Global Bifurcations and Chaos

1. Introduction: Background for Ordinary Differential Equations and Dynamical Systems.- 1.1. The Structure of Solutions of Ordinary Differential Equations.- 1.1a. Existence and Uniqueness of Solutions.- 1.1b. Dependence on Initial Conditions and Parameters.- 1.1c. Continuation of Solutions.- 1.1d. Autonomous Systems.- 1.1e. Nonautonomous Systems.- 1.1f. Phase Flows.- 1.1g. Phase Space.- 1.1h. Maps.- 1.1 i. Special Solutions.- 1.1j. Stability.- 1.1k. Asymptotic Behavior.- 1.2. Conjugacies.- 1.3. Invariant Manifolds.- 1.4. Transversality, Structural Stability, and Genericity.- 1.5. Bifurcations.- 1.6. Poincaré Maps.- 2. Chaos: Its Descriptions and Conditions for Existence.- 2.1. The Smale Horseshoe.- 2.1a. Definition of the Smale Horseshoe Map.- 2.1b. Construction of the Invariant Set.- 2.1c. Symbolic Dynamics.- 2.1d. The Dynamics on the Invariant Set.- 2.1e. Chaos.- 2.2. Symbolic Dynamics.- 2.2a. The Structure of the Space of Symbol Sequences.- 2.2b. The Shift Map.- 2.2c. The Subshift of Finite Type.- 2.2d. The Case of N = ?.- 2.3. Criteria for Chaos: The Hyperbolic Case.- 2.3a. The Geometry of Chaos.- 2.3b. The Main Theorem.- 2.3c. Sector Bundles.- 2.3d. More Alternate Conditions for Verifying Al and A2.- 2.3e. Hyperbolic Sets.- 2.3f. The Case of an Infinite Number of Horizontal Slabs.- 2.4. Criteria for Chaos: The Nonhyperbolic Case.- 2.4a. The Geometry of Chaos.- 2.4b. The Main Theorem.- 2.4c. Sector Bundles.- 3. Homoclinic and Heteroclinic Motions.- 3.1. Examples and Definitions.- 3.2. Orbits Homoclinic to Hyperbolic Fixed Points of Ordinary Differential Equations.- 3.2a. The Technique of Analysis.- 3.2b. Planar Systems.- 3.2c. Third Order Systems.- i) Orbits Homoclinic to a Saddle Point with Purely Real Eigenvalues.- ii) Orbits Homoclinic to a Saddle-Focus.- 3.2.d. Fourth Order Systems.- i) A Complex Conjugate Pair and Two Real Eigenvalues.- ii) Silnikov’s Example in ?4.- 3.2e. Orbits Homoclinic Fixed Points of 4-Dimensional Autonomous Hamiltonian Systems.- i) The Saddle-Focus.- ii) The Saddle with Purely Real Eigenvalues.- iii) Devaney’s Example: Transverse Homoclinic Orbits in an Integrable Systems.- 3.2f. Higher Dimensional Results.- 3.3. Orbits Heteroclinic to Hyperbolic Fixed Points of Ordinary Differential Equations.- i) A Heteroclinic Cycle in ?3.- ii) A Heteroclinic Cycle in ?4.- 3.4. Orbits Homoclinic to Periodic Orbits and Invariant Tori.- 4. Global Perturbation Methods for Detecting Chaotic Dynamics.- 4.1. The Three Basic Systems and Their Geometrical Structure.- 4.1a. System I.- i) The Geometric Structure of the Unperturbed Phase Space.- ii) Homoclinic Coordinates.- iii) The Geometric Structure of the Perturbed Phase Space.- iv) The Splitting of the Manifolds.- 4.1b. System II.- i) The Geometric Structure of the Unperturbed Phase Space.- ii) Homoclinic Coordinates.- iii) The Geometric Structure of the Perturbed Phase Space.- iv) The Splitting of the Manifolds.- 4.1c. System III.- i) The Geometric Structure of the Unperturbed Phase Space.- ii) Homoclinic Coordinates.- iii) The Geometric Structure of the Perturbed Phase Space.- iv) The Splitting of the Manifolds.- v) Horseshoes and Arnold Diffusion.- 4.1d. Derivation of the Melnikov Vector.- i) The Time Dependent Melnikov Vector.- ii) An Ordinary Differential Equation for the Melnikov Vector.- iii) Solution of the Ordinary Differential Equation.- iv) The Choice of SP,?S and SP,?u.- v) Elimination of t0.- 4.1e. Reduction to a Poincaré Map.- 4.2. Examples.- 4.2a. Periodically Forced Single Degree of Freedom Systems.- i) The Pendulum: Parametrically Forced at O (?) Amplitude, O (1) Frequency.- ii) The Pendulum: Parametrically Forced at O (1) Amplitude, O (?) Frequency.- 4.2.b. Slowly Varying Oscillators.- i) The Duffing Oscillator with Weak Feedback Control.- ii) The Whirling Pendulum.- 4.2c. Perturbations of Completely Integrable, Two Degree of Freedom Hamiltonian System.- i) A Coupled Pendulum and Harmonic Oscillator.- ii) A Strongly Coupled Two Degree of Free