Dynamics of One-Dimensional Maps

268 Pages·1997·5.87 MB·English

by  A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak, V. V. Fedorenko (auth.)

Dynamies of One-Dimensional Maps
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centrefor Mathematics antI Computer Science, Amsterdam, The NetherlantIs
Volume 407
Dynamics of
One-Dimensional Maps
by
A.N. Sharkovsky
S.F. Kolyada
A.G. Sivak
and
v. v. Fedorenko
Institute o[ Mathematics.
Ukrainian Academy o[ Sciences.
Kiev. Ukraine
Springer-Science+Business Media, B.Y.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-4846-2 ISBN 978-94-015-8897-3 (eBook)
DOI 10.1007/978-94-015-8897-3
This is a completely revised and updated translation of
the original Russian work of the same title,
published by Naukova Dumka, Kiev, 1989.
Translated by A.G. Sivak, P. Malyshev and D. Malyshev
Printed on acid-free paper
All Rights Reserved
@1997 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1997.
Sot1:cover reprint of the hardcover 1s t edition 1997
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
Introduction vii
1. Fundamental Concepts of the Theory of Dynamical Systems.
Typical Examples and Some Results 1
1.1. Trajectories of One-Dimensional Dynamical Systems 1
1.2. ffi-Limit and Statistically Limit Sets. Attractors and Quasiattractors 18
1.3. Return of Points and Sets 25
2. Elements of Symbolic Dynamics 35
2.1. Concepts of Symbolic Dynamics 35
2.2. Dynamical Coordinates and the Kneading Invariant 40
2.3. Periodic Points, 1;-Function, and Topological Entropy 44
2.4. Kneading Invariant and Dynamics of Maps 49
3. Coexistence of Periodic Trajectories 55
3.1. Coexistence of Periods of Periodic Trajectories 55
3.2. Types of Periodic Trajectories 64
4. Simple Dynamical Systems 69
4.1. Maps without Periodic Points 69
4.2. Simple Invariant Sets 74
4.3. Separation of All Maps into Simple and Complicated 78
4.4. Return for Simple Maps 86
4.5. Classification of Simple Maps According to the Types of Return 100
4.6. Properties of Individual Classes 107
v
vi Contents
5. Topological Dynamics of Unimodal Maps 117
5.1. Phase Diagrams of Unimodal Maps 117
5.2. Limit Behavior of Trajectories 124
5.3. Maps with Negative Schwarzian 137
5.4. Maps with Nondegenerate Critical Point 150
6. Metrie Aspeets of Dynamics 161
6.1. Measure ofthe Set ofLyapunov Stable Trajectories 161
6.2. Conditions far the Existence of Absolutely Continuous Invariant
Measures 165
6.3. Measure of Repellers and Attractors 170
7. Loeal Stability of Invariant Sets. Struetural Stability of Unimodal Maps 183
7.1. Stability of Simple Invariant Sets 183
7.1.1. Stability of Periodic Trajectories 183
7.1.2. Stability of Cycles of Intervals 187
7.2. Stability ofthe Phase Diagram 190
7.2.1. Classification of Cycles of Intervals and Their Coexistence 190
7.2.2. Conditions far the Preservation of Central Vertices 194
7.3. Structural Stability and Q-stability of Maps 196
8. One-Parameter Families ofUnimodal Maps 201
8.1. Bifurcations of Simple Invariant Sets 201
8.2. Properties of the Set of Bifurcation Values. Monotonicity Theorems 205
8.3. Sequence of Period Doubling Bifurcations 207
8.4. Rate of Period Doubling Bifurcations 216
8.5. Universal Properties of One-Parameter Families 223
Referenees 239
Subject Index 259
Notation 261
In TRODUCTIon
Last decades are marked by the appearance of a permanently increasing number of scien-
tific and engineering problems connected with the investigation of nonlinear processes
and phenomena. It is now dear that nonlinear processes are not exceptional; on the con-
trary, they can be regarded as a typical mode of existence of matter. At the same time,
independently of their nature, these processes are often characterized by similar intrinsic
mechanisms and admit universal approaches to their description.
As a result, we observe fundamental changes in the methods and tools used for math-
ematical simulation. Today, parallel with well-known methods studied in textbooks and
special monographs for many years, mathematical simulation often employs the results
of nonlinear dynarnics-a new rapidly developing field of natural sciences whose math-
ematical apparatus is based on the theory of dynamical systems.
The extensive development of nonlinear dynamics observed nowadays is explained
not only by increasing practical needs but also by new possibilities in the analysis of a
great variety of nonlinear models discovered for last 20 years. In this connection, a deci-
sive role was played by simple nonlinear systems, discovered by physicists and mathem-
aticians, which, on the one hand, are characterized by quite complicated dynamics but,
on the other hand, admit fairly complete qualitative analysis. The analysis of these sys-
tems (both qualitative and numerical) revealed many common regularities and essential
features of nonlinearity that should be kept in mind both in constructing new nonlinear
mathematical models and in analyzing these models. Among these features, one should,
first of all, mention stochastization and the emergence of structures (the relevant
branches of science are called the theory of strange attractors and synergetics, respecti-
vely).
The theory of one-dimensional dynamical systems is one of the most efficient tools
of nonlinear dynamics because, on the one hand, one-dimensional systems can be de-
scribed fairly completely and, on the other hand, they exhibit all basic complicated non-
linear effects. The investigations in the theory of one-dimensional dynamical systems
gave absolutely new results in the theory of difference equations, difference-differential
equations, and some dasses of differential equations. Thus, significant successes were
attained in constructing new types of solutions, which can be efficiently used in simulat-
