Global Bifurcations and Chaos: Analytical Methods

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. Poincare 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 Poincare 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 Freedom System. - 4.2d. Perturbation of a Completely Integrable Three Degree of Freedom System: Arnold Diffusion. - 4.2e. Quasiperiodically Forced Single Degree of Freedom Systems. - i) The Duffing Oscillator: Forced at O (? ) Amplitude with N O (1) Frequencies. - ii) The Pendulum: Parametrically Forced at O (? ) Amplitude, O (1) Frequency and O (1) Amplitude, O (? ) Frequency. - 4.3. Final Remarks. - i) Heteroclinic Orbits. - ii) Additional Applications of Melnikov's Method. - iii) Exponentially Small Melnikov Functions. - References.