DYNAMICAL SYSTEMS. I Clark: Robinson. Stability, Symbolic Dynamics, and Chaos. CRC Press Boca Raton Ann Arbor London Tokyo

Transcription

1 DYNAMICAL SYSTEMS Stability, Symbolic Dynamics, and Chaos I Clark: Robinson CRC Press Boca Raton Ann Arbor London Tokyo

2 Contents Chapter I. Introduction Population Growth Models, One Population Iteration of Real Valued Functions as Dynamical Systems Higher Dimensional Systems Outline of the Topics of the Chapters 9 Chapter II. One Dimensional Dynamics by Iteration Calculus Prerequisites 13 *2.2 Periodic Points 15 *2.2.1 Fixed Points for the Quadratic Family 20 *2.3 Limit Sets and Recurrence for Maps 22 *2.4 Invariant Cantor Sets for the Quadratic Family 26 *2.4,1 Middle Cantor Sets 26 *2.4.2 Construction of the Invariant Cantor Set The Invariant Cantor Set for n > 4 33 *2.5 Symbolic Dynamics for the Quadratic Map 37 *2.6 Conjugacy and Structural Stability 40 *2.7 Conjugacy and Structural Stability of the Quadratic Map Homeomorphisms of the Circle Exercises 57 Chapter III. Chaos and Its Measurement Sharlravskii's Theorem Examples for Sharkovskii's Theorem Subshifts of Finite Type Zeta Function Period Doubling Cascade Chaos Liapunov Exponents Exercises 88 Chapter IV. Linear Systems Review: Linear Maps and the Real Jordan Canonical Form 93 *4.2 Linear Differential Equations 95 *4.3 Solutions for Constant Coefficients 97 *4.4 Phase Portraits ' 102 *4.5 Contracting Linear Differential Equations 106 *4.6 Hyperbolic Linear Differential Equations 111 *4.7 Topologically Conjugate Linear Differential Equations 113 *4.8 Nonhomogeneous Equations 115 *4.9 Linear Maps Perron-Frobenius Theorem Exercises 127 * Core Sections

3 Chapter V. Analysis Near Fixed Points and Periodic Orbits 131 *5.1 Review: Differentiation in Higher Dimensions 131 *5.2 Review: The Implicit Function Theorem 134 *5.2.1 Higher Dimensional Implicit Function Theorem 136 *5.2.2~ The Inverse Function Theorem 137 *5.2.3 Contraction Mapping Theorem 138 *5.3 Existence of Solutions for Differential Equations 140 *5.4 Limit Sets and Recurrence for Flows 146 *5.5 Fixed Points for Nonlinear Differential Equations 149 *5.5.1 Nonlinear Sinks 150 *5.5.2 Nonlinear Hyperbolic Fixed Points 152 *5.5.3 Liapunov Functions Near a Fixed Point 154 *5.6 Stability of Periodic Points for Nonlinear Maps 156 *5.7 Proof of the Hartman-Grobman Theorem 158 *5.7.1 Proof of the Local Theorem Proof of the Hartman-Grobman Theorem for Flows 165 *5.8 Periodic Orbits for Flows The Suspension of a Map An Attracting Periodic Orbit for the Van der Pol Equations Poincare Map for Differential Equations in the Plane 176 *5.9 Poincare-Bendixson Theorem 179 *5.10 Stable Manifold Theorem for a Fixed Point of a Map Proof of the Stable Manifold Theorem Center Manifold 197 * Stable Manifold Theorem for Flows 199 *5.11 The Inclination Lemma Exercises 202 Chapter VI. Bifurcation of Periodic Points Saddle-Node Bifurcation Saddle-Node Bifurcation in Higher Dimensions Period Doubling Bifurcation Andronov-Hopf Bifurcation for Diffeomorphisms Andronov-Hopf Bifurcation for Differential Equations Exercises 231 Chapter VII. Examples of Hyperbolic Sets and Attractors 235 *7.1 Definition of a Manifold 235 *7.1.1 Topology on Space of Differentiable Functions 237 *7.1.2 Tangent Space 238 *7.1.3 Hyperbolic Invariant Sets 241 *7.2 Transitivity Theorems. 244 *7.3 Two Sided Shift Spaces \ Subshifts for Nonnegative Matrices 247 *7.4 Geometric Horseshoe Horseshoe for the Henon Map 255 *7.4.2 Horseshoe from a Homoclinic Point Melnikov Method for Homoclinic Points Fractal Basin Boundaries 274 * Core Sections

4 *7.5 Hyperbolic Toral Automorphisms Markov Partitions for Hyperbolic Toral Automorphisms JThe Zeta Function for Hyperbolic Toral Automorphisms 288 *7.6 Attractors 292 *7.7 The Solenoid Attractor Conjugacy of the Solenoid to an Inverse Limit The DA Attractor The Branched Manifold 303 *7.9 Plykin Attractors in the Plane Attractor for the Henon Map t Lorenz Attractor Geometric Model for the Lorenz Equations Homoclinic Bifurcation to a Lorenz Attractor 318 *7.12 Morse-Smale Systems Exercises 326 Chapter VIII. Measurement of Chaos in Higher Dimensions Topological Entropy Proof of Two Theorems on Topological Entropy Entropy of Higher Dimensional Examples Liapunov Exponents Sinai-Ruelle-Bowen Measure for an Attractor Fractal Dimension Exercises 362 Chapter IX. Global Theory of Hyperbolic Systems Fundamental Theorem of Dynamical Systems Fundamental Theorem for a Homeomorphism Stable Manifold Theorem for a Hyperbolic Invariant Set Shadowing and Expansiveness Ariosov Closing Lemma Decomposition of Hyperbolic Recurrent Points Markov Partitions for a Hyperbolic Invariant Set Local Stability and Stability of Anosov Diffeomorphisms Stability of Anosov Flows Global Stability Theorems Exercises.407 Chapter X. Generic Properties Kupka-Smale Theorem Transversality Proof of the Kupka-Smale Theorem Necessary Conditions for Structural Stability Nondensity of Structural Stability Exercises 430 Core Sections

5 Chapter XI. Smoothness of Stable Manifolds and x Applications Differentiable Invariant Sections for Fiber Contractions Differentiability of Invariant Splitting Differentiability of the Center Manifold Persistence of Normally Contracting Manifolds Exercises 448 References 451 Index 463