ing the processes of emergence of ordered coherent structures, the phenomenon of inter-
mittence, and self-stochastic modes. Significant achievements in this field led to the ap-
pearance of a new direction in the mathematical theory of turbulence based on the use of
vii
viii lntroduction
nonlinear difference equations and other equations (c1ose to nonlinear difference equa-
tions) as mathematical tools.
It is c1ear that iterations of continuous maps of an interval into itself are very simple
dynamical systems. It may seem that the use of one-dimensional dynamical systems
substantially restricts our possibilities and the natural ordering of points in the real line
may result in the absence of some types of dynamical behavior in one-dimensional sys-
tems. However, it is weIl known that even quadratic maps from the family x ~ x 2 + A.
may have infinitely many periodic points for some values of the parameter A.. Further-
more, for A. = - 2, the map possesses an invariant measure absolutely continuous with
respect to the Lebesgue measure, i.e., for this map, "stochastic" behavior is a typical be-
havior of bounded sequences of iterations. ActuaIly, the trajectories of one-dimensional
maps exhibit an extremely rich picture of dynamical behavior characterized, on the one
hand, by stable fixed points and periodic orbits and, on the other hand, by modes which
are practically indistinguishable from random processes being, at the same time, abso-
lutely deterministic.
This book has two principal goals: First, we try to make the reader acquainted with
the fundamentals of the theory of one-dimensional dynamical systems. We study, as a
rule, very simple nonlinear maps with a single point of extremum. Maps of this sort are
usually called unimodal. It turns out that unimodality imposes practically no restric-
tions on the dynamical behavior.
The second goal is to equip the reader with a more or less comprehensive outlook on
the problems appearing in the theory of dynamical systems and describe the methods us-
ed for their solution in the case of one-dimensional maps.
To understand distinctive features of topological dynamics on an interval on a more
profound level, the reader must not only study the formulations of the results but also
carefully analyze their proofs. Unfortunately, the size of the book is limited and, there-
fore, some theorems are presented without proofs.
This book does not contain special historical notes; only basic facts given in the form
of theorems contain references to their authors. Almost all results are achievements of
the last 20-30 years. The interest to the qualitative investigation of iterations of continu-
ous and discontinuous functions of a real variable was growing since 1930 s when ap-
plied problems requiring the study of such iterations appeared. However, these investi-
gations were not carried out systematically till 1970 s. The results of many authors
worked at that time are now weIl known. We would like to mention here less known
works of Barna [1], Leonov [1-3], and Pulkin [1, 2], which also contain many important
results.
In Chapter 1, following Sharkovsky, Maistrenko, and Romanenko [2], we give an
elementary introduction to the theory of one-dimensional maps. This chapter contains an
exposition of basic concepts of the theory of dynamical systems and numerous examples
illustrating various situations encountered in the investigation of one-dimensional maps.
Chapter 2 deals with the methods of symbolic dynamics. In particular, it contains a
presentation of the basic concepts and results of the theory of kneading invariants for
unimodal maps.
In Chapters 3 and 4, we prove theorems on coexistence of periodic trajectories. The
Introduction ix
maps whose topological entropy is equal to zero (i.e., maps that have only cyeles of pe-
riods 1,2,22, ... ) are studied in detail and elassified.
Various topological aspects of the dynamics of unimodal maps are studied in Chap-
ter 5. We analyze the distinctive features of the limiting behavior of trajectories of
smooth maps. In particular, for some elasses of smooth maps, we establish theorems on
the number of sinks and study the problem of existence of wandering intervals.
In Chapter 6, for a broad elass of maps, we prove that almost all points (with respect
to the Lebesgue measure) are attracted by the same sink. Our attention is mainly focused
on the problem of existence of an invariant measure absolutely continuous with respect
to the Lebesgue measure. We also study the problem of Lyapunov stability of dynamical
systems and determine the measures of repelling and attracting invariant sets.
The problem of stability of separate trajectories under perturbations of maps and the
problem of structural stability of dynamical systems as a whole are discussed in Chap-
ter 7.
In Chapter 8, we study one-parameter families of maps. We analyze bifurcations of
periodic trajectories and properties of the set of bifurcation values of the parameter, in-
eluding universal properties such as Feigenbaum universality.
Unfortunately, in the present book, we do not consider the maps of a cirele onto itself
and the maps of the complex plane onto itself. Some results established for maps of an
interval onto itself are related to the dynamics of rational endomorphisms of the Riemann
sphere: The beauty of the dynamics of the considered maps ofthe real line onto itself
from the family x ~ x 2 + A, A E lR, becomes visible (in the direct meaning of this
word) if we pass to the farnily z ~ Z2 + A, where z is a complex variable and A is a
complex parameter (see Peitgen and Richter [1]).
We hope that our book will be useful for everybody who is interested in nonlinear
dynarnics.