Robinson, C.

Dynamical Systems : Stability, Symbolic Dynamics, and Chaos

Chapter I. Introduction 1(12) 1.1 Population Growth Models, One Population 2(1) 1.2 Iteration of Real Valued Functions as Dynamical Systems 3(2) 1.3 Higher Dimensional Systems 5(4) 1.4 Outline of the Topics of the Chapters 9(4) Chapter II. One-Dimensional Dynamics by Iteration 13(52) 2.1 Calculus Prerequisites 13(2) *2.2 Periodic Points 15(7) *2.2.1 Fixed Points for the Quadratic Family 20(2) *2.3 Limit Sets and Recurrence for Maps 22(4) *2.4 Invariant Cantor Sets for the Quadratic Family 26(12) *2.4.1 Middle Cantor Sets 26(4) *2.4.2 Construction of the Invariant Cantor Set 30(3) *2.4.3 The Invariant Cantor Set for Mu is Greater than 4 33(5) *2.5 Symbolic Dynamics for the Quadratic Map 38(3) *2.6 Conjugacy and Structural Stability 41(6) *2.7 Conjugacy and Structural Stability of the Quadratic Map 47(3) 2.8 Homeomorphisms of the Circle 50(8) 2.9 Exercises 58(7) Chapter III. Chaos and Its Measurement 65(30) 3.1 Sharkovskii's Theorem 65(9) 3.1.1 Examples for Sharkovskii's Theorem 72(2) 3.2 Subshifts of Finite Type 74(6) 3.3 Zeta Function 80(2) 3.4 Period Doubling Cascade 82(2) 3.5 Chaos 84(4) 3.6 Liapunov Exponents 88(3) 3.7 Exercises 91(4) Chapter IV. Linear Systems 95(38) 4.1 Review: Linear Maps and the Real Jordan Canonical Form 95(2) *4.2 Linear Differential Equations 97(2) *4.3 Solutions for Constant Coefficients 99(5) *4.4 Phase Portraits 104(4) *4.5 Contracting Linear Differential Equations 108(5) *4.6 Hyperbolic Linear Differential Equations 113(2) *4.7 Topologically Conjugate Linear Differential Equations 115(2) *4.8 Nonhomogeneous Equations 117(1) *4.9 Linear Maps 118(11) 4.9.1 Perron-Frobenius Theorem 125(4) 4.10 Exercises 129(4) Chapter V. Analysis Near Fixed Points and Periodic Orbits 133(82) *5.1 Review: Differentiation in Higher Dimensions 133(3) *5.2 Review: The Implicit Function Theorem 136(6) *5.2.1 Higher Dimensional Implicit Function Theorem 138(1) *5.2.2 The Inverse Function Theorem 139(1) *5.2.3 Contraction Mapping Theorem 140(2) *5.3 Existence of Solutions for Differential Equations 142(6) *5.4 Limit Sets and Recurrence for Flows 148(3) *5.5 Fixed Points for Nonlinear Differential Equations 151(7) *5.5.1 Nonlinear Sinks 153(2) *5.5.2 Nonlinear Hyperbolic Fixed Points 155(1) *5.5.3 Liapunov Functions Near a Fixed Point 156(2) *5.6 Stability of Periodic Points for Nonlinear Maps 158(2) 5.7 Proof of the Hartman-Grobman Theorem 160(8) *5.7.1 Proof of the Local Theorem 166(1) 5.7.2 Proof of the Hartman-Grobman Theorem for Flows 167(1) *5.8 Periodic Orbits for Flows 168(13) 5.8.1 The Suspension of a Map 173(1) 5.8.2 An Attracting Periodic Orbit for the Van der Pol Equations 174(5) 5.8.3 Poincare Map for Differential Equations in the Plane 179(2) *5.9 Poincare-Bendixson Theorem 181(2) *5.10 Stable Manifold Theorem for a Fixed Point of a Map 183(20) 5.10.1 Proof of the Stable Manifold Theorem 187(13) 5.10.2 Center Manifold 200(2) *5.10.3 Stable Manifold Theorem for Flows 202(1) *5.11 The Inclination Lemma 203(1) 5.12 Exercises 204(11) Chapter VI. Hamiltonian Systems 215(22) 6.1 Hamiltonian Differential Equations 215(5) 6.2 Linear Hamiltonian Systems 220(3) 6.3 Symplectic Diffeomorphisms 223(4) 6.4 Normal Form at Fixed Point 227(4) 6.5 KAM