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Chaos and Hyperchaos in Two Coupled Identical Hindmarsh – Rose Systems

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Abstract

The dynamics of two coupled neuron models, the Hindmarsh – Rose systems, are studied. Their interaction is simulated via a chemical coupling that is implemented with a sigmoid function. It is shown that the model may exhibit complex behavior: quasi-periodic, chaotic and hyperchaotic oscillations. A phenomenological scenario for the formation of hyperchaos associated with the appearance of a discrete Shilnikov attractor is described. It is shown that the formation of these attractors leads to the appearance of in-phase bursting oscillations.

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Notes

  1. Using the symbol \(C\)(m,n) we denote the limit cycle, the indices in brackets indicate the dimension of the stable (m) and unstable (n) manifolds of the corresponding point in the Poincaré section

  2. In our numerical experiments, we used the fixed initial conditions: \(x_{10}=1.12\), \(y_{10}=2.67\), \(z_{10}=-0.42\), \(x_{20}=-0.91\), \(y_{20}=3.54\), \(z_{20}=-0.38\)

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ACKNOWLEDGMENTS

The authors thank Dr. Alexey Kazakov and Prof. Igor Belykh for useful discussions of this problem.

Funding

This work is supported by the Russian Science Foundation (project no. 20-71-10048, Sections 2.2, 3, 5). NVS, AAB are partially supported by the Laboratory of Dynamical Systems and Applications NRU HSE, grant of the Ministry of Science and Higher Education of the RF, ag. no. 075-15-2022-1101 (Sections 2.1, 4).

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  1. HSE University, ul. Bolshaya Pecherskaya 25/12, 603155, Nizhny Novgorod, Russia

    Nataliya V. Stankevich, Andrey A. Bobrovskii & Natalya A. Shchegoleva

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  1. Nataliya V. Stankevich
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  2. Andrey A. Bobrovskii
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  3. Natalya A. Shchegoleva
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Correspondence to Nataliya V. Stankevich, Andrey A. Bobrovskii or Natalya A. Shchegoleva.

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65P20, 92B25

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Stankevich, N.V., Bobrovskii, A.A. & Shchegoleva, N.A. Chaos and Hyperchaos in Two Coupled Identical Hindmarsh – Rose Systems. Regul. Chaot. Dyn. 29, 120–133 (2024). https://doi.org/10.1134/S1560354723540031

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