Global Bifurcation of Periodic Solutions with Symmetry

Bernold Fiedler (auth.)9780387192345, 0-387-19234-4

This largely self-contained research monograph addresses the following type of questions. Suppose one encounters a continuous time dynamical system with some built-in symmetry. Should one expect periodic motions which somehow reflect this symmetry? And how would periodicity harmonize with symmetry? Probing into these questions leads from dynamics to topology, algebra, singularity theory, and to many applications. Within a global approach, the emphasis is on periodic motions far from equilibrium. Mathematical methods include bifurcation theory, transversality theory, and generic approximations. A new homotopy invariant is designed to study the global interdependence of symmetric periodic motions. Besides mathematical techniques, the book contains 5 largely nontechnical chapters. The first three outline the main questions, results and methods. A detailed discussion pursues theoretical consequences and open problems. Results are illustrated by a variety of applications including coupled oscillators and rotating waves: these links to such disciplines as theoretical biology, chemistry, fluid dynamics, physics and their engineering counterparts make the book directly accessible to a wider audience.

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Table of contents :
Introduction….Pages 1-14
Main results….Pages 15-26
No symmetry — a survey….Pages 27-34
Virtual symmetry….Pages 35-47
Generic local theory….Pages 48-67
Generic global theory….Pages 68-83
General global theory….Pages 84-91
Applications….Pages 92-105
Discussion….Pages 106-115
Appendix on genericity….Pages 116-134