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Analysis of Three-Dimensional Autonomous Van der Pol–Duffing Type Oscillator and Its Synchronization in Bistable Regime

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Nonlinear Dynamical Systems with Self-Excited and Hidden Attractors

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 133))

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Abstract

This chapter proposes a three-dimensional autonomous Van der Pol-Duffing (VdPD) type oscillator which is designed from a nonautonomous VdPD two-dimensional chaotic oscillator driven by an external periodic source through replacing the external periodic drive source with a direct positive feedback loop. The dynamical behavior of the proposed autonomous VdPD type oscillator is investigated in terms of equilibria and stability, bifurcation diagrams, Lyapunov exponent plots, phase portraits and basin of attraction plots. Some interesting phenomena are found including for instance, period-doubling bifurcation, symmetry recovering and breaking bifurcation, double scroll chaos, bistable one scroll chaos and coexisting attractors. Basin of attraction of coexisting attractors is computed showing that is associated with an unstable equilibrium. So the proposed autonomous VdPD type oscillator belongs to chaotic systems with self-excited attractors. A suitable electronic circuit of the proposed autonomous VdPD type oscillator is designed and its investigations are performed using ORCAD-PSpice software. Orcard-PSpice results show a good agreement with the numerical simulations. Finally, synchronization of identical coupled proposed autonomous VdPD type oscillators in bistable regime is studied using the unidirectional linear feedback methods. It is found from the numerical simulations that the quality of synchronization depends on the coupling coefficient as well as the selection of coupling variables.

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Author information

Authors and Affiliations

  1. Department of Physics, Higher Teacher Training College, University of Bamenda, PO Box 39, Bamenda, Cameroon

    Gaetan Fautso Kuiate

  2. Department of Telecommunication and Network Engineering, IUT-Fotso Victor of Bandjoun, University of Dschang, P. O. Box: 134, Bandjoun, Cameroon

    Victor Kamdoum Tamba

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

    Sifeu Takougang Kingni

Authors
  1. Gaetan Fautso Kuiate
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  2. Victor Kamdoum Tamba
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  3. Sifeu Takougang Kingni
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Corresponding author

Correspondence to Sifeu Takougang Kingni .

Editor information

Editors and Affiliations

  1. School of Electronics and Telecommunications, Hanoi University of Science and Technology, Hanoi, Vietnam

    Viet-Thanh Pham

  2. Research and Development Centre, Vel Tech University, Chennai, Tamil Nadu, India

    Sundarapandian Vaidyanathan

  3. Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece

    Christos Volos

  4. Division of Dynamics, Faculty of Mechanical Engineering, Lodz University of Technology, Łódź, Poland

    Tomasz Kapitaniak

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Kuiate, G.F., Tamba, V.K., Kingni, S.T. (2018). Analysis of Three-Dimensional Autonomous Van der Pol–Duffing Type Oscillator and Its Synchronization in Bistable Regime. In: Pham, VT., Vaidyanathan, S., Volos, C., Kapitaniak, T. (eds) Nonlinear Dynamical Systems with Self-Excited and Hidden Attractors. Studies in Systems, Decision and Control, vol 133. Springer, Cham. https://doi.org/10.1007/978-3-319-71243-7_7

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  • DOI: https://doi.org/10.1007/978-3-319-71243-7_7

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-71242-0

  • Online ISBN: 978-3-319-71243-7

  • eBook Packages: EngineeringEngineering (R0)

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