Theorem 231(2) 6.6 Exercises 233(4) Chapter VII. Bifurcation of Periodic Points 237(26) 7.1 Saddle-Node Bifurcation 237(2) 7.2 Saddle-Node Bifurcation in Higher Dimensions 239(5) 7.3 Period Doubling Bifurcation 244(5) 7.4 Andronov-Hopf Bifurcation for Differential Equations 249(7) 7.5 Andronov-Hopf Bifurcation for Diffeomorphisms 256(3) 7.6 Exercises 259(4) Chapter VIII. Examples of Hyperbolic Sets and Attractors 263(106) *8.1 Definition of a Manifold 263(10) *8.1.1 Topology on Space of Differentiable Functions 265(1) *8.1.2 Tangent Space 266(3) *8.1.3 Hyperbolic Invariant Sets 269(4) *8.2 Transitivity Theorems 273(2) *8.3 Two-Sided Shift Spaces 275(2) 8.3.1 Subshifts for Nonnegative Matrices 275(2) *8.4 Geometric Horseshoe 277(31) 8.4.1 Horseshoe for the Henon Map 283(4) *8.4.2 Horseshoe from a Homoclinic Point 287(10) 8.4.3 Nontransverse Homoclinic Point 297(2) *8.4.4 Homoclinic Points and Horseshoes for Flows 299(3) 8.4.5 Melnikov Method for Homoclinic Points 302(5) 8.4.6 Fractal Basin Boundaries 307(1) *8.5 Hyperbolic Toral Automorphisms 308(18) 8.5.1 Markov Partitions for Hyperbolic Toral Automorphisms 313(7) 8.5.2 Ergodicity of Hyperbolic Toral Automorphisms 320(2) 8.5.3 The Zeta Function for Hyperbolic Toral Automorphisms 322(4) *8.6 Attractors 326(2) *8.7 The Solenoid Attractor 328(6) 8.7.1 Conjugacy of the Solenoid to an Inverse Limit 333(1) 8.8 The DA Attractor 334(4) 8.8.1 The Branched Manifold 338(1) *8.9 Plykin Attractors in the Plane 338(3) 8.10 Attractor for the Henon Map 341(3) 8.11 Lorenz Attractor 344(9) 8.11.1 Geometric Model for the Lorenz Equations 347(6) 8.11.2 Homoclinic Bifurcation to a Lorenz Attractor 353(1) *8.12 Morse-Smale Systems 353(8) 8.13 Exercises 361(8) Chapter IX. Measurement of Chaos in Higher Dimensions 369(34) 9.1 Topological Entropy 369(18) 9.1.1 Proof of Two Theorems on Topological Entropy 379(7) 9.1.2 Entropy of Higher Dimensional Examples 386(1) 9.2 Liapunov Exponents 387(5) 9.3 Sinai-Ruelle-Bowen Measure for an Attractor 392(1) 9.4 Fractal Dimension 393(5) 9.5 Exercises 398(5) Chapter X. Global Theory of Hyperbolic Systems 403(46) 10.1 Fundamental Theorem of Dynamical Systems 403(8) 10.1.1 Fundamental Theorem for a Homeomorphism 410(1) 10.2 Stable Manifold Theorem for a Hyperbolic Invariant Set 411(3) 10.3 Shadowing and Expansiveness 414(4) 10.4 Anosov Closing Lemma 418(1) 10.5 Decomposition of Hyperbolic Recurrent Points 419(6) 10.6 Markov Partitions for a Hyperbolic Invariant Set 425(10) 10.7 Local Stability and Stability of Anosov Diffeomorphisms 435(3) 10.8 Stability of Anosov Flows 438(3) 10.9 Global Stability Theorems 441(3) 10.10 Exercises 444(5) Chapter XI. Generic Properties 449(20) 11.1 Kupka-Smale Theorem 449(4) 11.2 Transversality 453(2) 11.3 Proof of the Kupka -- Smale Theorem 455(6) 11.4 Necessary Conditions for Structural Stability 461(3) 11.5 Nondensity of Structural Stability 464(2) 11.6 Exercises 466(3) Chapter XII. Smoothness of Stable Manifolds and Applications 469(18) 12.1 Differentiable Invariant Sections for Fiber Contractions 469(8) 12.2 Differentiability of Invariant Splitting 477(3) 12.3 Differentiability of the Center Manifold 480(1) 12.4 Persistence of Normally Contracting Manifolds 480(4) 12.5 Exercises 484(3) References 487(14) Index 501 * Core Sections