Research On The Dynamic Behavior Of Delayed Neural Networks With External Stimuli And Inertial Terms

As we all know, the introduce of time delay often brings great changes on the dynamical behavior in the fields of network systems. But the time delay has to be considered into the factors which influence systems due to the limitation of the signal transformation velocity. The delayed neural network is an important part of the delayed large-scale system, which exhibits the rich and colorful dynamical behavior. Due to the important applications in signal processing, moving image processing as well as optimizing problems, the dynamical behavior of the delayed nueral network has been studied by many researchers all the time, especially the dynamical behaviors of the global stability, the local stability (including asymptotic stability, exponential stability and absolute stability) as well as bifurcation and chaos, and many intensive and valuable results have been derived. This thesis mainly focuced on two types of neural network(Hopfield neural network and Inertial neural network). On the one hand, we also studied the global stability, the local stability, the Hopf bifurcation, the resonant codimension two bifurcation and chaos for autonomous system; On the other hand, we studied the local stability and the Hopf bifurcation for the nonautonomous system.The main contents and originalities in this paper can be summarized as follows:①Delay-dependent asymptotic stability for neural networks with distributed delays and discrete delaysSignal transformation can not be finished in time due to the finite signal propogation time in biological networks, or to the finite switching speed of amplifiers in electronic neural networks, so the models depending the discrete time delays are not complete, the more exact models should include the distributed time delays. Moreover, we study the delay-dependent asymptotic stability for neural networks with distributed delays and discrete delays via constructing suitable Lyapunov-Krasovskii functions. Our studies are very important, on the one hand, the stability criteria for two kinds of delays are derived, and it is significant to give the more exact models and to predict the dynamical behavior of the models; on the other hand, comparing with the delay-independent stability criteria, the delay-dependent stability criteria are easier to obtain for the small delay neural network and have less limitations for the parameters of system.②Local stability and Hopf bifurcation in an inertial two-neuron system with time delay Under certain conditions, neurons exhibite a quasi-active membrane, which can be modeled by a phenonmenokogical inductance that allows the membrane to behave like a bandpass filter, enabling electrical tuning, or spatio-temporal filtering. So there are some strong biological backgrounds for the inertial delayed neural networks. In this thesis, we have studied the local stability and the existence of Hopf bifurcation at first, then we derived the direction of Hopf bifurcation and stability criterias of bifurcating periodic solutions by applying the center manifold theorem and the normal form theory. These studies are very important. On the one hand, bifurcations, which involve emergence of oscillatory behaviors, may provide an explanation for the parameter sensitivity observed in practice in many realistic small-scale networks such as the Internet, the electrical power grids, and the biological neural networks; and on the other hand, if we understand more about the bifurcation behaviors of small-scale networks, we can apply the existing effective bifurcation control method to achieve some desirable system behaviors that benefit the networks.③Local stability and resonant codimension-two bifurcation in an inertial two-neuron system with time delayIt is a very important topic to predict and avoid resonant in delayed neural network. In this dissertation, we focuse on the discussion for the roots of the system's characteristic equation based on the studies of the above problem, then using the delay as bifurcation parameter, we derive the conditions for the occurrence of two pairs of pure imaginary roots(i.e.±iω1 ,±iω2) are derived. When the frequency ratioω1 :ω2 is rational number, the phenomena of resonant will occure. This result and analytical method can provide the theoretical basis for the couple of system amplitudes, the synchronization of frequency, and can also reduced the occurrence of system resonant.④Local stability and Hopf bifurcation for delayed inertial neural network under periodic excitationThere is evidence from the experimental studies that assemblies of cells in the visual cortex oscillate synchronously in response to external stimuli. So we have studied the local stability, the existence and the direction for bifurcating periodic solutions of delayed inertial neural model with periodic stimuli (non-autonomous system). By using the center manifold theorem and the averaged method in the theory of nonlinear oscillation, first we derived the center manifold of the system is derived. Then the averaged equations of the system are obtained. Finally the bifurcating equations via analyzing the Jacobian matrix for the averaged equations are obtained, furthermore we got the direction of the periodic solution and the bifurcating points from the bifurcation equations. Since most of the existing literature on theoretical studies of bifurcation problem is predominantly concerned with autonomous systems, Literature dealing with bifurcation for non-autonomous systems appears to be scarce. The analyzing method for the system's dynamical behavior in this dissertation may be helpful to solve these problems.⑤Local stability and Hopf bifurcation in Hopfield neural network with time delay and periodic excitationThe application of the Hopfield neural network has permeated many fields such as biology, physics, geology, etc. And it has also been widely applied to intelligent control, pattern recognition and nonlinear optimization. Moreover it is significant to study the dynamic behavior of this model which has wide use. In this dissertation, we study the dynamic behavior of the Hopfield model under the influence of time delay and periodic stimuli by using the center manifold theorem as well as the perturbation techniques.