Separatrix Crossing: Time-Invariant Potentials with Dissipation

Strongly nonlinear oscillations in a double-well potential that cross a separatrix due to dissipation are analyzed. Away from the separatrix the nearly periodic oscillations are found using standard asymptotic techniques that are known to fail near the separatrix. The solution near the separatrix is represented as a large sequence of perturbed solitary pulses. The asymptotic behavior of this sequence is matched (forward and backward in time) to the nearly periodic oscillations. In this manner, the oscillations in the post-crossing region are connected, for the first time, to initial conditions. In particular, we determine the amplitude and the phase of the nonlinear oscillator after crossing the separatrix and show them to be sensitively dependent on the initial conditions. The shift between the critical times associated with the slow variation theory before and after crossing the separatrix is derived.