Global Bifurcations and Chaos

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.

Klappentext

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.

Inhalt

1. Introduction: Background for Ordinary Differential Equations and Dynamical Systems.- 1.1. The Structure of Solutions of Ordinary Differential Equations.- 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.- 2.2. Symbolic Dynamics.- 2.3. Criteria for Chaos: The Hyperbolic Case.- 2.4. Criteria for Chaos: The Nonhyperbolic Case.- 3. Homoclinic and Heteroclinic Motions.- 3.1. Examples and Definitions.- 3.2. Orbits Homoclinic to Hyperbolic Fixed Points of Ordinary Differential Equations.- 3.3. Orbits Heteroclinic to Hyperbolic Fixed Points of Ordinary Differential Equations.- 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.- 4.2. Examples.- 4.3. Final Remarks.- References.