Dynamics Of Two Different Kinds Of Nonlinear Stochastic Systems

Based on the Langevin equation of the nonlinear dynamical system, the Fokker-Planck equation is derived by using some approximative approaches. Then, discuss the effects of noises and time delays on the dynamical behaviors of the system by analyzing some corresponding numerical simulation curves. Much attention is paid on the steady-state properties of the system, the mean first passage time (MFPT) and stochastic resonance (SR). The main contents of this paper are listed as following:Firstly, the steady-state properties of a cancer development system driven by non-Gaussian coloured noise and a bistable system with different kinds of time delays driven by coloured noises are investigated. The non-Gaussian coloured noise is simplified to Gaussian coloured noise by the path integral method, then, the expression of the stationary probability distribution (SPD) of the system is derived and the effects of noise parameters on the SPD are discussed. By employing the probability density approach which is based on the approximation of probability density distributions, the expression of the SPD of the bistable system is obtained. The effects of the noise parameters and the delays on the SPD as well as the P-bifurcation behaviors are discussed.Secondly, the MFPTs in a cancer development system driven by noises with coloured correlations and a time-delayed bistable system driven by noises with white correlations are investigated. By using the fastest decline approach, the MFPT of the cancer development system is obtained. Then, the influences of non-Gaussian colored noise on the MFPT are discussed. In the time-delayed bistable system, we mainly study the effects of noise intensities, cross-correlation strength and two delayed times on the MFPT.Finally, the stochastic resonance in a nonlinear system with two different kinds of time delays driven by coloured noises is investigated. In the part of theoretical analysis of general model, we simplify the non-Markov stochastic process caused by coloured noises and time delays to Markov process. Then, apply the general theory to a bistable system. Through the two-state theory, the expression of signal-to-noise ratio (SNR) is derived. The influences of two delayed times on the SNR are discussed and the SNR as a function of multiplicative or additive noise intensity with different values of the self-correlation times and delayed times are studied.