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Revealing Chaos Synchronization Below the Threshold in Coupled Mackey–Glass Systems

Author

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  • Marat Akhmet

    (Department of Mathematics, Middle East Technical University, Dumlupınar Boulevard, 06800 Ankara, Turkey)

  • Kağan Başkan

    (Department of Physics, Middle East Technical University, Dumlupınar Boulevard, 06800 Ankara, Turkey)

  • Cihan Yeşil

    (Department of Physics, Middle East Technical University, Dumlupınar Boulevard, 06800 Ankara, Turkey)

Abstract

This study presents a novel concept in chaos synchronization, delta synchronization of chaos, which reveals the presence of chaotic models evolving in unison even in the absence of generalized synchronization. Building upon an analysis of unpredictability in Poincaré chaos, we apply this approach to unilaterally coupled time-delay Mackey–Glass models. The main novelty of our investigation lies in unveiling the synchronization phenomenon for a coupling constant below the synchronization threshold, an unattainable domain for conservative methods. Furthermore, we rigorously examine the coexistence of generalized synchronization and complete synchronization of unpredictability, which is a special case of delta synchronization, above the threshold. Therefore, the threshold is no longer a requirement for the synchronization of chaos in view of the present results. Additionally, transitions to fully chaotic regimes are demonstrated via return maps, phase portraits, and a bifurcation diagram. The findings are substantiated by tables and novel numerical characteristics.

Suggested Citation

  • Marat Akhmet & Kağan Başkan & Cihan Yeşil, 2023. "Revealing Chaos Synchronization Below the Threshold in Coupled Mackey–Glass Systems," Mathematics, MDPI, vol. 11(14), pages 1-15, July.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:14:p:3197-:d:1198995
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    References listed on IDEAS

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    1. Akhmet, Marat & Yeşil, Cihan & Başkan, Kağan, 2023. "Synchronization of chaos in semiconductor gas discharge model with local mean energy approximation," Chaos, Solitons & Fractals, Elsevier, vol. 167(C).
    2. Li, Li, 2015. "Bifurcation and chaos in a discrete physiological control system," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 397-404.
    3. Shahverdiev, E.M. & Nuriev, R.A. & Hashimov, R.H. & Shore, K.A., 2006. "Chaos synchronization between the Mackey–Glass systems with multiple time delays," Chaos, Solitons & Fractals, Elsevier, vol. 29(4), pages 854-861.
    4. Caneco, Acilina & Grácio, Clara & Leonel Rocha, J., 2009. "Kneading theory analysis of the Duffing equation," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1529-1538.
    5. Ma, Jun & Mi, Lv & Zhou, Ping & Xu, Ying & Hayat, Tasawar, 2017. "Phase synchronization between two neurons induced by coupling of electromagnetic field," Applied Mathematics and Computation, Elsevier, vol. 307(C), pages 321-328.
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