受周期和白噪声激励的分段线性系统的吸引域与离出问题研究

离出行为是随机非线性系统的重要现象之一，而离出问题是除随机动力系统理论以外考察随机非线性系统随机稳定性的另一种重要的方法.分段线性系统是一个经典的非线性动力学模型，受随机激励后成为随机系统，但并不是严格的随机动力系统，因而此时随机动力系统理论也不适用.为了研究同时受周期和白噪声激励的分段线性系统，首先使用Poincaré截面模拟其在无噪声时确定性的动力学行为，然后使用Monte Carlo模拟对其在白噪声激励下的离出行为进行了数值仿真分析.其次，为了考察离出问题中的重要参数，系统的平均首次通过时间（mean first-passage time，MFPT），使用van der Pol变换，随机平均法，奇异摄动法和射线方法进行了量化计算.通过对理论结果与模拟结果的对比分析，得到结论:当系统吸引子对应的吸引域边界出现碎片化时，理论结果与模拟结果的误差极大；而当吸引域边界足够光滑的以后，理论结果与模拟结果才会相当吻合. 

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.