Preface for the Second Edition
Preface
1 Introduction
1.1 Dynamical Systems
1.1.1 Phase Space and Orbits
1.1.2 Parameters and Bifurcation of Dynamical Behavior
1.1.3 Examples of Dynamical Systems
1.2 Symbolic Dynamics as Coarse—Grained Description of Dynamics
1.2.1 Fine—Grained and Coarse—Grained Descriptions
1.2.2 Symbolic Dynamics as the Simplest Dynamics
1.3 Abstract versus Applied Symbolic Dynamics
1.3.1 Abstremt Symbolic Dynamics
1.3.2 Applied Symbolic Dynamics
1.4 Literature on Symbolic Dynamics
2 Symbolic Dynamics of Unimodal Maps
2.1 Symbolic Sequences in Unimodal Maps
2.1.1 Numerical Orbit and Symbolic Sequence
2.1.2 Symbolic Sequence and Functional Composition
2.1.3 The Word—Lifting Technique
2.2 The Quadratic Map
2.2.1 An Over—Simplified Population Model
2.2.2 Bifurcation Diagram of the Quadratic Map
2.2.3 Dark Lines in the Bifurcation Diagram
2.3 Ordering of Symbolic Sequences and the Admissibility
Condition
2.3.1 Property of Monotone Functions
2.3.2 The Ordering Rule
2.3.3 Dynamical Invariant Range and Kneading Sequence
2.3.4 The Admissibility Condition
2.4 The Periodic Window Theorem
2.4.1 The Periodic Window Theorem
2.4.2 Construction of Median Words
2.4.3 The MSS Table of Kneading Sequences
2.4.4 Nomenclature of Unstable Periodic Orbits
2.5 Composition Rules
2.5.1 The *—Composition
2.5.2 Generalized Composition Rule
2.5.3 Proof of the Generalized Composition Rule
2.5.4 Applications of the Generalized Composition Rule
2.5.5 Further Remarks on Composition Rules
2.6 Coarse—Grained Chaos
2.6.1 Chaos in the Surjective Unimodal Map
2.6.2 Chaos in Maps
2.7 Topological Entropy
2.8 Piecewise Linear Maps and Metric Representation of Symbolic Sequences
2.8.1 The Tent Map and Shift Map
2.8.2 The A—Expansion of Real Numbers
2.8.3 Characteristic Function of the Kneading Sequence
2.8.4 Mapping of Subintervals and the Stefan Matrix
2.8.5 Markov Partitions and Generating Partitions
2.8.6 Metric Representation of Symbolic Sequences
2.8.7 Piecewise Linear Expanding Map
3 Maps with Multiple Critical Points
4 Symbolic Dynamics of Circle Maps
5 Symbolic Dynamics of Two—Dimensional Maps
6 Application to Ordinary Differential Equations
7 Counting the Number of Periodic Orbits
8 Symbolic Dynamics and Grammatical Complexity
9 Symbolic Dynamics and Knot Theory
10 Appendix
References
Index
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