Probability Density Evolution Of Nonlinear Systems Induced By Gaussian Noise

The Fokker-Planck equation is one of the most important equation in nonlinear scientific research, which has been widely applied in the fields of biosystem, process of chemistry and Nonequilibrium statistical physics during the past decades. What’s more, the investigation of the evolution on the probability distribution is one of the most studied and utilized in nonlinear dynamics and stochastic theory. One reason is that the long time behavior of system is reflected by the stationary probability distribution, and the system will be ruled by the long time behavior after all sorts of different length of time in the transient process; Another reason is that many sig-nificant properties are determined by the evolution process of the time-dependent solution. Many works have been done on the study of the stationary probability distribution in nonlinear systems around the stable state, while the time-dependent solution of the multidimensional system is rarely studied and the time-dependent solution, together with the stationary probability distribution of the multidimen-sional system is also barely involved. Based on the theory and technique of random process and nonlinear dynamical systems, the evolution problem of the nonlinear system subject to Gaussian noises excitation is further investigated, the important effects and application are also revealed in the evolution process. The main contents and conclusions are as follows:1The influences of uncorrelated Gaussian noises on the evolution problem around unstable state of the nonlinear system are investigated in detailed. Focusing on the phenomena of the probability distribution induced by the intensity and the self-correlation time of the Gaussian noises. The Langevin equation of the general nonlinear dynamic system which is excited by stochastic force reads x＝f(x)＋9(x)ξ(t)＋η(t), where f(x), g(x) is function of x, f(x) is particularly nonlinear function. ξ(t) is Gaussian colored noise, while η(t) is Gaussian white noise. As is known to all, the exact analytical solution of some special Fokker-Planck equation such as Ornstein-Uhlenbeck process and Kramers equation is easily obtained, while difficult and chal-lenging is known for general equation. So all kinds of approximate methods play an important role in solving this problem. There are two difficulties solved in this model:one is the limited self-correlation time of the Gaussian colored noises, which make the system remember the history, so that lose the character of Markov. An-other is the character of nonlinear. Firstly, the non-Markovian process induced by the limited self-correlation time of the Gaussian colored noises is transformed into the Markovian process by using the method of enlarge dimensional, the above model can be expressed in a couple of first-order differential equations: Secondly, based on the Taylor expansion, the nonlinear system is changed for linear system. Finally, applying the approximate result to the logistic model, the influ-ence and variation are discussed by analyzing figures. The results reveal that the approximate method for the system not only can respond to vivid changes but also can verify the rapid of the evolution around unstable state.2A procedure for predicting the response of the second-order Duffing system with damping force under Gaussian noises excitation by using the Ω-expansion of the Green function is proposed. The evolution of probability distribution form unstable state to stable state under the effects of uncorrelated Gaussian noises and correlated Gaussian noises, respectively, are explored in detailed. A special bistable system that are characterized by the second-order Duffing system is very relevant as a bright paradigm for the evolution of probability distribution. The equation of motion of this system is of the form x+γx-αx+βx3=xξ1(t)+ξ2(t).On the basis of the above existing approximate results by using Ω-expansion of the Green function and the method of enlarge dimensional, the influence of Gaussian noises and damping force on the evolution of the Duffing system are analyzed by simulating the approximate results. The reason for this is that the time-dependent solution around unstable state can’t reflect the characteristics of this system. Fur-ther the approximate method not only reveal the evolution process of the multidi-mensional system but also exhibit dynamic process for the evolution of probability distribution form unstable state to stable state. What’s more, a good agreement is found by comparing analytical and numerical simulations when system parameters are equal.