Some Dynamical Issues For Systems Driven By Symmetric Lévy Processes

Stochastic dynamical systems arise naturally as mathematical models for com-plex systems under the influences of random fluctuations. The mean exit time, station-ary and time-dependent probability densities are among deterministic tools to quantify stochastic dynamical behaviors.An analytical and computational analysis is conducted to investigate the mean exit time and bifurcations for nonlinear dynamical systems driven by additive as well as multiplicative symmetric non-Gaussianα—stable Levy noises. Moreover, the mean exit problem for some two dimensional dynamical systems driven by ad-ditiveα—stable Levy noises has also been studied. Various examples are presented to demonstrate the efficiency of the adopted numerical methods and the validity of the results about dynamical issues of these stochastic systems. As a primary example, the following double-well system under additive symmetric Levy noises is considered The corresponding mean exit time, u(x), for a solution orbit starting at x in a bounded domain, is described by a differential-integral equation. A computational analysis is conducted to investigate the relative importance of jump measure and diffusion coefficient in affecting the mean exit time. A numerical approximation scheme for this differential-integral equation with an exterior condition is devised and validated. Moreover, a similar problem for the same system under multiplicative symmetric Levy noises is discussed. Various results are obtained about the mean exit time and proba-bility density evolution, for different system parameters.Stochastic bifurcation is a phenomenon for stochastic dynamical systems, and it is poorly understood for non-Gaussian stochastic dynamical systems. It is a very im-portant topic for understanding "qualitative changes" under stochastic influences. A computational analysis is conducted to investigate bifurcations of a simple dynamical system under non-Gaussian symmetric Levy processes, by examining the changes in stationary probability density functions for the solution orbits of this stochastic sys tem. The stationary probability density functions are obtained by numerically solv-ing a nonlocal Fokker-Planck equation. This allows numerically investigating phe-nomenological bifurcation, or P-bifurcation, for stochastic differential equations with non-Gaussian Levy noises. Various bifurcations are observed as system parameters change, and are compared with the bifurcation under (Gaussian) Brownian motion. Furthermore, the (time-dependent) Fokker-Planck equation are considered to high-light the system evolution when system parameters change.In fact, the above numerical scheme and the methodology for examining the stochastic dynamical behaviors work for more general stochastic systems with addi-tive or multiplicativeα-stable Levy noises. In order to show that the above research methodology indeed works for higher dimensional systems, the following two dimen-sional non-Gaussian stochastic system is considered: The corresponding mean exit time, u(x,y), is described by an elliptic partial differential-integral equation. Numerical analysis and simulations are conducted to examine how the mean exit time changes with the system parameters.