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Codimension two bifurcation in a coupled FitzHugh–Nagumo system with multiple delays

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  • Achouri, Houssem
  • Aouiti, Chaouki
  • Hamed, Bassem Ben

Abstract

In this paper, a coupled FitzHugh–Nagumo system with multiple delays is considered. In a first step, the critical point at which a zero root of multiplicity two occurs in the characteristic equation is constructed. In a second step, in order to ensure that all the roots of the characteristic equation except for the double zero root have negative real parts, the zeros of a third and a fourth degree exponential polynomial are studied. Moreover, the critical values where the Bogdanov–Takens bifurcation occurs are derived. By using the normal form theory and reduction on the centre manifold, the truncated normal form is obtained, and throughout the bifurcation diagram, its dynamical behaviours are studied. Finally, a numerical example is given to demonstrate our results.

Suggested Citation

  • Achouri, Houssem & Aouiti, Chaouki & Hamed, Bassem Ben, 2022. "Codimension two bifurcation in a coupled FitzHugh–Nagumo system with multiple delays," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    • Handle: RePEc:eee:chsofr:v:156:y:2022:i:c:s0960077922000352
    DOI: 10.1016/j.chaos.2022.111824
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    References listed on IDEAS

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    1. Shengwei Yao & Huonian Tu, 2014. "Stability Switches and Hopf Bifurcation in a Coupled FitzHugh-Nagumo Neural System with Multiple Delays," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-13, July.
    2. Chaouki Aouiti & El abed Assali & Jinde Cao & Ahmed Alsaedi, 2018. "Global exponential convergence of neutral-type competitive neural networks with multi-proportional delays, distributed delays and time-varying delay in leakage delays," International Journal of Systems Science, Taylor & Francis Journals, vol. 49(10), pages 2202-2214, July.
    3. Anderson Hoff & Juliana Santos & Cesar Manchein & Holokx Albuquerque, 2014. "Numerical bifurcation analysis of two coupled FitzHugh-Nagumo oscillators," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 87(7), pages 1-9, July.
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    Cited by:

    1. Li, Jin & Guo, Ying & Liu, Xiaotong & Zhang, Yifan, 2024. "Stabilization of highly nonlinear stochastic coupled systems with Markovian switching under discrete-time state observations control," Chaos, Solitons & Fractals, Elsevier, vol. 178(C).
    2. Chen, Jing & Xiao, Min & Wu, Xiaoqun & Wang, Zhengxin & Cao, Jinde, 2022. "Spatiotemporal dynamics on a class of (n+1)-dimensional reaction–diffusion neural networks with discrete delays and a conical structure," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    3. Chesebro, Anthony G. & Mujica-Parodi, Lilianne R. & Weistuch, Corey, 2023. "Ion gradient-driven bifurcations of a multi-scale neuronal model," Chaos, Solitons & Fractals, Elsevier, vol. 167(C).

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