S-shaped bifurcations in a two-dimensional Hamiltonian system

Abstract

We study the solutions to the following Dirichlet boundary problem: \begin{equation*}\frac{d^2x(t)}{dt^2}+\lambda f(x(t))=0,\end{equation*} where $x \in \mathbb{R}$, $t \in \mathbb{R}$, $\lambda \in \mathbb{R}^+$, with boundary conditions: \begin{equation*} x(0)=x(1)=A \in \mathbb{R}. \end{equation*} Especially we focus on varying the parameters $\lambda$ 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 $\lambda$ 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.

Keywords

• ordinary differential equations
• boundary value problem
• period function