S-shaped bifurcations in a two-dimensional Hamiltonian system

Abstract

We study the solutions to the following Dirichlet boundary problem: d(2)x(t)/dt(2) + lambda f(x(t)) = 0, where x is an element of R, t is an element of R, lambda is an element of R+, with boundary conditions: x(0) = x(1) = A is an element of R. Especially we focus on varying the parameters A and A in the case where the phase plane representation of the equation contains a saddle loop filled with a period annulus surrounding a center. We introduce the concept of mixed solutions which take on values above and below x = A, generalizing the concept of the well-studied positive solutions. This leads to a generalization of the so-called period function for a period annulus. We derive expansions of these functions and formulas for the derivatives of these generalized period functions. The main result is that under generic conditions on f (x) so-called S-shaped bifurcations of mixed solutions occur. As a consequence there exists an open interval for sufficiently small A for which A can be found such that three solutions of the same mixed type exist. We show how these concepts relate to the simplest possible case f (x) = x(x + 1) where despite its simple form difficult open problems remain.