Dynamics Research On Several Neural Network Models With Discontinuous Activation Functions

In this paper, the dynamical behaviors of several classes neural networks with not only mixed time delays, i.e., time varying delays and distributed delays, but also discontinuous activation functions have been investigated by using topological degree theory, Leray-Schauder alternative theorem in multi-valued analysis, fixed point theorem, technique of inequality, generalized Lyapunov-like approach and matrix theory. The contents mainly include the existence and uniqueness of equilibrium point or almost periodic solution, global asymptotic stability, convergence of the associated output of the solution, global convergence in finite time, and so on. Our works mainly include three parts.Firstly, we investigate the dynamical behaviors of a novel class of CohenGrossberg neural networks with not only mixed time delays, i.e., time varying delays and distributed delays, but also discontinuous activations by Leray-Schauder alternative theorem in multi-valued analysis, generalized Lyapunov-like approach and technique of inequality. We derive some sufficient conditions for the existence,uniqueness of the equilibrium, global exponential asymptotic stability of the solution, and study the convergence of output solution. It should be mentioned that the activation functions in the model can be unbounded, non-monotonic, even at the discontinuous point the left limit mustn’t lower right limit, which has seldom been found in other study on Cohen-Grossberg neural networks with discontinuous activations. Some recent results in the literature are generalized and significantly improved. The theoretical analysis are verified by numerical illustration.Secondly, the global stability has been investigated for a novel class of competitive neural networks not only mixed time delays, i.e., time varying delays and distributed delays, but also discontinuous activations. Without presuming the boundedness, monotonic of activation functions and the left limit lower right limit of activation function at its discontinuous point, we firstly get a set of sufficient condition ensuring the existence, uniqueness of the equilibrium, global asymptotic stability of the solution by Leray-Schauder alternative theorem in multi-valued analysis, general Lyapunov method and study the convergence of output solution;Second, by M-matrix, topological degree theory of set-valued map and general Lyapunov method, we get another set of sufficient condition; At last, as the discontinuity of the activation function, we investigate the convergence in finite time,which has seldom been found in existing literature. We also got the sufficient condition ensuring global exponential stability of equilibrium point when the activation function is monotone nondecreasing. Our results extend and improve some results in existing literature.Finally, we study the dynamical behaviors of the almost solution of a class of Cohen-Grossberg neural networks with not only mixed time delays, i.e., time varying delays and distributed delays, but also discontinuous activation functions, which may be unbounded and monotone nondecreasing, by Matrix theorem,fixed point theorem and generalized Lyapunov method, including existence, global exponential stability and so on. The new results in this paper generalize and significantly improve some recent results in the literature. Several examples are presented to demonstrate the effectiveness of the theoretical results.