On the Global Analysis of a Piecewise Linear System that is excited by a Gaussian White Noise

Abstract

The global analysis is very important for a nonlinear dynamical system which possesses a chaotic saddle and a nonchaotic attractor, especially for the one that is driven by a noise. For a random dynamical system, within which, chaotic saddles exist, it is found that if the noise intensity exceeds a critical value, the so called “noise-induced chaos” is observed. Meanwhile, the exit behavior is also found to be influenced significantly by the existence of chaotic saddles. In the present paper, based on the generalized cell-mapping digraph (GCMD) method, the global dynamical behaviors of a piecewise linear system, wherein a chaotic saddle exists and consists of subharmonic solutions in a wide frequency range, are investigated numerically. Further, in order to simplify the system that is driven by a Gaussian white noise excitation, the stochastic averaging method is applied and through which, a five-dimensional Itô system is obtained. Some of the global dynamical behaviors of the original system are retained in the averaged one and then are analyzed. The researches in this paper show that GCMD method is a good numerical tool to investigate the global behaviors of a nonlinear random dynamical system, and the stochastic averaging method is effective for solving the global problems.