Random Slow Manifolds Of Multi-scale Stochastic Dynamical Systems And Applications

Lots of dynamical systems involve the interplay of two time scales.For example,the Lorenz-Krishnamurthy model,the FitzHugh-Nagumo system,the van der Pol oscillator,the settling of inertial particles under uncertainty,and the stiff stochastic chemical systems all involve two time scales.The dimension of a two-scale system can be reduced to that of the slow variables by a slow manifolds with exponential tracking property if it exists.The slow manifolds of slow-fast system is a special invariant manifolds which is studied extensively.Solutions on the slow manifolds evolve relatively slow compared to the fast variables.We study a type of Wong-Zakai approximation for the random slow manifolds of a slow-fast stochastic dynamical system.We choose the integrated Ornstein-Uhlenbeck process to substitute Brownian motion in the system as the Wong-Zakai approximation system.An Ornstein-Uhlenbeck process is introduced historically to depict the velocity of the particle in Brownian motion.Its integration is regarded as the displacement of the particle.Confining trajectories of the original system to the Wong-Zakai random slow manifolds,we get a reduced system for the original system.Based on Smoluchowski-Kramers approximation,we devise an estimation method for an unknown parameter in a high dimensional Newton equation of motion by a low dimensional equation.Further,based on Wong-Zakai approximation about the random slow manifolds,the reduced system for the original system captures some qualitative properties about the original system.Employing this reduced Wong-Zakai system,we devise an accurate estimate method for an unknown system parameter of the original system.From the perspective of random slow manifolds,we consider the reasonableness of the simulation results on the maximum likelihood transition pathway.Employing the reduced system by the random slow manifolds,we compare the stochastic bifurcation phenomenon between the reduced system and the original system.The slow reduction system can capture the stochastic bifurcation information of the original system qualitatively.