Dynamical Behavior of Complex-Valued Hopfield Neural Networks with Discontinuous Activation Functions

Abstract

This paper presents some theoretical results on dynamical behavior of complex-valued neural networks with discontinuous neuron activations. Firstly, we introduce the Filippov differential inclusions to complex-valued differential equations with discontinuous right-hand side and give the definition of Filippov solution for discontinuous complex-valued neural networks. Secondly, by separating complex-valued neural networks into real and imaginary part, we study the existence of equilibria of the neural networks according to Leray---Schauder alternative theorem of set-valued maps. Thirdly, by constructing appropriate Lyapunov function, we derive the sufficient condition to ensure global asymptotic stability of the equilibria and convergence in finite time. Numerical examples are given to show the effectiveness and merits of the obtained results.