Random Exponential Attractor For Non-autonomous Zakharov Lattice System With Multiplicative White Noise

The random exponential attractor is an important tool to describe the long-term asymptotic behavior of the state of random dynamical systems,and it is one of the frontier problems of dynamical systems at present.The random exponential attractor is a positive invariant measurable set with a finite fractal dimension and attracts any trajectories at an exponential rate.Therefore,the existence of random exponential attractor implies that the asymptotic behavior of the state of systems can be described by a finite number of independent parameters,which will bring the possibility for numerical simulation or practical applications.Lattice dynamical system has been widely used in materials science,physics,chemistry,biology and other fields,and is one of the important research objects in applied mathematics.In recent years,the research on the asymptotic behavior of the Zakharov lattice system with important practical backgrounds has obtained many results,but the dimension of its random attractors and the rate of attracting trajectories have not been studied until now.This thesis mainly consider the existence of random exponential attractor for nonautonomous Zakharov lattice systems with multiplicative white noise.The structure of this thesis is as follows: The first chapter presents the brief introduction of infinitedimensional dynamical systems and lattice dynamical systems,and then presents the background and research status of Zakharov lattice system.The second chapter introduces the concepts of continuous cocycles and random exponential attractors.In chapter three,firstly,the Zakharov random lattice system with multiplicative white noise is equivalently transformed through the Ornstein-Uhlenbeck process into a random system without noise term,and its solutions generate a continuous cocycle on the phase space of the infinite sequences.Then,we present the existence theorem of random exponential attractors for the continuous cocycle.Finally,we give the boundedness and tail estimate of the solution when the coefficients of random terms are appropriately small.In chapter four,it is further proved that the above continuous residues satisfy Lipschitz continuity and the random squeezing property on the tempered closed random absorption set.Then the boundedness of the expectation of some random variables are proved.Finally,the existence of random exponential attractor for Zakharov random lattice system isobtained.