Stability Analysis Of Several Classes Of Delayed Nonlinear Systems And Hopf Bifurcation

The dynamical analysis of nonlinear systems is one of the important issues ofcontrol theory, which includes complex contents, the corresponding theory underwentcontinual development. We often choose the differential equations (or differenceequations, corresponding to discrete state) as the mathematical tools to describe thenonlinear systems. In the paper, the author focuses on three important nonlinear systems,including Bidirectional Associative Memory (BAM)、Hopfiled neural network、onemodified epidemic model for computer viruses (SIRA). For the first two nonlinearsystems, the necessary conditions guaranteeing the global stability of the systems arederived while the conservatism is strived to be relaxed, with the full consideration of theeffect of the time delays、stochastic perturbations and impulse effects. For the SIRAmodel with significant applications, the paper not only investigates the virus-freeequilibrium point, the condition for the existence of the positive equilibrium point andthe global stability, but also it analyzes the Hopf bifurcation taking the time delay τ asbifurcation parameter. Finally, some formulae for determining the direction, stabilityand other properties of bifurcating periodic solutions are obtained. Specifically, themain contents are as follows:①Studying the complicated effects that the impulse effects have on the stabilityof the BAM neural networks under the stochastic perturbations. Based on a generalizedtwo-order Halanay inequality, we obtain the sufficient conditions guaranteeing theglobal stability of the systems. Besides, the application of the Halanay inequality solvesthe problem that an applicable Lyapunov functional is not easily to be found usually,that is to say, our method simplifies the chosen of Lyapunov functional.②By means of Lyapunov function method and Razumikhin technique, weinvestigate the global exponential stability of the Hopfield neural networks under thestochastic perturbations and impulse effects. The obtained results get rid of therestriction on frequency of the impulse, namely, not requiring the impulse interval to belarger than time delay, through which the results can be applicable to more situations.Furthermore, the results do not have any demand on the stability of the original system,that is to say, even the original system is not stable the impulse can stabilize the system.This finding provides us a new viewpoint to investigate the impact of the impulse on thestability of system. ③According to the unique properties of the computer viruses, the paper proposesa modified SIRA model originated from some famous epidemic models. We furtheranalyze the stability and the conditions for the occurrence of a Hopf bifurcation, andfinally acquire some formulae for determining other properties of bifurcating periodicsolutions.