Multiple Solutions of Boundary-Value Problems for Hamiltonian Systems

Abstract

We consider two-point boundary-value problems for a Hamiltonian system of the form x′ = f(k, y), y′ = g(x, ????), where k and ???? are parameters. The numbers of solutions, both positive and oscillatory, for the boundary-value problems are estimated. Our main tool is the analysis of the phase plane combined with the evaluation of time-map functions. The bifurcation diagram and solution curves are constructed for the Hamiltonian system. We also present examples illustrating the bifurcations with respect to the parameters k and ????: