Bifurcations of Periodic Orbits

This chapter and Chapter IX use the theory of normal forms developed in Chapter VI. They contain an introduction to generic bifurcation theory and its applications. Bifurcation theory has grown into a vast subject with a large literature; so, this chapter can only present the basics of the theory. The primary focus of this chapter is the study of periodic solutions—their existence and evolution. Periodic solutions abound in Hamiltonian systems. In fact, a famous Poincaré conjecture is that periodic solutions are dense in a generic Hamiltonian system, a point that was established in the C1 case by Pugh and Robinson (1977).