Dyanmical Systems by Clark Robinson Table of Contents Chapter 1 Introduction 1 1.1 Population Growth Models, One Population 2 1.2 Iteration of Real Valued Functions as Dynamical Systems 3 1.3 Higher Dimensional Systems 5 1.4 Outline of the Topics of the Chapters 9 Chapter 2 One Dimensional Dynamics by Iteration 13 2.1 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 30 2.4.3 The Invariant Cantor Set for $\mu > 4$ 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 46 2.8 Homeomorphisms of the Circle 49 2.9 Exercises 57 Chapter 3 Chaos and Its Measurement 63 3.1 Sharkovskii's Theorem 63 3.1.1 Examples for Sharkovskii's Theorem 70 3.2 Subshifts of Finite Type 72 3.3 Zeta Function 78 3.4 Period Doubling Cascade 79 3.5 Chaos 81 3.6 Liapunov Exponents 86 3.7 Exercises 88 Chapter 4 Linear Systems 93 4.1 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 116 4.9.1 Perron-Frobenius Theorem 123 4.10 Exercises 127 Chapter 5 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 163 5.7.2 Proof of the Hartman-Grobman Theorem for Flows 165 5.8 Periodic Orbits for Flows 165 5.8.1 The Suspension of a Map 171 5.8.2 An Attracting Periodic Orbit for the Van der Pol Equations 171 5.8.3 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 181 5.10.1 Proof of the Stable Manifold Theorem 185 5.10.2 Center Manifold 197 5.10.3 Stable Manifold Theorem for Flows 199 5.11 The Inclination Lemma 200 5.12 Exercises 202 Chapter 6 Bifurcation of Periodic Points 211 6.1 Saddle-Node Bifurcation 211 6.2 Saddle-Node Bifurcation in Higher Dimensions 213 6.3 Period Doubling Bifurcation 218 6.4 Andronov-Hopf Bifurcation for Diffeomorphisms 223 6.5 Andronov-Hopf Bifurcation for Differential Equations 224 6.6 Exercises 231 Chapter 7 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 247 7.3.1 Subshifts for Nonnegative Matrices 247 7.4 Geometric Horseshoe 249 7.4.1 Horseshoe for the Henon Map 255 7.4.2 Horseshoe from a Homoclinic Point 259 7.4.3 Melnikov Method for Homoclinic Points 268 7.4.4 Fractal Basin Boundaries 274 7.5 Hyperbolic Toral Automorphisms 275 7.5.1 Markov Partitions for Hyperbolic Toral Automorphisms 279 7.5.2 The Zeta Function for Hyperbolic Toral Automorphisms 288 7.6 Attractors 292 7.7 The Solenoid Attractor 294 7.7.1 Conjugacy of the Solenoid to an Inverse Limit 299 7.8 The DA Attractor 300 7.8.1 The Branched Manifold 303 7.9 Plykin Attractors in the Plane 304 7.10 Attractor for the Henon Map 306 7.11 Lorenz Attractor 309 7.11.1 Geometric Model for the Lorenz Equations 312 7.11.2 Homoclinic Bifurcation to a Lorenz Attractor 318 7.12 Morse-Smale Systems 318 7.13 Exercises 326 Chapter 8 Measurement of Chaos in Higher Dimensions 333 8.1 Topological Entropy 333 8.1.1 Proof of Two Theorems on Topological Entropy 343 8.1.2 Entropy of Higher Dimensional Examples 350 8.2 Liapunov Exponents 351 8.3 Sinai-Ruelle-Bowen Measure for an Attractor 356 8.4 Fractal Dimension 356 8.5 Exercises 362 Chapter 9 Global Theory of Hyperbolic Systems 367 9.1 Fundamental Theorem of Dynamical Systems 367 9.1.1 Fundamental Theorem for a Homeomorphism 374 9.2 Stable Manifold Theorem for a Hyperbolic Invariant Set 374 9.3 Shadowing and Expansiveness 377 9.4 Anosov Closing Lemma 381 9.5 Decomposition of Hyperbolic Recurrent Points 382 9.6 Markov Partitions for a Hyperbolic Invariant Set 388 9.7 Local Stability and Stability of Anosov Diffeomorphisms 398 9.8 Stability of Anosov Flows 401 9.9 Global Stability Theorems 403 9.10 Exercises 407 Chapter 10 Generic Properties 413 10.1 Kupka-Smale Theorem 413 10.2 Transversality 417 10.3 Proof of the Kupka--Smale Theorem 419 10.4 Necessary Conditions for Structural Stability 425 10.5 Nondensity of Structural Stability 428 10.6 Exercises 430 Chapter 11 Smoothness of Stable Manifolds and Applications 433 11.1 Differentiable Invariant Sections for Fiber Contractions 433 11.2 Differentiability of Invariant Splitting 441 11.3 Differentiability of the Center Manifold 444 11.4 Persistence of Normally Contracting Manifolds 444 11.5 Exercises 448 References 451 Index 465