Multistability Analysis of Fractional-Order State-Dependent Switched Competitive Neural Networks With Sigmoidal Activation Functions

Abstract

This work explores the issue of multistability for a competitive neural network (NN) class with sigmoidal activation functions (AFs) involving state-dependent switching and fractional-order derivative. Specifically, first, we consider three different switching point locations, and establish some sufficient criteria ensuring that NNs with n-neurons can have, and only have, $5^{n_{1}}\cdot 3^{n_{2}}$ equilibrium points (EPs) with $n_{1}+n_{2}=n$ , by utilizing the geometric features of the sigmoidal functions, the fixed point theorem, the Filippov’s EP definition, and the contraction mapping theorem. Then, based on novel Lyapunov functions and by applying the fractional-order calculus theory, it is demonstrated that $3^{n_{1}}\cdot 2^{n_{2}}$ out of $5^{n_{1}}\cdot 3^{n_{2}}$ total EPs are locally stable. This work’s investigation reveals that competitive NNs with switching afford more storage capacity compared to the nonswitching case. Additionally, our results are valid for the integer-order and fractional-order switched NNs, improving and generalizing current works. Furthermore, two numerical examples and an application example of associative memory are provided to validate the effectiveness of the theoretical findings, and the way various fractional orders affect the NNs’ convergence speed is shown through simulations.