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Chaos and Coexisting Bifurcations in a Novel 3D Autonomous System with a Non-Hyperbolic Fixed Point: Theoretical Analysis and Electronic Circuit Implementation

  • General and Applied Physics
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Abstract

A 3D autonomous chaotic system with the distinguishing feature of having a couple of fixed points, one of which is non-hyperbolic, is proposed. Interestingly, these fixed points become all hyperbolic in other parameters’ regions yielding diverse modes of oscillations in the system. The stability of the equilibria is discussed based on the Routh–Hurwitz stability criterion. The complex dynamics of the proposed system is numerically explored by using phase space trajectory plots, bifurcation diagrams, graphs of Lyapunov exponents, and basins of attraction. It is found that the system experiences period-doubling bifurcation, coexisting bifurcations, periodic windows, and chaos when monitoring its parameters. When the system is tuned to develop non-hyperbolic chaos, the basin of attraction of the strange attractor (coexisting with one of the fixed points) intersects with neighborhood of equilibria which is typical of self-excited oscillations. The coexistence between periodic and chaotic behaviors is found for specific parameter values. The analysis of the basins of attraction for the coexisting attractors reveals extremely complex structures. PSpice simulations based on a suitably designed electronic analog of the system confirm the results of theoretical analysis. The model proposed in this work shows “elegant” mathematical simplicity (i.e., only quadratic nonlinearities) and extremely rich modes of oscillations, and thus may be regarded as a prototypal member of the recently discovered and very restricted class of nonlinear systems developing non-hyperbolic chaos.

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Authors and Affiliations

  1. Unité de Recherche d’Automatique et Informatique Appliquée (URAIA), Department of Electrical Engineering, IUT-FV Bandjoun, University of Dschang (Cameroon), Dschang, Cameroon

    J. Kengne, V. Folifack Signing & G. H. Kom

  2. Department of Electrical Engineering, Golpayegan University of Technology, Golpayegan, Iran

    H. Abdolmohammadi

  3. Unité de Recherche de Matière Condensée, d’Electronique et de Traitement du Signal (URMACETS), Department of Physics, University of Dschang, P.O. Box 67, Dschang, Cameroon

    V. Folifack Signing & G. H. Kom

  4. Nuclear Technology Section, Institute of Geological and Mining Research, P.O. Box 4110, Yaounde, Cameroon

    V. Folifack Signing

  5. Biomedical Engineering Department, Amirkabir University of Technology, Tehran, 15875-4413, Iran

    S. Jafari

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  1. J. Kengne
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Correspondence to J. Kengne.

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Kengne, J., Abdolmohammadi, H., Signing, V.F. et al. Chaos and Coexisting Bifurcations in a Novel 3D Autonomous System with a Non-Hyperbolic Fixed Point: Theoretical Analysis and Electronic Circuit Implementation. Braz J Phys 50, 442–453 (2020). https://doi.org/10.1007/s13538-020-00758-7

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