Noise induced escape from stable invariant tori

Abstract

Abstract In the present paper, the noise-induced escape from stable invariant tori with the frame of general Langevin dynamics is investigated under weak random perturbations. Based on the large deviation theory, the quasi-potential, a quantity exponentially dominating the mean first escape time and the stationary or quasi-stationary probability density, is explored both analytically and numerically. The results show that whether the stochastic trajectories are ergodic on a torus has a crucial influence on the behavior of the quasi-potential. Specifically, there are two sources of ergodicity. One is the ergodicity of the deterministic flow, and the other requires the nondegeneracy of the noise in the tangential directions of a torus. It is found that if the ergodicity holds, the quasi-potential will be independent of the initial position on a torus, but not when the ergodicity is broken. In particular, it indicates that, for nonlinear systems driven by combined Gaussian white noise and multiple harmonic excitations, the mean first exit time varies discontinuously with respect to the frequency vector of these harmonic excitations as the noise intensity approaches zero. Adding noise to the phase of these harmonic excitations will eliminate the dependence on the initial position and thus, make the systems more robust. It reminds us that great care must be taken when dealing with noise-induced problems involving systems that possess unstable geometric structures.