Multistability of Fractional-Order Recurrent Neural Networks With Discontinuous and Nonmonotonic Activation Functions

Yujiao Huang,Tiaoyang Cai,Haixia Long,Xiaoyan Yuan,Xinggang Fan

doi:10.1109/access.2019.2935776

Abstract

The coexistence of multiple stable equilibria in recurrent neural networks is an important dynamic characteristic for associative memory and other applications. In this paper, the existence and local Mittag-Leffler stability of multiple equilibria are investigated for a class of fractional-order recurrent neural networks with discontinuous and nonmonotonic activation functions. By using Brouwer's fixed point theory, several conditions are established to ensure the existence of 5 n equilibria, in which all the coemponents of 4 n equilibria are located in the continuous intervals of the activation functions. and some of the components of 5 n - 4 n equilibria are located at some discontinuous points of the activation functions. The introduction of discontinuous activation functions makes the neural networks have more equilibria than those with continuous activation functions. Furthermore, some criteria are proposed to ensure local Mittag-Leffler stability of 3 n equilibria. The introduction of nonmonotonic activation functions makes the neural networks have more stable equilibria than those with monotonic activation functions. Two examples are given to illustrate the effectiveness of the results.

Highlights

In recent decades, dynamic characteristics of artificial neural networks have been deeply studied due to their wide application prospects in the fields of pattern recognition, associative memory, optimization and so on [1]–[6]
When neural networks are used in pattern recognition and associative memory, neural networks are expected to have multiple locally stable equilibria, which are designed as ideal patterns for pattern recognition and associative memory [12]–[14]
MAIN RESULTS When neural networks are used in pattern recognition, associative memory and other applications, the stable equilibria of neural networks are designed as ideal patterns

Summary

INTRODUCTION

Dynamic characteristics of artificial neural networks have been deeply studied due to their wide application prospects in the fields of pattern recognition, associative memory, optimization and so on [1]–[6]. It is noteworthy that few literatures have discussed the multistability of fractional-order recurrent neural networks with discontinuous activation functions. It is necessary to investigate multistability of fractional-order neural networks with discontinuous and nonmonotonic activation functions. 1. We will establish sufficient conditions to guarantee the existence of 4n equilibria for fractional-order recurrent neural networks with the characteristic that all the components of the equilibria are located in the continuous intervals of discontinuous and nonmonotonic activation functions. The number of equilibria of neural networks can be greatly increased by setting a discontinuous point in the activation function.

PRELIMINARIES

NUMERICAL EXAMPLES

CONCLUSION