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Multiple scales and normal forms in a ring of delay coupled oscillators with application to chaotic Hindmarsh–Rose neurons

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Abstract

We study the appearance and stability of spatiotemporal periodic patterns like phase-locked oscillations, mirror-reflecting waves, standing waves, in-phase or antiphase oscillations, and coexistence of multiple patterns, in a ring of bidirectionally delay coupled oscillators. Hopf bifurcation, Hopf–Hopf bifurcation, and the equivariant Hopf bifurcation are studied in the viewpoint of normal forms obtained by using the method of multiple scales which is a kind of perturbation technique, thus a clear bifurcation scenario is depicted. We find time delay significantly affects the dynamics and induces rich spatiotemporal patterns. With the help of the unfolding system near Hopf–Hopf bifurcation, it is confirmed in some regions two kinds of stable oscillations may coexist. These phenomena are shown for the delay coupled limit cycle oscillators as well as for the delay coupled chaotic Hindmarsh–Rose neurons.

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Acknowledgements

The authors wish to express their gratitude to the editors and the reviewers for the helpful comments. This work is supported in part by NNSF of China (No. 11031002), by the Heilongjiang Provincial Natural Science Foundation (No. A200806), and by the Fund of Education Department of Heilongjiang Province (No. 12521085).

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Authors and Affiliations

  1. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

    Yuxiao Guo, Weihua Jiang & Ben Niu

  2. Department of Applied Mathematics, Harbin University of Science and Technology, Harbin, 150080, China

    Ben Niu

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  1. Yuxiao Guo
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  2. Weihua Jiang
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Correspondence to Weihua Jiang.

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Guo, Y., Jiang, W. & Niu, B. Multiple scales and normal forms in a ring of delay coupled oscillators with application to chaotic Hindmarsh–Rose neurons. Nonlinear Dyn 71, 515–529 (2013). https://doi.org/10.1007/s11071-012-0678-6

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