BREATHER COMPETITION AND PULSE ORBITS IN THE DAMPED DRIVEN SINE–GORDON EQUATION

Abstract

The generalized asymptotic inertial manifold (GAIM) of the damped driven Sine–Gordon equation is put forward to govern the long-term behavior of the full partial differential equation (PDE). We then study qualitatively the ordinary differential equation (ODE) by the singular perturbation-theory, which results from restricting the damped driven Sine–Gordon equation to its GAIM. Firstly an analytical criterion for the existence of the homoclinic orbit resulting in chaos is given. Further, the existence of the pulse orbits is showed under the same parametric values as those used in the previous numerical experiments. In our viewpoint these results reflect just the breather competition behavior observed numerically in the Sine–Gordon equation. By comparing with the earlier results obtained in the two-mode Fourier truncation system of the damped driven Sine–Gordon equation, we think that a reasonable discretization reduction from PDE to ODE is very important in the study of dynamics in the infinite dimensional dynamical systems.