Study on critical conditions and transient behavior in noise-induced bifurcations

Abstract

In this work, the stochastic sensitivity function method, which can describe the probabilistic distribution range of a stochastic attractor, is extended to the non-autonomous dynamical systems by constructing a 1/N-period stroboscopic map to discretize a continuous cycle into a discrete one. With confidence ranges of a stochastic attractor and the global structure of the deterministic nonlinear system, like chaotic saddle in basin of attraction and/or saddle on basin boundary aswell as its stable and unstablemanifolds, the critical noise intensity for the occurrence of transition behavior due to noise-induced bifurcations may be estimated. Furthermore, to efficiently capture the stochastic transient behaviors after the critical conditions, an idea of evolving probabilistic vector (EPV) is introduced into the Generalized Cell Mapping method (GCM) in order to enhance the computation efficiency of the numerical method. A Mathieu-Duffing oscillator under external and parametric excitation as well as additive noise is studied as an example of application to show the validity of the proposed methods and the interesting phenomena in noise-induced explosive and dangerous bifurcations of the oscillator that are characterized respectively by an abrupt enlargement and a sudden fast jump of the response probability distribution are demonstrated. The insight into the roles of deterministic global structure and noise as well as their interplay is gained.