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Ergodic and resonant torus doubling bifurcation in a three-dimensional quadratic map

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Abstract

We consider the rich dynamics and bifurcations exhibited by a three-dimensional quadratic map. Torus doubling bifurcations are central to bifurcation theory. Such bifurcations can only occur in a higher dimensional map of dimensions greater than or equal to three. After subsequent doublings, formation of Shilnikov attractors and hyperchaotic attractors are usually observed. It is shown that the map under consideration shows both resonant and ergodic torus doubling bifurcation. We show that the system exhibits resonant torus doubling bifurcation in which the doubled mode-locked periodic orbits lies on a Möbius strip. Additionally, we also illustrate the doubling of ergodic tori and analyze the bifurcation via the use of the second Poincaré section and the method of Lyapunov bundles. The analysis involves the techniques of construction of one-dimensional manifolds, a one-parameter continuation of saddle periodic orbits, and multi-dimensional Newton–Raphson method to locate the saddle periodic orbits.

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The data that support the findings of this study are available within the article.

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Acknowledgements

S.S.M acknowledges fruitful discussions with Professor Motomasa Komuro on the method of Lyapunov bundles. S.S.M also acknowledges insightful discussions with Professor Alexey Kazakov on two-parameter continuation techniques.

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  1. School of Digital Sciences, Digital University Kerala, Technopark Phase IV, Thiruvananthapuram, Kerala, 695317, India

    Sishu Shankar Muni

  2. Indian Institute of Information Technology and Management Kerala, Technopark Road, Thiruvananthapuram, Kerala, 695581, India

    Sishu Shankar Muni

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Muni, S.S. Ergodic and resonant torus doubling bifurcation in a three-dimensional quadratic map. Nonlinear Dyn 112, 4651–4661 (2024). https://doi.org/10.1007/s11071-024-09284-6

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