Global Bifurcations and Chaos: Analytical Methods

ISBN-13: 9781461210412 / Angielski / Miękka / 2014 / 495 str.

Stephen Wiggins

Opis

Global Bifurcations and Chaos: Analytical Methods is unique in the literature of chaos in that it not only defines the concept of chaos in deterministic systems, but it describes the mechanisms which give rise to chaos (i.e., homoclinic and heteroclinic motions) and derives explicit techniques whereby these mechanisms can be detected in specific systems. These techniques can be viewed as generalizations of Melnikov's method to multi-degree of freedom systems subject to slowly varying parameters and quasiperiodic excitations. A unique feature of the book is that each theorem is illustrated with drawings that enable the reader to build visual pictures of global dynamcis of the systems being described. This approach leads to an enhanced intuitive understanding of the theory.

Spis treści

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 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.