Stability analysis of multiple equilibria for recurrent neural networks with discontinuous Mexican-hat-type activation function

This paper is concerned with stability analysis of multiple equilibria for recurrent neural networks. A new type of activation function, namely, discontinuous Mexican-hat-type activation function, is proposed for recurrent neural networks. Then with the aid of the fixed point theorem, some sufficient conditions for coexistent multiple equilibria are obtained to guarantee that such n-neuron recurrent neural networks can have at least 4ⁿ equilibria. In view of the theory of strict diagonal dominance matrix, further stability analysis reveals that 3ⁿ equilibria are locally exponentially stable. The new results considerably improve the existing multistability results in the literature.