10 - Hamiltonian Bifurcation

Summary

In this chapter, we study some examples of bifurcations in the Hamiltonian context. A lot of the ideas from the previous chapters come into this discussion, and links with new ones get established, such as connections with chaotic dynamics and solution spaces in relativistic field theories. Our discussion will be by no means complete; it will focus on certain results of personal interest and results that fit in with the rest of the chapters. Some additional information on bifurcation theory in the Hamiltonian context may be found in the references cited below and in Abraham and Marsden [1978], Arnold [1978], Meyer and Hall [1991] and the references therein.

Some Introductory Examples

Bifurcation theory deals with the changes in the phase portrait structure of a given dynamical system as parameters are varied. One usually begins by focussing on the simplest features of the phase portrait, such as equilibrium points, relative equilibria, periodic orbits, relative periodic orbits, homoclinic orbits, etc., and studies how they change in number and stability characteristics as the system parameters are changed. Often these changes lead to new structures, such as more equilibria, periodic orbits, tori, or chaotic solutions, and the way in which stability or instability is transfered to these new structures from the old ones is of interest.