Research On Dynamical Behavior Of Nonlinear Stochastic System With Delays

Applications of the determinate system described by ordinary differential equation in the domain of physics, engineering technology, biology and econimic system etc. are well known. But, with development of science and technology, description on practical problem is required to be better and better. Therefore, effects of the stochastic factors cann't be neglected, and analysis for practical processes should be necessary to convert from usual determinate method to stochastic approach, then description on practical systems should be naturally made from determinate differential equation to stochastic one, that is to say, determinate systems become stochastic systems. Stability is an important dynamic characteristic and one of the main target of engineering design. The complexity of environment that practical engineering systems is in and action which is required to accomplish are not completely same, so the paper shall go deep into the dynamic behaviors of nonlinear stochastic systems. Lasalle theorems is an important tool to investigate the stability of stochastic systems. It cancells the requirement of the positive Lyapunov function, and so is abroadly adopted in practical engineering problems. This paper has established firstly the Lasalle-type theorems for general neutral stochastic functional differential equations by using It? formula, semi-martingale convergence theorem, kolmogorov-?entsov theorem etc. and inequality technique . The results includes the stochastic-type Lasalle theorem on stochastic differential systems and stochastic functional differential systems in the existing references. Meanwhile, for the rigor conditions about pth moment and the linear growth conditons in existing stochastic-type Lasalle theorem on stochastic functional differential systems, the improved stochastic-type Lasalle theorem on stochastic functional differential systems is established, and an example is given to verify the stochastic-type Lasalle theorem of the paper. In the existing references about one Lyapunov function to dicuss the stability of stochastic functional differential systems, this paper investigates stability of general stochastic functional differential systems. Meanwhile, applying It? formula, semimartingale convergence theorem and H?lder inequality, the paper established criterions on asymptotic stabilities, polynomial stabilities and exponential stabilities of general neutral stochastic functional differential systems . Compared with the classical ones about stochastic stability, the results in the paper made best use of the beneficial work of stochastic perturbation item, and show that the unstable systems is on the contrary stable if it gets proper stochastic perturbation. The stochastic hybrid systems with Markovian switching was first systemically investigated. The paper established the existence-uniqueness theorem and estimation of the solution of stochastic hybrid systems with Markovian switching by the existence-uniqueness theorem and estimation of the solution of the neutral stochastic differential systems with delay and Burkholder-Davis-Gundy inequality. Furthermore, the existence-uniqueness theorem of the solution of the stochastic hybrid systems without the linear growth condition was given by the truncation technique. In the end, sufficient criterions on asymptotic stabilities, polynomial stabilities and exponential stabilities of the stochastic hybrid systems were establised by the generalized It? formula, semimartingale convergence theorem. With the time evolution, Hopfield neural networks will converge to the equilibrium point set of the networks or the minima in the energy function. The networks deal with optimization computation and association memory is indeed based on the characteristic of the networks. Because high-order Hopfield neural networks have more extensive applications than the usual Hopfield neural networks anda neural network in practice is often subject to environmental noise, the paper will dicuss the convergence problem on the high-order Hopfield neural networks under the perturbation of environmental noise. The paper shows that under the perturbation of environmental noise high-order Hopfield neural networks still converge to the equilibrium point set of the networks or the minima in the energy function. Meanwhile, the bounds of noise intensity were given, and our result cancels the requirement of symmetry of the networks.