Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

Applied Mathematical Sciences #42

by John Guckenheimer, Philip Holmes

From the "This book is concerned with the application of methods from dynamical systems and bifurcation theories to the study of nonlinear oscillations. Chapter 1 provides a review of basic results in the theory of dynamical systems, covering both ordinary differential equations and discrete mappings. Chapter 2 presents 4 examples from nonlinear oscillations. Chapter 3 contains a discussion of the methods of local bifurcation theory for flows and maps, including center manifolds and normal forms. Chapter 4 develops analytical methods of averaging and perturbation theory. Close analysis of geometrically defined two-dimensional maps with complicated invariant sets is discussed in chapter 5. Chapter 6 covers global homoclinic and heteroclinic bifurcations. The final chapter shows how the global bifurcations reappear in degenerate local bifurcations and ends with several more models of physical problems which display these behaviors." #Book Review - Engineering Societies Library, New York#1 "An attempt to make research tools concerning `strange attractors' developed in the last 20 years available to applied scientists and to make clear to research mathematicians the needs in applied works. Emphasis on geometric and topological solutions of differential equations. Applications mainly drawn from nonlinear oscillations." #American Mathematical Monthly#2

Genres: Mathematics, Physics

478 pages, Hardcover

First published August 1, 1983

Reviews

[Michael Weaver]

Guckenheimer and Holmes really explain in systematic detail dynamical system thermal turbulence in planetary interiors and its evolution in time dependent models from that of planetary convection codes. I particularly found the role of k-jets fascinating via the explanation of the mathematical constructions and how those techniques are employed.

[DJ]

apparently THE book on chaos and bifurcation theory