Synopses & Reviews
The theory of one-dimensional systems is one of the most efficient tools of nonlinear dynamics, as, on the one hand, it describes one-dimensional systems fairly completely, and on the other hand exhibits all basic complicated nonlinear effects. This volume has two main goals. Firstly, it acquaints the reader with the fundamentals of the theory of one-dimensional dynamical systems. Very simple nonlinear maps with a single point of extremum, also called unimodal maps, are studied. Unimodality is found to impose hardly any restrictions on the dynamical behaviour. Secondly, it equips the reader with a comprehensive view of the problems appearing in the theory of dynamical systems and describes the methods used for their solution in the case of one-dimensional maps. Audience: This book will be of interest to researchers and postgraduate students whose work involves nonlinear dynamics.
Synopsis
maps whose topological entropy is equal to zero (i.e., maps that have only cyeles of pe- 2 riods 1,2,2, ... ) 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.
Table of Contents
Introduction.
1. Fundamental Concepts of the Theory of Dynamical Systems. Typical Examples and Some Results.
2. Elements of Symbolic Dynamics.
3. Coexistence of Periodic Trajectories.
4. Simple Dynamical Systems.
5. Topological Dynamics of Unimodal Maps.
6. Metric Aspects of Dynamics.
7. Local Stability of Invariant Sets. Structural Stability of Unimodal Maps.
8. One-Parameter Families of Unimodal Maps. References. Subject Index. Notation.