Dynamical SystemsStability, Symbolic Dynamics, and Chaos: (Studies in Advanced Mathematics)

This new text/reference treats dynamical systems from a mathematical perspective, centering on multidimensional systems of real variables. Background material is carefully reviewed as it is used throughout the book, and ideas are introduced through examples. Numerous exercises help the reader understand presented theorems and master the techniques of the proofs and topic under consideration. 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. A proof of the existence and continuity of solutions with respect to initial conditions is included. Explicit formulas for the various bifurcations are included, and a treatment of the Hénon map and the Melnikov method is provided. The dynamical systems approach of the book concentrates on properties of the whole system or subsets of the system rather than individual solutions. Even the more local theory which is treated deals with characterizing types of solutions under various hypothesis. Later chapters deal more directly with more global aspects, with one chapter discussing various examples and later chapters giving the global theory.

Table of Contents:
Introduction Population Growth Models, One Population Iteration of Real Valued Functions as Dynamical systems Higher Dimensional Systems Outline of the Topics of the Chapters One Dimensional Dynamics by Iteration Calculus Prerequisites Periodic Points Limit Sets and recurrence for Maps Invariant Cantor Sets for the Quadratic Family Symbolic Dynamics for the Quadratic Map Conjugacy and Structural Stability Conjugacy and Structural Stability of the Quadratic Map Homeomorphisms of the Circle Exercises Chaos and Its Measurement Sharkovskii's Theorem Subshifts of Finite Type Zeta Function Period Doubling Cascade Chaos Liapunov Exponents Exercises Linear Systems Review: Linear Maps and the Real Jordan Canonical Form Linear Differential Equations Solutions for Constant Coefficients Phase Portraits Contracting Linear Differential Equations Hyperbolic Linear Differential Equations Topologically Conjugate Linear Differential Equations Nonhomogeneous Equations Linear Maps Exercises Analysis Near Fixed Points and Periodic Orbits Review: Differentiation in Higher Dimensions Review: The Implicit Function Theorem Existence of Solutions for Differential Equations Limit Sets and Recurrence for Flows Fixed Points for Nonlinear Differential Equations Stability of Periodic Points for Nonlinear Maps Proof of the Hartman-Grobman Theorem Periodic Orbits for Flows Poincaré-Bendixson Theorem Stable Manifold Theorem for a Fixed Point of a Map The Inclination Lemma Exercises Bifurcation of Periodic Points Saddle-Node Saddle-Node Bifurcation in Higher Dimensions Period Doubling: Bifurcation Andronov-Hopf Bifurcation for Diffeomorphisms Andronov-Hopf Bifurcation for Differential Equations Exercises Examples of Hyperbolic Sets and Attractors Definition of a Manifold Transitivity Theorems Geometric Horseshoe Toral Anosov Diffeomorphisms Attractors The Solenoid Attractor The DA Attractor Plykin Attractors in the Plane Attractor for the Hénon Map Lorenz Attractor Morse-Smale Systems Exercises Measurement of Chaos in Higher Dimensions Topological Entropy Liapunov Exponents Sinai-Bowen-Ruelle Measure for an Attractor Fractal Dimension Exercises Global Theory of Hyperbolic Systems Fundamental Theorem of Dynamical Systems Stable Manifold Theorem for a Hyperbolic Invariant Set Shadowing and Expansiveness Anosov Closing Lemma Decomposition of Recurrent Points Markov Partition fro a Hyperbolic Invariant Set Local Stability and Stability of Anosov Diffeomorphisms Stability of Anosov Flows Global Stability Theorems Exercises Generic Properties Kupka-Smale Theorem Transversality Proof of the Kupka-Smale Theorem Necessary Conditions for Structural Stability Nondensity of Structural Stability Exercises Smoothness of Stable Manifolds and Applications Fiber Contractions Differentiability of Invariant Splitting Differentiability of the Center Manifold Persistence of Normally Contracting Manifolds Exercises References Index