Infinitely many coexisting hidden attractors in a new hyperbolic-type memristor-based HNN

Isaac Sami Doubla; Balamurali Ramakrishnan; Zeric Njitacke Tabekoueng; Jacques Kengne; Karthikeyan Rajagopal

doi:10.1140/epjs/s11734-021-00372-x

2024 Impact factor 2.3

Special Topics

Bursting Oscillations, Bifurcation Delay and Multi Stability in Complex Nonlinear Systems

Eur. Phys. J. Spec. Top. (2022) 231: 2371-2385
https://doi.org/10.1140/epjs/s11734-021-00372-x

Regular Article

Infinitely many coexisting hidden attractors in a new hyperbolic-type memristor-based HNN

Isaac Sami Doubla¹^,2, Balamurali Ramakrishnan⁴, Zeric Njitacke Tabekoueng³, Jacques Kengne¹ and Karthikeyan Rajagopal⁴

¹ Research Unit of Automation and Applied Computer (URAIA), Electrical Engineering Department of IUT-FV, University of Dschang, P.O. Box 134, Bandjoun, Cameroon
² Research Unit of Condensed Matter, Electronics and Signal Processing (UR-MACETS) Department of Physics, Faculty of Sciences, University of Dschang, P.O. Box 67, Dschang, Cameroon
³ Department of Electrical and Electronic Engineering, College of Technology (COT), University of Buea, P.O. Box 63, Buea, Cameroon
⁴ Center for Nonlinear Systems, Chennai Institute of Technology, Chennai, Tamilnadu, India

Received: 1 August 2021
Accepted: 7 December 2021
Published online: 3 January 2022

Abstract

In this article, a new model of Hopfield Neural Network (HNN) with two neurons considering a synaptic weight with a hyperbolic-type memristor is studied. Equilibrium points analysis shows that the system has an unstable line of equilibrium in the absence of the external stimuli (i.e. $I_{1} = 0)$ $I_{1} =0)$ and presents no equilibrium point in the presence of the external stimuli (i.e. $I_{1} \neq 0)$ $I_{1} \ne 0)$ ; hence the model admits hidden attractors. Analyses are carried out for both cases $I_{1} = 0$ $I_{1} =0$ and $I_{1} \neq 0$ $I_{1} \ne 0$ using appropriate tools (bifurcation diagrams and the Lyapunov exponents, phase portraits, etc.). For both modes of operations, the system exhibits complex homogeneous and heterogeneous bifurcations, respectively marked by a large number of coexisting attractors. The roads to chaos unfold in the same scenario of period doubling. The Hamiltonian plot for the case $I_{1} = 0$ $I_{1} =0$ allows us to observe an increase in the energy of the neuronal structure when it migrates from regular oscillations to irregular ones. Moreover, the existence of infinitely many coexisting homogeneous solutions (chaotic or periodic) is revealed for case $I_{1} = 0$ $I_{1} =0$ . In contrast, for $I_{1} \neq 0$ $I_{1} \ne 0$ (i.e $I_{1} = 0.1)$ $I_{1} =0.1)$ the new model presents infinitely many coexisting hidden heterogeneous attractors (periodic and chaotic). An electronic circuit design of the new hyperbolic memristor enables the analog computer of the whole system to be designed for future engineering applications. Simulation results based on this analog computer in PSpice confirm those of the numerical investigations.