The Analysis Of Existence And Stability Of Periodic Solution For Competitive Neural Networks

The competitive neural networks as one of the popular artificial neural networks havereceived significant attention in the study of neural network dynamics. This paper isconcerned with the existence and the stability of periodic solution for competitive neuralnetworks.The paper is divided into five chapters.In the first chapter, there are some main introduction on the research meaning of neuralnetworks and the biological mechanisms, the research background and the research results ofthe competitive neural networks. Meanwhiles, the definition and the research significance ofthe time delays is explained in the chapter1.The second, three, four chapters are the focus of this article. In chapter2, the existenceand the stability of periodic solution for competitive neural networks with time-varying delaysand distributed delays on time scales is studied. As far as we know, there is still no workdedicated to studying the stability of competitive neural networks on time scales. Therefore,this chapter make some promotion for the stability of competitive neural networks. Under thecondition without assuming the boundedness of the activation functions, we prove theexistence of periodic solution for competitive neural networks with varying delays anddistributed delays on time scales by using the theory of time scales and the contractionmapping theory. The stability of periodic solution for the system is discussed by constructinga suitable Lyapunov function. Finally, two numerical simulations are given to illustrate thevalidity of the theoretical results.In chapter3, the existence and the stability of anti-periodic solution for competitiveneural networks with delays in the leakage terms on time scales is investigated. By means ofthe method of coincidence degree and M-matrices, the existence of anti-periodic solution ofthe system on time scales is discussed. Then, based on Lyapunov functional methods and thetheory of time scales, some sufficient conditions are obtained to guarantee the globalexponential stability of anti-periodic solutions of the model. Finally, two examples are givento illustrate the effectiveness of the theoretical results. It is worth mentioning that the resultsof this chapter not only remove the hypotheses that the activation functions are bounded, evenand odd, but also remove the requirement that the activation functions are zero at the zero.Therefore, the results in this chapter give some improvements to previous ones. In addition,The method that combine the method of coincidence degree and M-matrices is used in theproof of the existence of anti-periodic solution for the system, which enrich the context of thechapter and diversify the problem-solving skills in the paper. Therefore, the work has certainresearch value. In chapter4, based on the quadratic convex combination technique, some sufficientconditions are gained to ensure the stability for competitive neural networks withtime-varying delays and delays in the leakage term by constructing Lyapunov function andcalculating its derivation. According to our survey, this is the first paper to discuss thestability of competitive neural networks by utilizing the quadratic convex combinationtechnique. Therefore, it is meaning to dedicate to this work.In chapter5, we summarize the innovation of this paper, at the same time we put forwardthe improvement direction of this article.In the end, all the references of this paper are listed.