Analytic solution of stability-instability transitions in a two-dimensional Hamiltonian system.

Abstract

We present a Hamiltonian system of two nonlinearly coupled oscillators which exhibits, as the coupling parameter is varied, a finite or infinite number of unstable and stable regions, depending on the initial condition. For two particular types of initial conditions, we present an analytic solution that describes the nature of the stable and unstable motions of this system, locates precisely the stability-instability transition points, and determines the behavior of the largest Lyapunov exponent near these transition points.