CONNECTION ACROSS A SEPARATRIX WITH DISSIPATION

Dissipative perturbations of strongly nonlinear oscillators that corre spond to slowly varying double-well potentials are considered. The met hod of averaging, which describes the solution as nearly periodic, fai ls as the trajectory approaches the unperturbed separatrix, a homoclin ic orbit of the saddle point, significantly before it is captured in e ither well. Nevertheless, perturbed initial conditions corresponding t o the boundary of the basin of attraction for each well, which are the perturbed stable manifolds of the saddle point, are accurately determ ined using only the method of averaging modified by Melnikov energy id eas near the separatrix. To determine the amplitude and phase of the c aptured oscillations after crossing the separatrix, a transition regio n is constructed consisting of a large sequence of nearly solitary pul ses along the separatrix. The amplitude and phases of the slowly varyi ng nonlinear oscillations away from the separatrix, both before and af ter capture, are matched to this transition region. In this way, analy tic connection formulas across the separatrix are obtained and are sho wn to depend on the perturbed initial conditions.