corresponding solution concept) we are going to study in this paper. Definition 2.3 Given an interval J, a Banach space X and a topological space V, let f : I X X x V —> X be a mapping such that for every choice ofp£V the mapping / ( • , -,p) : I x X —> X has the Caratheodory property. Then an equation of the form
x = f{t,x,p)
(1)
is called a nonautonomous (parameter dependent) differential equation and a contin uous mapping X : J —► X defined on some nonempty interval J C I is called a solution of (1) (to the parameter value p), if the mapping / ( • , A(-),p) is locally integrable3 and if the identity X(t)-\(S)
=
Jj{T,\{T),p)dT
is valid for all t,s € J. If in addition the mapping X satisfies the condition A(T 0 ) = £0 for some TQ G J and £o € X, then X is said to satisfies the initial condition X{T0)
= £o
(2)
or we say that X solves the initial value problem (1), (2). As in the theory of ordinary differential equations in RN it is possible to prove local existence and uniqueness results for nonautonomous differential equations in arbitrary Banach spaces, provided the right-hand side / satisfies certain conditions (see Amann [1, Ch. 11,7], Dieudonne [9, Ch. X,4], or Zeidler [28, Ch. 3,3]). However, the equations occuring in these notes are of quite a particular structure which allows us to prove stronger results on existence, uniqueness and continuous dependence of solutions on initial values and parameters. The corresponding results are contained in the following theorem. T h e o r e m 2.4 Given an interval I, a Banach space X and a topological space V, let f : I X X X V -* X be such that for any p G V the mapping /(•*,-, p) has the 3
In this context we have to use the notion of Bochner integral which is the straightforward generalization of the Lebesgue integral to Banach space valued strongly measurable mappings. For more details we refer to Appendix A or to the books of Cohn [5, Appendix E] or Craven [6, Section 7.4]. Note also that a continuous mapping A is strongly measurable, so Lemma 2.2 furnishes the strong measurability of the mapping /(•, X(-),p) appearing in this definition.
Bernd Aulbach & Thomas Wanner
50
Caratheodory property. Furthermore let £,£0 : I —► RJ be locally integrable functions such that the two estimates
\\f(t,xi,p)-f(t,xi,p)\\ < e(t)||x, -si||,
(3) (4)
||/(«,0,p)|| < to{t)
are valid for almost all t £ I and for all x\,x2 € X and p € V. Finally, suppose that the mapping / ( T , Z , - ) is continuous for any t £ I and x € X. Then for any (To,£o,Po) € / x X xV the differential equation
x =
f{t,x,p)
(5)
has a unique solution A(-; 7b,£o>Po) : I —* X to the parameter value po satisfying the initial condition x(r 0 ) = £o; and the so-defined mapping \:IxIxXxV—>X is continuous. Proof: Since every interval is the union of a nested, increasing sequence of compact intervals we only have to consider the special case of a compact interval I = [a,b]. Let B denote the Banach space of continuous mappings ft : I —> X with norm \\n\\\ := max{||/i(t)|| : t € / } , and define the mapping T: BxIxXxV —> B by T{lJ.,To,Zo,po){t) := & + / f(s,n(s),pQ)ds
for all t € J ,
•'7X)
where the existence of the integral is an immediate consequence of Lemma 2.2 and the estimate ||/(a,fiW,j*)|| < \\f(s,n(s),po)-f(s,0,po)\\ + \\f(syOypo)\\ < £(s)-\\\p\\\ + £o(s) for almost all 5 G / , together with the required local integrability of t and £0. The proof of Theorem 2.4 is divided into three parts. (I) We begin by inductively verifying the estimate \\Tn(fi,ro^o,Po)(t)-Tn(u,T0^0,po)(t)\\
< ±1
fi(s)ds
— v\
(6)
for all //, v G # , r, T0 € / , £o € X, p0 6 V and n € N, where the iterated operator Tn : B x I x X XV -> B is denned as in relation (101) of Appendix B. For n = 1 assumption (3) provides the estimate \\T(fi,T0^o,Po)(t)-T(u,rQ^o,Po){t)\\ If I
\\f{s,Ks),Po)-f{s,v{s),po)\\ds\
< I
JTQ
\fte(s)\\fi(s)-u(s)\\ds I
JTQ
) ,
(8)
which we call the cocycle property. This identity is valid since both sides of it axe solutions of (5) satisfying the initial condition x(ti) = A ^ j r , £ , p ) . The cocycle property (8), which obviously generalizes the well known group property for flows of autonomous differential equations, will prove to be crucial for the analysis in these notes. Let us now turn our attention to Gronwall's Lemma which is known to be fun damental for estimating the growth of solutions of differential equations. Since we study differential equations whose right-hand sides are only measurable with respect to t we have to use the following slightly generalized form of this lemma.
