Explore Stochastic Instabilities of Periodic Points by Transition Path Theory

We consider the noise-induced transitions in the randomly perturbed discrete logistic map from a linearly stable periodic orbit consisting of T periodic points. The traditional large deviation theory and asymptotic analysis for small noise limit as well as the derived quasi-potential can not distinguish the quantitative difference in noise-induced stochastic instabilities of these T periodic points. We generalize the transition path theory to the discrete-time continuous-space stochastic process to attack this problem. As a first criterion of quantifying the relative instability among T periodic points, we compare the distribution of the last passage locations in the transitions from the whole periodic orbit to a prescribed set far away. This distribution is related to the contributions to the transition rate from each periodic points. The second criterion is based on the competency of the transition paths associated with each periodic point. Both criteria utilise the reactive probability current in the transition path theory. Our numerical results for the logistic map reveal the transition mechanism of escaping from the stable periodic orbit and identify which peri- odic point is more prone to lose stability so as to make successful transitions under random perturbations.