Synchronization Of Random Dynamical System With Ornstein-Uhlenbeck Noise

This paper mainly study synchronization phenomena of random dynamical systemdriven by Ornstein-Uhlenbeck process, it is a generalization of the synchronizationphenomena of random dynamical system driven by Gaussian noises or Lévy noises.Thispaper mainly focus on the following three aspects:Firstly, the definition of Ornstein-Uhlenbeck process is extended to the form of mean-reverting, when the initial meet a special distribution, Ornstein-Uhlenbeck process is aninvariant distribution; Secondly, discusses the synchronization phenomena of randomdynamical system with additive and multiplicative Ornstein-Uhlenbeck noises, when thesystem exists additive Ornstein-Uhlenbeck noises, by using the conclusions proved in theresearch of randomdynamical system with additive Gaussian noises, it proves the existenceof random attractors of random dynamical system, while the corresponding differencebetween any pair of solutions of the equation tends to zero, thus, the random attractorsconsistsofsingletonsets,whichprovestheexistenceanduniquenessofstochasticstationarysolutions of dissipatively coupled stochastic differential equations, when the system existslinear multiplicative Ornstein-Uhlenbeck noises, the stochastic differential equations aretransformed to random ordinary differential equations using a transformation whichinvolving the corresponding Ornstein-Uhlenbeck processes, and then by calculatingcorresponding difference between any pair of solutions of the random ordinary differentialequations and the distance from the origin, prove the existence and uniqueness of stochasticstationary solution of dissipatively coupled stochastic differential equations; Finally, weverify the stochastic stationary solutions of dissipatively coupled stochastic differentialequations converge to the stochastic stationary solutions of "average" equation, which givesthe results of synchronization of random dynamical system driven by additive and linearmultiplicative Ornstein-Uhlenbeck noises.These concludes have a good generalization, when coefficient of viscosity of Langevin’ s equation equal to zero, it can be extended to synchronization phenomenon ofrandom dynamical system driven by Gaussian noises, Lévy noise or fractional Brownianmotion noises.