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Three-Dimensional Chaotic Autonomous System with a Circular Equilibrium: Analysis, Circuit Implementation and Its Fractional-Order Form

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Abstract

A three-dimensional autonomous chaotic system with a circular equilibrium is investigated in this paper. Some dynamical properties and behaviors of this system are described in terms of equilibria, eigenvalue structures, bifurcation diagrams, Lyapunov exponents, time series and phase portraits. For specific parameters, the system displays periodic and chaotic attractors. The physical existence of the chaotic behavior found in the proposed system is verified by using the Orcad-PSpice software and experimental verification. A good qualitative agreement is shown between the experimental results, PSpice and numerical simulations. Furthermore, the commensurate fractional-order version of the system with a circular equilibrium is numerically studied. It is found that chaos exists in this system with order less than three. By tuning the commensurate fractional order, the system with a circular equilibrium displays chaotic and periodic attractors, respectively. Finally, chaos synchronization of identical fractional-order chaotic systems with a circular equilibrium is achieved by using the unidirectional linear error feedback coupling. It is shown that the fractional-order chaotic system can achieve synchronization for appropriate coupling strength.

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Acknowledgments

S.T.K. and G.R.K. thank Prof. Elisabeth Ngo Bum, the Director of the Institute of Mines and Petroleum Industries (University of Maroua, Cameroon), for creating a good environment with perfect balance between teaching and research time.

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Authors and Affiliations

  1. Department of Mechanical and Electrical Engineering, Institute of Mines and Petroleum Industries, University of Maroua, P.O. Box 46, Maroua, Cameroon

    Sifeu Takougang Kingni & Guy Richard Kol

  2. School of Electronics and Telecommunications, Hanoi University of Science and Technology, 01 Dai Co Viet, Hanoi, Vietnam

    Viet-Thanh Pham

  3. Department of Biomedical Engineering, Amirkabir University of Technology, 424 Hafez Ave., Tehran, 15875-4413, Iran

    Sajad Jafari

  4. School of Geology and Mining Engineering, University of Ngaoundéré, P.O. Box 454, Ngaoundéré, Cameroon

    Guy Richard Kol

  5. Laboratory of Modelling and Simulation in Engineering, Biomimetics and Prototypes (LaMSEBP) and TWAS Research Unit, Department of Physics, Faculty of Science, University of Yaoundé I, P.O. Box 812, Yaoundé, Cameroon

    Sifeu Takougang Kingni & Paul Woafo

  6. Applied Physics Research Group (APHY), Vrije Universiteit Brussel, Pleinlaan 2, 1050, Brussels, Belgium

    Sifeu Takougang Kingni

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  1. Sifeu Takougang Kingni
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Correspondence to Sifeu Takougang Kingni.

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Kingni, S.T., Pham, VT., Jafari, S. et al. Three-Dimensional Chaotic Autonomous System with a Circular Equilibrium: Analysis, Circuit Implementation and Its Fractional-Order Form. Circuits Syst Signal Process 35, 1933–1948 (2016). https://doi.org/10.1007/s00034-016-0259-x

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  • DOI: https://doi.org/10.1007/s00034-016-0259-x

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