External Disturbance to Non-Integrable Hamiltonian System

Abstract

The response of the regular and irregular regions to a periodic external force in two-dimensional Hamiltonian systems are studied, through power spectrum, energy change, Poincare map and Lyapunov number. In the regular case, three-dimensional tori are constructed for small perturbation. It is shown that there exists a certain critical value a c of the amplitude of the external force below which the trajectories are regular but above which they become irregular. In the irregular region, main attention is paid to the Lyapunov number, which is shown to vary sensitively to the variation of the amplitude, indicating the difference of the nature of the orbits. This suggests that perturbation theory is not applicable to the irregular case, as is tusually done in the linear response theory.