Dynamical Systems: Stability, Symbolic Dynamics, and Chaos 2nd Edition (Studies in Advanced Mathematics)

Description

Several distinctive aspects make Dynamical Systems unique, including: treating the subject from a mathematical perspective with the proofs of most of the results included providing a careful review of background materials introducing ideas through examples and at a level accessible to a beginning graduate student focusing on multidimensional systems of real variables The book treats the dynamics of both iteration of functions and solutions of ordinary differential equations. Many concepts are first introduced for iteration of functions where the geometry is simpler, but results are interpreted for differential equations. The dynamical systems approach of the book concentrates on properties of the whole system or subsets of the system rather than individual solutions. The more local theory discussed deals with characterizing types of solutions under various hypothesis, and later chapters address more global aspects. 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 µ > 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 Sharkovskii’s Theorem 3.1.1 Examples for Sharkovskii’s 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 Poincaré Map for Differential Equations in the Plane 5.9 Poincaré-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 Hénon 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 Hénon 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 – 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