Cited by 81
Huazhong University of Science and Technology
Publishes on Mathematical and Theoretical Epidemiology and Ecology Models, Nonlinear Differential Equations Analysis, Fiscal Policy and Economic Growth. 64 papers and 725 citations.
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SUMMARY This paper establishes Razumikhin‐type theorems on general decay stability for stochastic functional differential equations. This improves existing stochastic Razumikhin‐type theorems and can make us examine the stability with general decay rate in the sense of the p th moment and almost sure. These stabilities may be specialized as the exponential stability and the polynomial stability. When the almost sure stability is examined, the conditions of this paper may defy the linear growth condition for the drift term, which implies that the theorems of this paper can work for some cases to which the existing results cannot be applied. This paper also examines some sufficient criteria under which this stability is robust. To illustrate applications of our results clearly, this paper also gives two examples and examines the exponential stability and the polynomial stability, respectively. Copyright © 2011 John Wiley & Sons, Ltd.
In this article, we investigate the stochastic suppression and stabilisation of nonlinear systems. Given an unstable differential equation , in which f satisfies the one-sided polynomial growth condition, we introduce two Brownian noise feedbacks and therefore stochastically perturb this system into the nonlinear stochastic differential equation . This article shows that appropriate β may guarantee that this stochastic system exists as a unique global solution although the corresponding deterministic may explode in a finite time. Then sufficiently large q may ensure that this stochastic system is almost surely exponentially stable.
Hopfield (1984 Proc. Natl Acad. Sci. USA 81 3088–92) showed that the time evolution of a symmetric neural network is a motion in state space that seeks out minima in the system energy (i.e. the limit set of the system). In practice, aneuralnetwork is often subject to environmental noise. It is therefore useful and interesting to find out whether the system still approaches some limit set under stochastic perturbation. In this paper, we will give a number of useful bounds for the noise intensity under which the stochastic neural network will approach its limit set.