Hopfield Neuronal Network of Fractional Order: A note on its numerical integration

TL;DR

This paper investigates the fractional-order Hopfield neural network using ABM numerical integration, revealing spurious bifurcation branches that depend on initial conditions and step size, with implications for stability analysis.

Contribution

It introduces a novel analysis of fractional-order Hopfield networks, highlighting the impact of integration step size on bifurcation branch accuracy and stability.

Findings

Spurious bifurcation branches depend on initial conditions.

Reducing step size aligns spurious branches with true branches.

Spurious branches are delayed and persist across parameter space.

Abstract

In this paper, the commensurate fractional-order variant of an Hopfield neuronal network is analyzed. The system is integrated with the ABM method for fractional-order equations. Beside the standard stability analysis of equilibria, the divergence of fractional order is proposed to determine the instability of the equilibria. The bifurcation diagrams versus the fractional order, and versus one parameter, reveal a strange phenomenon suggesting that the bifurcation branches generated by initial conditions outside neighborhoods of unstable equilibria are spurious sets although they look similar with those generated by initial conditions close to the equilibria. These spurious sets look ``delayed'' in the considered bifurcation scenario. Once the integration step-size is reduced, the spurious branches maintain their shapes but tend to the branches obtained from initial condition within neighborhoods of equilibria. While the spurious branches move once the integration step size reduces, the branches generated by the initial conditions near the equilibria maintain their positions in the considered bifurcation space. This phenomenon does not depend on the integration-time interval, and repeats in the parameter bifurcation space.