Homoclinic chaos in systems perturbed by weak Langevin noise

Abstract

We consider the effect of weak additive noise on the homoclinic threshold of a driven dissipative nonlinear system. A new ``generalized'' Melnikov function is derived for the system and is seen to be the Melnikov function for the corresponding noise-free system plus a correction term that depends on the second-order noise characteristics. The correction term is explicitly calculated for three model systems [Duffing oscillator, Josephson junction, and rf superconducting quantum interference device (SQUID)]. The effect of a distribution of dc driving terms on the chaotic attractor of a dissipative system is also examined via numerical simulation of the rf SQUID.