Extreme Multistability of a Fractional-Order Discrete-Time Neural Network

A Othman Almatroud

doi:10.3390/fractalfract5040202

Abstract

At present, the extreme multistability of fractional order neural networks are gaining much interest from researchers. In this paper, by utilizing the fractional ℑ-Caputo operator, a simple fractional order discrete-time neural network with three neurons is introduced. The dynamic of this model are experimentally investigated via the maximum Lyapunov exponent, phase portraits, and bifurcation diagrams. Numerical simulation demonstrates that the new network has various types of coexisting attractors. Moreover, it is of note that the interesting phenomena of extreme multistability is discovered, i.e., the coexistence of symmetric multiple attractors.

Highlights

We present the dynamics of the fractional-order discrete-time neural network (FoDtNN) (13) by considering some numerical simulation
In order to reveal the extreme multistability phenomena of the FoDtNN (13), the maximum LEs and bifurcation diagrams of the state variable x3 are calculated as shown in Figures 7 and 9, where the fractional order = are chosen as 0.98 and 0.1, respectively
Referring to fractional-order discrete-time neural networks, this paper has introduced a new FoDtNN with an extreme multistability property

Summary

Introduction

Coexisting bifurcations and attractors have recently been observed in discrete-time systems (maps). Fractional order discrete time systems with the left Caputo operator were investigated and used in a variety of disciplines since [12,13,14]. It provides a new way to describe fractional order systems resulting by replacing the Caputo fractional derivative with Caputo difference operator, without any loss in the memory effects, which leads to better results. It is the first fractional-order discrete-time neural network to exhibit extreme multistability, or the coexistence of several attractors given the same variety of system parameters and initial conditions.

Preliminaries

The Mathematical Model

Dynamical Analysis and Numerical Simulations

Coexisting of Symmetric Attractors Depending on System Parameter T1

Extreme Multistability Analysis

Conclusions