Numerical Stability Analysis On Several Classes Of Stochastic Systems

In recent years, with the advancement of technology, the stochastic systems theory has been continuously developed and improved. Stochastic differential equations as the mathematical description of stochastic systems, has been widely used in various fields,such as physics, biology, finance, electrical engineering and control. The stability is one of the important properties in the theory of stochastic differential equations, and it is also the necessary condition to maintain stochastic systems operating normal, so the research on stability is very important valuable both in theoretical significance and practical application. Several classes of stochastic systems are researched in this dissertation, the stability of the numerical methods and systems are analyzed. The main contents contain the following several aspects:The problem of the mean-square stability for a class of semi-linear stochastic pantograph differential equations is studied. The exponential Euler method is constructed, the conditions on mean-square stability of the analytical solution are given, it is proved that under the conditions the exponential Euler method can preserve the mean-square stability for any non-zero step size. Finally, numerical examples verify the obtained conclusions.The stability for a class of stochastic delay differential equations under Poisson white noise excitations is discussed. The sufficient conditions on stability of the analytical solution are obtained for linear stochastic delay differential equations under Poisson white noise excitations, the exponential Euler method can reproduce the mean square stability with the sufficiently small step-size. Further, the compensated exponential Euler method is constructed for semi-linear stochastic differential delay equations under Poisson white noise excitations, the conditions which guarantees the mean square stability of the analytical solution are established, and it is confirmed that the numerical method preserves the mean square stability for underlying systems with any step size. The corresponding numerical examples are given.The stability for a class of two mass relative rotation system with Mathieu-Duffing oscillator under Gauss white noise excitations is considered. The mathematical model is established in the physical background and practical significance. According to the Melnikov method, the system will appear chaotic dynamic behavior. Under the parametric excitation of Gauss white noise, the systems can be changed unstable state to stable state,and the stabilization of the systems is achieved. Numerical simulation further verifies the conclusion.