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Dynamical analysis of a simple autonomous jerk system with multiple attractors

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Abstract

In recent years, tremendous research efforts have been devoted to simple chaotic oscillators based on jerk equation that involves a third-time derivative of a single variable. In the present paper, we perform a systematic analysis of a simple autonomous jerk system with cubic nonlinearity. The system is a linear transformation of Model MO5 first introduced in Sprott (Elegant chaos: algebraically simple flow. World Scientific Publishing, Singapore, 2010) prior to the more detailed study by Louodop et al. (Nonlinear Dyn 78:597–607, 2014). The basic dynamical properties of the model are investigated including equilibria and stability, phase portraits, frequency spectra, bifurcation diagrams, and Lyapunov exponent plots. It is shown that the onset of chaos is achieved via the classical period-doubling and symmetry-restoring crisis scenarios. One of the key contributions of this work is the finding of a window in the parameter space in which the jerk system experiences the unusual and striking feature of multiple attractors (e.g. coexistence of four disconnected periodic and chaotic attractors). Basins of attraction of various coexisting attractors are computed showing complex basin boundaries. Among the very few cases of lower-dimensional systems (e.g. Newton–Leipnik system) capable of displaying such type of behaviour reported to date, the jerk system with cubic nonlinearity considered in this work represents the simplest and the most ‘elegant’ prototype. An appropriate electronic circuit describing the jerk system is designed and used for the investigations. Results of theoretical analyses are perfectly traced by laboratory experimental measurements.

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Authors and Affiliations

  1. Laboratoire d’Automatique et Informatique Appliquée (LAIA), Department of Electrical Engineering, IUT-FV Bandjoun, University of Dschang, Dschang, Cameroon

    J. Kengne & Z. T. Njitacke

  2. Laboratory of Electronics and Signal Processing, Department of Physics, University of Dschang, P.O. Box 67, Dschang, Cameroon

    Z. T. Njitacke & H. B. Fotsin

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  1. J. Kengne
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  2. Z. T. Njitacke
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  3. H. B. Fotsin
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Correspondence to J. Kengne.

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Kengne, J., Njitacke, Z.T. & Fotsin, H.B. Dynamical analysis of a simple autonomous jerk system with multiple attractors. Nonlinear Dyn 83, 751–765 (2016). https://doi.org/10.1007/s11071-015-2364-y

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  • DOI: https://doi.org/10.1007/s11071-015-2364-y

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