Difference Equations and Their Applications

364 Pages·1993·9.796 MB·English

by  A. N. Sharkovsky, Yu. L. Maistrenko, E. Yu. Romanenko (auth.)

Difference Equations and Their Applications
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 250
Difference Equations
and Their Applications
by
A. N. Sharkovsky,
Yu. L. Maistrenko
and
E. Yu. Romanenko
Institute_of Mathematics,
Ukrainian Academy of Sciences,
Kiev, Ukraine
translated by
D.V. Malyshev. P.V. Malyshev and Y.M. Pestryakov
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
ISBN 978-94-010-4774-6 ISBN 978-94-011-1763-0 (eBook)
DOI 10.1007/978-94-011-1763-0
Printed on acid-free paper
This is an updated translation of the original work
Pa3HOCTHhle ypaBHeHHA H HX npUJlOlKeHHlI, Naukova Dumka © 1986
Ali Rights Reserved
© 1993 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1993
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
Contents
Notation IX
Preface Xl
Introduction 1
Part I. ONE-DIMENSIONAL DYNAMICAL SYSTEMS 15
1. Introduction to the Theory of Dynamical Systems 15
§1. Are One-Dimensional Dynamical Systems Simple? 17
§2. What May Occur in One-Dimensional Dynamical Systems.
Some Notions and Examples 23
§3. Intermixing (Strange) Attractors 37
2. Periodic Trajectories 45
§ 1. Attracting Fixed Points 45
§2. Coexistence of Cycles 52
§3 Bifurcations of Cycles 63
3. Behavior of Trajectories 71
§ 1. Trajectories of Simple Dynamical Systems 71
§2. Return of Points and Sets 75
§3. Criteria of Simplicity and Complexity for Maps 81
§4 Stability of Trajectories and Dynamical Systems 84
4. Dynamical Systems/or V-Maps 95
§1. Unimodal Maps 95
§2. Schwarzian and Attracting Cycles 98
§3. Periodic Intervals 101
§4. Spectral Decomposition of the Set of Non-Wandering Points 111
§5. Bifurcations of the Periodic Intervals and Stability
of the Spectral Decomposition 119
VI Contents
Part II. DIFFERENCE EQUATIONS WITH
CONTINUOUS TIME 125
1. Nonlinear Difference Equations 125
§ 1. Statement of the Problem 125
§2. Asymptotically Discontinuous Solutions 134
§3. Separator of a Map. The Simplest Properties of
Asymptotically Discontinuous Solutions 139
§4. The Limiting Semi-Group 143
§5. Asymptotic Behavior of Asymptotically
Discontinuous Solutions 152
§6. Stability of Asymptotically Discontinuous Solutions 155
2. Difference Equations with V-Nonlinearity 159
§ 1. Limiting Semi-Group, Separator, Spectrum of Jumps 159
§2. Spectrum of Asymptotic Jumps. Solutions of
Relaxation and Turbulent Types 163
§3. Stability and Bifurcations of Solutions 175
§4. Emergence of Ordered Structures 183
Part III. DIFFERENTIAL-DIFFERENCE EQUATIONS 187
1. Completely Integrable Differential-Difference Equations 188
§ 1. What Kind of Asymptotic Behavior of Solutions of
Differential-Difference Equations One May Expect 188
§2. Equations with a Decomposable Operator. Completely
Integrable Differential-Difference Equations 193
§3. Connection Between Solutions of Completely Integrable
Differential-Difference Equations and Solutions of the
Corresponding Difference Equations. The Phase Plane Method 197
§4. Periodic Solutions of Completely Integrable Differential
Difference Equations. Their Exceptional Character 206
§5. Asymptotically Periodic Solutions of Completely Integrable
Equations. Their Typicalness 211
2. Differential-Difference Equations Close To Difference Ones 223
§1. General Definitions and Properties 223
§2. Asymptotic Behavior of Solutions of the Perturbed Equation 227
§3. Stability of Solutions 234
Contents vii
3. Singularly Perturbed Differential-Difference Equations 239
§ 1. Statement of the Problem. Continuous Dependence of
Solutions on the Parameter v on a Finite Interval 239
§2. Invariance of the Asymptotic Properties of Solutions 243
§3. Behavior of Solutions as t --7 00 248
Part IV. BOUNDARY- VALUE PROBLEMS FOR
HYPERBOLIC SYSTEMS OF PARTIAL
DIFFERENTIAL EQUATIONS 273
1. Reduction of Boundary-Value Problems to Difference and
Differential-Difference Equations 274
§ 1. Reduction to a Nonlinear Difference Equation 274
§2. Reduction to Differential-Difference Equations 277
§3. Reduction Procedure in More General Situations 279
2. Boundary-Value Problem for a System with Small Parameter 285
§1. Boundary Value Problem for a Non-Perturbed System 285
§2. The Case of E > O. Existence of Solution and Extendability 290
§3. Stability in the Hausdorff Metric 292
§4. Stability in the Skorokhod Metric and Asymptotic Periodicity 297
3. Boundary-Value Problem for Systems with Two Spatial Variables 305
§ 1. Statement of the problem. Its Correctness. Reduction to a
Difference Equation 305
§2. Solution of the Boundary-Value Problem in the Case of
Linear Boundary Conditions 310
§3. Nonlinear Boundary Conditions. Exclusion of the "Mean Row" 313
§4. Asymptotic Behavior of Solutions as t --7 00 318
§5 Self-Stochasticity 330
References 335
Index 357
nOTRTlon
Pi the.set of positive integers;
jl(jl+) the set of integer (nonnegative integer) numbers;
JR(JR +) the set of real (nonnegative real) numbers;
if the closure of the set A;
