Continuation of Periodic Solutions

Abstract

In the last chapter, some local results about periodic solutions of Hamiltonian systems were presented. The systems contain a parameter, and the conditions under which a periodic solution can be continued in the parameter were discussed. Since Poincaré used these ideas extensively, it has become known as Poincaré’s continuation method. By Lemma V.E.2, a solution ø(t, ξ′) of an autonomous differential equation is T-periodic if and only if ø(T’) = ø is the general solution. This is a finite-dimensional problem since is a function defined in a domain of ℝm+1 into ℝm. Thus, periodic solutions can be found by the finite-dimensional methods, i.e., the finite-dimensional implicit function theorem, the finite-dimensional fiixed point theorems, the finite-dimensional degree theory, etc. This chapter will present results which depend only on the finite-dimensional implicit function theorem. Chapter X will present a treatment of fixed point methods as they apply to Hamiltonian systems. In this chapter the periodic solutions vary continuously with the parameter (“can be continued”), but Chapter VII will discuss the bifurcations of periodic solutions.