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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Persistence and Doubling of Chaotic Attractors in Coupled 3-Cell Hopfield Neural Networks

Journal of Applied Nonlinear Dynamics 15(4) (2026) 915--931 | DOI:10.5890/JAND.2026.12.008

Mehmet Onur Fen$^{1}$, Fatma Tokmak Fen$^2$

$^1$ Department of Mathematics, TED University, 06420 Ankara, Turkey

$^2$ Department of Mathematics, Gazi University, 06560 Ankara, Turkey

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Abstract

Two novel phenomena for unidirectionally coupled $3$-cell Hopfield neural networks (HNNs) are investigated. The first one is the persistence of chaos, which means the permanency of sensitivity and infinitely many unstable periodic oscillations in the response HNN even if the networks are not synchronized in the generalized sense. The doubling of chaotic attractors is the second phenomenon realized in this study. It can be achieved when the response network possesses two stable point attractors in the absence of the driving. This feature leads to the formation of two coexisting chaotic attractors with disjoint basins. Lyapunov functions are utilized to deduce the presence of an invariant region, and the sensitivity is rigorously proved. The absence of synchronization is approved via the auxiliary system approach and analysis of conditional Lyapunov exponents. Additionally, quadruple and octuple coexisting chaotic attractors are demonstrated, and the formation of hyperchaos is discussed.

Acknowledgments

The authors would like to thank the editor and anonymous reviewers for the insightful comments which helped to improve the paper.

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