2. Basic Definitions and Results
53
Lemma 2.5 (Gronwall's Lemma) Suppose we are given a nonempty interval I, a point TQ G / and locally integrable functions u,a,b : i" —► RQ . If then the inequality u(t) < a(t)+
I
b(s)u(s)ds
(9)
is valid for all t £ I, then we get the estimate u{t) < a(t) +
/ 0(5)6(5) exp ( / b(a) da
ds
for all t € I
(10)
\ I J8
JTQ
Proof: Basically, the proof of the lemma follows the lines of the continuous case considered e.g. in Amann [1, pp. 89f ]. For the sake of completeness, however, we give a detailed proof which goes as follows. Set v{t) := fL b(s)u(s)ds for all t £ I. Then according to Craven [6, Proposition 5.4.5] the mapping v is differentiate for almost all i £ J with derivative v(t) = b(t)u(t). Furthermore, (9) furnishes for these values of t the inequality v(t) < a(t)b(t) + sgn(* - TQ)b(t)v(t) . Multiplying this estimate by the function c(t) := exp ( — / b(s) ds j = exp ( — / sgn(5 — TQ)b(s) ds) we get the inequality c(t)v(t) < a(t)b(t)c(t) for almost all t £ / , and therefore we have j ( a , ) ( t ) - a(t)b(t)c(t)
c(t)v(t)
< 0
for almost all t £ / . Integrating this inequality from r 0 to t and using the relation V(TO) = 0 we get the estimate c(t)v(t)
= j —{cvYs)ds
< I
a(s)b{s)c{s)ds
for alH £ / , and with the help of (9) we finally arrive at the desired estimate u(t)
< a(t) + sgn(t - r 0 ) • v(t) < a{t) +
sgn(t-T0)'J^a{s)b{s)^ds
= a{t) + / a(s)6(5)exp ( / b(a)da for every t £ I.
jds
O
As a first consequence of Gronwall's lemma we state and prove the following result which will be needed in the proof of our fundamental Theorem 4.1.
Bernd AuJbach k Thomas Wanner
54
Lemma 2.6 Consider the parameter dependent differential equation x =
f(t,x,p)
(11)
satisfying all the assumptions of Theorem 2.4, and let X(t; r, £,p) denote the cocycle of (11) guaranteed by this theorem. Moreover, let r 0 € / be arbitrary and let fio : V —> X be a bounded mapping, i.e. assume there is a constant M > 0 with IMiP)|| < M
for all peV
.
Then for every e > 0 there exists a 6 > 0 such that for any r E I with \T — T0\ < 6 and any p € V the estimate IWT;T0,/*O(P),P)-/*OO>)||
0 such that P(r;ro,/*o(p) } p)-A(f;r 0 ,/io(p),p)||
)||
^
IMP)|| + |
r\\fM*,VhP),p)\\d*
I •'TO
A{t)x + b(t),
IxX
-+ X
has the Caratheodory property, because A and b are strongly measurable and because the mapping (A,x) » Ax, L(X) xX -► X
Bernd Aulbach & Thomas Wanner
56 is even continuous. Since the relations lllA{t)Xl
+m -
[ A { t ) x 2
and
+ h m
\
< \\A(t)\\ • \\Xl - x2\\
\\A(tyO+b(t)\\
= \\b(t)\\
axe true for all i G I and xi,x2 G X, and since the functions ||^(-)ll a n ( l IIK")II a r e locally integrable, Theorem 2.4 can be applied with £(t) := ||A(t)|| and £0(t) := ||6(i)||. Thus, to every initial condition there exists exactly one solution of (12) on I. In order to gain a clearer picture of the cocycles of linear differential equations we make use of a construction generalizing the principal fundamental matrix of linear systems in RN. Definition 2.8 Given the framework of the preceding Definition 2.7, let A(f;r, £) denote the cocycle of the homogeneous equation (18). Then the mapping $(i,s) :=X(t]s,-)
:
X^X
which is defined for any t,s £ I is called the evolution operator of (13). The basic properties of the evolution operator axe described in the following lemma. L e m m a 2.9 Consider a homogeneous linear differential equation x = A(t)x
(14)
satisfying the assumptions of Definition 2.7. Then the corresponding evolution oper ator §(t,s)
:
X^X
