Stability Analysis of Fractional Order Memristor Synapse-coupled Hopfield Neural Network with Ring Structure

TL;DR

This paper investigates the stability of a fractional-order memristor-based Hopfield neural network with ring structure, analyzing how fractional order and network size influence equilibrium stability and potential chaotic behavior.

Contribution

It introduces a novel fractional-order memristor synapse-coupled Hopfield neural network with ring topology and provides stability conditions dependent on fractional order and network size.

Findings

Stability depends on fractional order and number of neurons.

Numerical simulations confirm theoretical stability conditions.

Potential for chaos increases with higher fractional order.

Abstract

A memristor is a nonlinear two-terminal electrical element that incorporates memory features and nanoscale properties, enabling us to design very high-density artificial neural networks. To enhance the memory property, we should use mathematical frameworks like fractional calculus, which is capable of doing so. Here, we first present a fractional-order memristor synapse-coupling Hopfield neural network on two neurons and then extend the model to a neural network with a ring structure that consists of n sub-network neurons, increasing the synchronization in the network. Necessary and sufficient conditions for the stability of equilibrium points are investigated, highlighting the dependency of the stability on the fractional-order value and the number of neurons. Numerical simulations and bifurcation analysis, along with Lyapunov exponents, are given in the two-neuron case that substantiates the theoretical findings, suggesting possible routes towards chaos when the fractional order of the system increases. In the n-neuron case also, it is revealed that the stability depends on the structure and number of sub-networks.