The Dynamic Behavior For Several Types Of Stochastic Lattice Dynamical Systems With Time Delays

Firstly,a type of random cellular neural networks with unbounded distributed time delays is considered in this paper.More concretely,we investigate the local and global existence of the solution to the random lattice dynamical system,and prove the uniqueness for the solution.Then the long time behavior of the random lattice dynamical system is analyzed.Moreover,we establish the comparison theorem for the cellular neural networks with unbounded distributed time delays,and derive the existence of the extremal D-complete quasi-trajectories for the cocycle generated by the random cellular neural networks with unbounded distributed time.Secondly,an infinite lattice model of a recurrent neural network with random connection strengths between neurons is developed and analyzed.To incorporate the presence of various type of delays in the neural networks,both discrete and distributed time varying delays are considered in the model.For the existence of random pullback attractors and periodic attractors,the nonlinear terms of the resulting system are not expected to be Lipschitz continuous,but only satisfy a weaker continuity assumption along with growth conditions,under which the uniqueness of the underlying Cauchy problem may not hold.Then after extending the concept and theory of monotone multivalued semiflows to the random context,the structure of random pullback attractors with or without periodicity is investigated.In particular,the existence and stability properties of extremal random complete trajectories are studied.Thirdly,we investigate a class of stochastic recurrent neural networks with discrete and distributed delays for both biological and mathematical interests.We do not assume any Lipschitz condition on the nonlinear term,just a continuity assumption together with growth conditions so that the uniqueness of the Cauchy problem fails to be true.Moreover,the existence of pullback attractors with or without periodicity is presented for the multi-valued noncompact random dynamical system.In particular,a new method for checking the asymptotical compactness of solutions to the class of nonautonomous stochastic lattice systems with infinite delay is used.Next,the dynamic behavior for a general class of stochastic lattice dynamical system with unbounded distributed time delays is considered in Chapter 5.In fact,we present the local existence,global existence and the uniqueness of the solution to this type of stochastic lattice differential equation,and illustrate the asymptotic stability of the stochastic delayed lattice dynamical system by proving the existence of the pullbackattractor to the cocycle generated by the stochastic lattice differential equation.Finally,a type of abstract stochastic nonlinear lattice differential equation with delays is presented.Assuming the nonlinear terms noncompact conditions and growth conditions,we obtain the existence of periodic orbits for the stochastic lattice dynamical system by making use of the fixed-point theorem.The theoretical results can be applied to some practical and valuable equations such as stochastic nonlinear cellular neural networks with delays.