RESEARCH FOR ATTRACTING REGION AND EXIT PROBLEM OF A PIECEWISE LINEAR SYSTEM UNDER PERIODIC AND WHITE NOISE EXCITATIONS

Exit behaviour is one of the significant phenomena of stochastic nonlinear systems, other than the theory of random dynamical system, the exit problem is an another way to investigate the stochastic stability for a stochastic nonlinear system. The piecewise linear system is a classical model in non-linear dynamics, for which, the stochastic excitation leads to a stochastic system, not a rigorous random dynamical system, and then the theory of random dynamical system is not applicable. Thus, in order to learn the stochastic dynamical behaviours for a piecewise linear system that is under a periodic and a Gaussian white noise excitations, its exit behaviour is examined in the present paper via investigating the mean first-passage time which is one of the most important quantities within exit problem and is also used to quantify the global stability of a stochastic system. Some numerical experiments are designed to investigate the deterministic dynamical behaviors in the case that only the periodic excitations are added, and based upon the Monte Carlo simulation, the other numerical procedures are designed to reveal the exit behavior of the system that is under both periodic and Gaussian white noise excitations. In order to obtain the analytical expression of the mean first-passage time, van der Pol transition and stochastic averaging method are firstly applied to simplify the system, then singular perturbation method and ray method are used to quantify the mean first-passage time. Comparing the analytical results with the analog ones, we conclude that if the attracting boundary is fractal, the two results are far different; otherwise if the attracting boundary is smooth enough, the two results match very well.