U/)(A) &-neighborhood of the set A;
mesA the Lebesgue measure of the set A;
I closed bounded interval;
r
nth iteration of the map f;
flA restriction of the map f onto the set A;
2x the space of closed subsets of the space X;
ro(x) the set of ro-limiting points of a trajectory which passes through
the point X;
Fixf the set of fIxed points;
Perf the set of periodic points;
Qif) the set of non-wandering points;
B if) the set of weakly non-wandering points;
D if) the separator;
pr (x) prolongation of the point x with respect to the initial data;
pr (x,f) prolongation of the point x with respect to the dynamical system;
Q(x) ro-prolongation (domain of influence) of the point x;
10m the set of CO-maps having cycles with period m;
C(X, y) the space of Cr -smooth functions from X onto Y with
uniform metric for derivatives;
t\ (A, B) the Hausdorff distance between the sets A and B;
grf the graph of the function f;
t\{f, g} the Hausdorff distance between grf and gr g;
Cll(X, y) the space of upper semi-continuous functions from X onto 2 y
with the metric given by the Hausdorff distance ll{· , .};
sif, g) the Skorokhod distance.
ix
Preface
The theory of difference equations is now enjoying a period of Renaissance. Witness the
large number of papers in which problems, having at first sight no common features, are
reduced to the investigation of subsequent iterations of the maps f· IR. m ~ IR. m, m > 0, or
(which is, in fact, the same) to difference equations
The world of difference equations, which has been almost hidden up to now, begins to
open in all its richness. Those experts, who usually use differential equations and, in fact,
believe in their universality, are now discovering a completely new approach which re
sembles the theory of ordinary differential equations only slightly. Difference equations,
which reflect one of the essential properties of the real world-its discreteness-rightful
ly occupy a worthy place in mathematics and its applications.
The aim of the present book is to acquaint the reader with some recently discovered
and (at first sight) unusual properties of solutions for nonlinear difference equations.
These properties enable us to use difference equations in order to model complicated os
cillating processes (this can often be done in those cases when it is difficult to apply
ordinary differential equations). Difference equations are also a useful tool of syn
ergetics- an emerging science concerned with the study of ordered structures. The
application of these equations opens up new approaches in solving one of the central
problems of modern science-the problem of turbulence.
Our presentation is mainly based on the modern theory of one-dimensional dynami
cal systems interest in which has considerably grown recently. The first part is devoted to
this theory. We consider the coexistence of periodic trajectories and their bifurcations,
the spectral decomposition of the set of non-walking points, strange attractors, repellers,
and the stability both of certain trajectories and of dynamical systems as a whole.
In the second part, we describe the asymptotic properties of the solutions of diffe
rence equations with continuous argument. We select solutions of relaxation and turbu
lent types; the characteristic feature of the latter is their convergence in the Hausdorff
metric to generalized solutions whose graphs are fractal sets (according to Mandelbrot).
The third part is devoted to differential-difference equations which are close to diffe
rence ones. Here we study the following problems: to what extent do the equations of
this sort inherit the properties of difference equations, and how do they change under
regular and singular perturbations ?
xi
xii Preface
In the fourth part, we develop the method of investigation of nonlinear boundary-va
lue problems for hyperbolic systems which is based on the reduction of these to diffe
rence and differential-difference equations. We study the stability and asymptotic beha
vior of solutions. We also consider classes of problems with chaotic dynamics, i.e., prob
lems whose behavior for large time can be described only in terms of stochastic differen
tial equations, but not in terms of deterministic equations.
Many results presented in this monograph are published for the first time. To our
regret, our efforts to arrange the presentation systematically are not always successful. In
fact, some sections resemble a review, where statements are often given without proofs,
or where the proofs are only outlined.
The reader needs no deep knowledge of special branches of mathematics. Despite
this, however, it will be helpful for the reader to know the fundamentals of the qualitative
theory of differential equations.
The authors are grateful to Drs. A.G. Sivak, V.V. Fedorenko, S.Ya. Aliev, A.F. Iva
nov, and S.F. Kolyada who contributed to the preparation and writing of some sections.
We also thank Dr. V.L. Maistrenko for the numerical calculations and computer gra
phics.
The authors