is an element of GL(X), the group of invertible operators in L{X), and the mapping $ : / x I -+ GL{X) is continuous. Furthermore, for all i , s , r 6 I the identities *(*,*) = $(*,s) =
idx, $(t,r)o$(r,s),
•ft*)" 1 = *(M)
(15) (16)
(17)
are valid and the following estimate holds:
P M | | < exp(|jfVMII(X). Due to the continuity of the mapping {A,X)
H+ AX,
L{X)xL{X)
-»• L(X)
the mapping (t,X)
■-> A(t)X,
IxL(X)
-► L(X)
has the Caratheodory property and satisfies all assumptions of Theorem 2.4. Hence the initial value problem (19) has a uniquely determined solution $(•, s) : I —> L(X), and the induced mapping $ : I X J —» L(A') is continuous. Furthermore, for all t,s £ I and £ G # the properties of the Bochner integral (see Appendix A) imply Ht,s)Z-t
= [*{*,*)-*(*,j)]e = [ ^ A ( r ) i ( r , 3 ) d r ] e = jf
A(T)*(T,3)£ : R x X —* X is called the general solution or flow of the autonomous differential equation (22). For a pictorial representation of the solutions of autonomous equations it therefore suffices to draw, for given £o G X, the orbit or the trajectory through £o, i.e. the subset {ip{t,£o) € X : t £ R} of the phase space X. It follows immediately from the uniqueness of solutions and the above-mentioned "translation in time" that two orbits are either identical or disjoint.
3
Quasibounded Solutions
For the theory to come it is essential to study a special class of solutions of nonautonomous differential equations which are characterized by their asymptotic growth rates, so-called quasibounded solutions. In order to motivate the notion of quasiboundedness we consider finite-dimensional autonomous homogeneous linear differential equations x = Ax\
(24)
where A denotes an M x M-matrix over K = R or K = C. The corresponding evolution operator $(i,s) is then a mapping taking its values in the space L(K M ) which will be identified with the space KMxM of all M x M-matrices over K in the canonical way. For the explicit form of this evolution matrix or principal fundamental matrix we refer e.g. to the book of Hirsch & Smale [16]. With the aid of the matrix exponential function the evolution matrix $(£, s) has the form eA^~^ and therefore the cocycle of (24) reads A(*;T,{) =
e^"^.
Furthermore, for any real number a > max{ReA : A € cr(A)} , where a(A) C C denotes the spectrum of the matrix A and ReA the real part of the eigenvalue A, there is some constant K > 1 such that eMt-r)\\
T .
(25)
3. Quasibounded Solutions
61
Hence, for any r £ R and £ € K M we have ||A(t;r,OH < i n i £ | | e a ( t - r )
for all r > r .
So for a given a 6 R all solutions of (24) have the same exponential decay rate (depending on a) as t tends to infinity — provided the real parts of the eigenvalues of A satisfy certain conditions. This idea of quasiboundedness is to be generalized in the following definition, where we call an interval unbounded to the left, if it is of the form (—oo,r) or (—oo,r] for some r G E U o o . Intervals unbounded to the right are defined analogously. Definition 3.1 Suppose we are given an n 6 R, a Banach space X', an interval I and a mapping g : I —> X. Then (a) g is called n+-quasibounded or 77-quasibounded as t —* oo, if I is unbounded to the right and the inequality sup{||^(t)||e _,7t : t>r]
< oo
holds for some r £ I. In this case we define
llflllJ„:=sup{|| 9 (*)||e-":i>T}. (b) g is called ^"-quasibounded or 77-quasibounded as t —> — 00, if I is unbounded to the left and the inequality
sup{|| 0 (cf. Figure 1(a)). (b) For 77 = 0 the notion of 7/±-quasiboundedness coincides with the ordinary boundedness of ^-valued mappings. Furthermore, g is 0 + -quasibounded if and only if for some r £ R the restriction