Research On The Application Of Generalized Cell Mapping Method In Stochastic Response And Exit Problem

The complex dynamical behaviors of nonlinear stochastic systems have been an important project in the scientific research.The generalized cell mapping method was proposed as an effective numerical method,which has been widely applied in the global analysis of nonlinear systems,and shows a good prospect of application in the study of stochastic dynamical systems.With the generalized cell mapping method,this dissertation aims to investigate the response probability density functions,and the first-passage time and exit location distribution in the exit problem of nonlinear stochastic dynamical systems.The main contents and results are as follows.The response probability density functions of nonlinear stochastic dynamical systems have been studied by the generalized cell mapping method.The N-dimensional nonlinear system excited by Gaussian white noise is considered,and the method for obtaining the transient and steady-state response probability density functions is introduced.In the generalized cell mapping,the Monte Carlo simulation method is applied to compute the transient probability between any two state cells,then the transient response probability distribution is obtained with the algorithm of sparse matrix,and the steady-state probability distribution is obtained after the termination condition is given.In the illustrative examples,the steady-state response probability density functions of an excitable Fitz Hugh-Nagumo model are firstly studied,which shows the effectiveness of the generalized cell mapping method.And the effect of noise intensity and time scale ratio one the excited state has been analyzed respectively.Then the transient and steady-state response of a noisy system with non-negative real-power restoring force driven by periodic and Gaussian white noise is then considered.Combined with the global properties of the noise-free system,the evolutionary process of the transient response probability density functions is revealed.It is found in the steady-state response analysis that the stochastic P-bifurcation is induced by the change of the frequency of periodic excitation,and the chaotic response occurs when the non-negative power is increased.Consider the nonlinear systems with both periodic and Gaussian white noise excitations,the elements in transient probability matrix are computed by using the short-time Gaussian approximation,which can greatly improve the effectiveness of the generalized cell mapping method for the response probability density function.Firstly the whole period of system is divided into several small intervals,so that the short-time transient probability density function on each of them can be approximated as a Gaussian one,in which the mean vector and covariance matrix are obtained by numerically solving the moment equations with Gaussian closure.Consequently,a set of short-time transient probability matrix can be computed,and then we can construct the mapping of the period.Secondly,we give computational considerations on the issues in the implementation of the generalized cell mapping method,including the choice of the division parameter and integral region,and the quantitative assessment of the proposed method.At last,in the examples we consider the steady-state response of a system with the non-viscous exponential damping.The moment equations with Gaussian closure are derived for the equivalent three-dimensional system,and the effect of damping coefficient and relaxation parameter on the probability density functions has been discussed,respectively.We also investigate the transient and steady-state response of the smooth and discontinuous(SD)oscillator.Its moment equations are given for both smooth and discontinuous cases.The results of response analysis show that the chaotic saddle of deterministic system affects the shape of stochastic response probability density function,and stochastic P-bifurcation can be induced by the change of the smooth parameter.The first-passage time statistics of nonlinear stochastic dynamical systems has been studied by the generalized cell mapping method.Using a signal-degree-of-freedom system as an example,the problem of first-passage time statistics is described in combination of the attractors and basins of attraction of the corresponding noise-free system.The method for obtaining the probability density of first-passage time and the mean first-passage time is introduced by adding proper absorbing boundary condition and initial condition in the framework of the generalized cell mapping.The first example applies the proposed method to the first-passage time problem of the inversed Van der Pol oscillator under Gaussian white noise excitation.The validity of the proposed method is demonstrated by the results from direct Monte Carlo simulation.In the second example,we have studied the first-passage time statistics properties of a second order bistable system with multiplicative Poisson white noise.Both symmetric and asymmetric cases have been investigated,and the effects of the noise intensity and mean arrival rate of impulse on the first-passage time statistics have been discussed respectively.It reveals the non-Gaussian effect on the first-passage time of Poison white noise.With the same noise intensity Poisson white noise can make for a faster firstpassage,and this effect is more obvious when the mean arrival rate is smaller.Based on the investigation of the response analysis and first-passage time,the generalized cell mapping method is extended to the study of exit location distribution in the stochastic exit problem.According to the global properties of the deterministic system,a proper region is chosen and divided into cell state space.All the cells are classified into three new categories based on their position.The one-step transient probability matrix is constructed by numerical integration and Monte Carlo simulation,in which the absorbing boundary condition for the exit location is added.Considering the initial position at the attractor with probability 1,we can obtain the steady-state distribution using the Markov theory and the algorithm of sparse matrix,then extract the exit location distribution of system.In the given examples,the exit location distribution in the Kramers exit problem,the Maier-Stein model and a prey-predator model have been studied.The accuracy of the proposed generalized cell mapping method is verified by the results from the direct simulation.It is found that the generalized cell mapping method has obvious advantage over the direct simulation when the noise intensity is small.