Cover
Title Page
Half Title
Copyright Page
Preface to the First Edition
Preface to the Second Edition
Dedication Page
Table of Contents
Chapter I: Introduction
1.1 Population Growth Models, One Population
1.2 Iteration of Real Valued Functions as Dynamical Systems
1.3 Higher Dimensional Systems
1.4 Outline of the Topics of the Chapters
Chapter II: One-Dimensional Dynamics by Iteration
2.1 Calculus Prerequisites
2.2 Periodic Points
2.2.1 Fixed Points for the Quadratic Family
2.3 Limit Sets and Recurrence for Maps
2.4 Invariant Cantor Sets for the Quadratic Family
2.4.1 Middle Cantor Sets
2.4.2 Construction of the Invariant Cantor Set
2.4.3 The Invariant Cantor Set for ยต > 4
2.5 Symbolic Dynamics for the Quadratic Map
2.6 Conjugacy and Structural Stability
2.7 Conjugacy and Structural Stability of the Quadratic Map
2.8 Homeomorphisms of the Circle
2.9 Exercises
Chapter III: Chaos and Its Measurement
3.1 Sharkovskiiโs Theorem
3.1.1 Examples for Sharkovskiiโs Theorem
3.2 Subshifts of Finite Type
3.3 Zeta Function
3.4 Period Doubling Cascade
3.5 Chaos
3.6 Liapunov Exponents
3.7 Exercises
Chapter IV: Linear Systems
4.1 Review: Linear Maps and the Real Jordan Canonical Form
4.2 Linear Differential Equations
4.3 Solutions for Constant Coefficients
4.4 Phase Portraits
4.5 Contracting Linear Differential Equations
4.6 Hyperbolic Linear Differential Equations
4.7 Topologically Conjugate Linear Differential Equations
4.8 Nonhomogeneous Equations
4.9 Linear Maps
4.9.1 Perron-Frobenius Theorem
4.10 Exercises
Chapter V: Analysis Near Fixed Points and Periodic Orbits
5.1 Review: Differentiation in Higher Dimensions
5.2 Review: The Implicit Function Theorem
5.2.1 Higher Dimensional Implicit Function Theorem
5.2.2 The Inverse Function Theorem
5.2.3 Contraction Mapping Theorem
5.3 Existence of Solutions for Differential Equations
5.4 Limit Sets and Recurrence for Flows
5.5 Fixed Points for Nonlinear Differential Equations
5.5.1 Nonlinear Sinks
5.5.2 Nonlinear Hyperbolic Fixed Points
5.5.3 Liapunov Functions Near a Fixed Point
5.6 Stability of Periodic Points for Nonlinear Maps
5.7 Proof of the Hartman-Grobman Theorem
5.7.1 Proof of the Local Theorem
5.7.2 Proof of the Hartman-Grobman Theorem for Flows
5.8 Periodic Orbits for Flows
5.8.1 The Suspension of a Map
5.8.2 An Attracting Periodic Orbit for the Van der Pol Equations
5.8.3 Poincarรฉ Map for Differential Equations in the Plane
5.9 Poincarรฉ-Bendixson Theorem
5.10 Stable Manifold Theorem for a Fixed Point of a Map
5.10.1 Proof of the Stable Manifold Theorem
5.10.2 Center Manifold
5.10.3 Stable Manifold Theorem for Flows
5.11 The Inclination Lemma
5.12 Exercises
Chapter VI: Hamiltonian Systems
6.1 Hamiltonian Differential Equations
6.2 Linear Hamiltonian Systems
6.3 Symplectic Diffeomorphisms
6.4 Normal Form at Fixed Point
6.5 KAM Theorem
6.6 Exercises
Chapter VII: Bifurcation of Periodic Points
7.1 Saddle-Node Bifurcation
7.2 Saddle-Node Bifurcation in Higher Dimensions
7.3 Period Doubling Bifurcation
7.4 Andronov-Hopf Bifurcation for Differential Equations
7.5 Andronov-Hopf Bifurcation for Diffeomorphisms
7.6 Exercises
Chapter VIII: Examples of Hyperbolic Sets and Attractors
8.1 Definition of a Manifold
8.1.1 Topology on Space of Differentiable Functions
8.1.2 Tangent Space
8.1.3 Hyperbolic Invariant Sets
8.2 Transitivity Theorems
8.3 Two-Sided Shift Spaces
8.3.1 Subshifts for Nonnegative Matrices
8.4 Geometric Horseshoe
8.4.1 Horseshoe for the Hรฉnon Map
8.4.2 Horseshoe from a Homoclinic Point
8.4.3 Nontransverse Homoclinic Point
8.4.4 Homoclinic Points and Horseshoes for Flows
8.4.5 Melnikov Method for Homoclinic Points
8.4.6 Fractal Basin Boundaries
8.5 Hyperbolic Toral Automorphisms
8.5.1 Markov Partitions for Hyperbolic Toral Automorphisms
8.5.2 Ergodicity of Hyperbolic Toral Automorphisms
8.5.3 The Zeta Function for Hyperbolic Toral Automorphisms
8.6 Attractors
8.7 The Solenoid Attractor
8.7.1 Conjugacy of the Solenoid to an Inverse Limit
8.8 The DA Attractor
8.8.1 The Branched Manifold
8.9 Plykin Attractors in the Plane
8.10. Attractor for the Hรฉnon Map
8.11 Lorenz Attractor
8.11.1 Geometric Model for the Lorenz Equations
8.11.2 Homoclinic Bifurcation to a Lorenz Attractor
8.12 Morse-Smale Systems
8.13 Exercises
Chapter IX: Measurement of Chaos in Higher Dimensions
9.1 Topological Entropy
9.1.1 Proof of Two Theorems on Topological Entropy
9.1.2 Entropy of Higher Dimensional Examples
9.2 Liapunov Exponents
9.3 Sinai-Ruelle-Bowen Measure for an Attractor
9.4 Fractal Dimension
9.5 Exercises
Chapter X: Global Theory of Hyperbolic Systems
10.1 Fundamental Theorem of Dynamical Systems
10.1.1 Fundamental Theorem for a Homeomorphism
10.2 Stable Manifold Theorem for a Hyperbolic Invariant Set
10.3 Shadowing and Expansiveness
10.4 Anosov Closing Lemma
10.5 Decomposition of Hyperbolic Recurrent Points
10.6 Markov Partitions for a Hyperbolic Invariant Set
10.7 Local Stability and Stability of Anosov Diffeomorphisms
10.8 Stability of Anosov Flows
10.9 Global Stability Theorems
10.10 Exercises
Chapter XI: Generic Properties
11.1 Kupka-Smale Theorem
11.2 Transversality
11.3 Proof of the Kupka โ Smale Theorem
11.4 Necessary Conditions for Structural Stability
11.5 Nondensity of Structural Stability
11.6 Exercises
Chapter XII: Smoothness of Stable Manifolds and Applications
12.1 Differentiable Invariant Sections for Fiber Contractions
12.2 Differentiability of Invariant Splitting
12.3 Differentiability of the Center Manifold
12.4 Persistence of Normally Contracting Manifolds
12.5 Exercises
References
Index