AccScience Publishing / NSCE / Online First / DOI: 10.36922/NSCE025440016
RESEARCH ARTICLE

Theoretical analysis and microcontroller-based design of a cyclic network of three mutually coupled Duffing oscillators

Mahirakhon Rakhmatullaeva1 Khabibullo Nosirov2 Kengne Jacques3* Chedjou Jean Chamberlain4
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1 Department of Information Technology, Faculty of Computer Engineering, Tashkent University of Information Technologies, Tashkent, Uzbekistan
2 Department of Television and Radio Broadcasting Systems, Faculty of Information Technologies, Tashkent University of Information Technologies, Tashkent, Uzbekistan
3 Research Unit in Automation and Applied Computer Science (UR-AIA), Department of Electrical Engineering, University Institute of Technology, Fotso Victor, Bandjoun, Koung-Khi, Cameroon
4 Transportation Informatics Group (TIG), Department of Information Technology, Faculty of Technical Sciences, University of Klagenfurt, Austria
Received: 2 November 2025 | Revised: 27 December 2025 | Accepted: 5 January 2026 | Published online: 4 February 2026
© 2026 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
Abstract

Recently, coupled oscillator systems have garnered significant interest due to their rich dynamics and applications in various fields. This article examines the collective dynamics of a ring network comprising three mutually interacting autonomous Duffing oscillators. Analytical and numerical methods are employed to elucidate the network’s overall behavior as a function of its parameters. We show that the route leading to multi-scroll chaos begins with a series of Hopf bifurcations (associated with 8 of the 27 equilibrium points), followed by a sequence of period-doubling bifurcations, boundary crises, and merging crises, ultimately giving rise to a multi-scroll attractor as one of the system parameters (e.g., a coupling strength) is varied. This route is characterized by several regions of multi-stability, where multiple attractors of different topologies coexist in varying numbers, depending on the exact value of the control parameter. This mechanism is elucidated using key analytical tools, such as bifurcation diagrams, phase portraits, and basins of attraction corresponding to competing attractors. Reducing the number of interactions within the network led to profound modifications in the locations of the equilibrium points, the mechanisms underlying the onset of chaos, and the topology of the resulting multi-scroll attractor. An experimental validation is carried out by considering a physical implementation of the model using the Arduino microcontroller. This study provides valuable insights that serve as an introduction to understanding the dynamics of significantly more complex networks of Duffing oscillators.

Graphical abstract
Keywords
Heterogeneous multi-stability
Microcontroller-based realization
Ring network of Duffing oscillators
Route to multi-scroll chaos
Funding
None.
Conflict of interest
Kengne Jacques is an Editorial Board Member of this journal, but was not in any way involved in the editorial and peer-review process conducted for this paper, directly or indirectly. The other authors declare no competing interests.
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