S-shaped bifurcations in a two-dimensional Hamiltonian system

摘要

We study the solutions to the following Dirichlet boundary problem:d2x(t)dt2+λf(x(t))=0,where x∈R, t∈R, λ∈R+, with boundary conditions:x(0)=x(1)=A∈R.Especially we focus on varying the parameters λ 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 λ 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.