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Dynamics of One-Dimensional Maps : Mathematics and Its Applications - A.N. Sharkovskii

Dynamics of One-Dimensional Maps

Mathematics and Its Applications


Published: 30th April 1997
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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.

Fundamental Concepts of the Theory of Dynamical Systems. Typical Examples and Some Resultsp. 1
Elements of Symbolic Dynamicsp. 35
Coexistence of Periodic Trajectoriesp. 55
Simple Dynamical Systemsp. 69
Topological Dynamics of Unimodal Mapsp. 117
Metric Aspects of Dynamicsp. 161
Local Stability of Invariant Sets. Structural Stability of Unimodal Mapsp. 183
One-Parameter Families of Unimodal Mapsp. 201
Referencesp. 239
Subject Indexp. 259
Notationp. 261
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780792345329
ISBN-10: 0792345320
Series: Mathematics and Its Applications
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 262
Published: 30th April 1997
Publisher: Springer
Country of Publication: NL
Dimensions (cm): 23.4 x 15.6  x 2.54
Weight (kg): 1.25