Transient Behaviors in Noise-Induced Bifurcations Captured by Generalized Cell Mapping Method with an Evolving Probabilistic Vector

Abstract

In this paper, an idea of evolving probabilistic vector (EPV) is introduced into the Generalized Cell Mapping (GCM) method to replace the classical fix-sized probabilistic vector in order to efficiently capture the transient behaviors in noise-induced bifurcations, by which an initial localized probability distribution around a deterministic attracting set of a nonlinear dynamical system may expand abruptly or escape with a jump as the noise intensity increases and exceeds some critical values. A Mathieu–Duffing oscillator under excitation of both additive and multiplicative noise is studied as an example of application to show the validity of the proposed method 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. The insight into the roles of deterministic global structure and noise as well as their interplay is gained.