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Bursting oscillations in a 3D system with asymmetrically distributed equilibria: Mechanism, electronic implementation and fractional derivation effect

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  • Kingni, S.T.
  • Nana, B.
  • Mbouna Ngueuteu, G.S.
  • Woafo, P.
  • Danckaert, J.

Abstract

In this paper, the dynamical behavior of a three-dimensional (3D) autonomous oscillator (Pehlivan and Uyarogˇlu, 2012) is further investigated in some detail (including local stability and bifurcation structures analysis) in order to reveal bursting dynamics. This system has the unusual feature of involving three equilibria asymmetrically distributed. We find that for specific parameters, the system exhibits periodic and chaotic bursting oscillations. By rescaling its dynamical variables, we show that this system evolves on two time scales, corresponding respectively to fast and slow dynamics that lie at the basis of bursting dynamics. The mechanism underlying this phenomenon is drawn via the bifurcation analysis of the fast subsystem with respect to the slow subsystem variable. This reveals that the bursting oscillations found there result from the system switching between the unstable and the stable states of the only equilibrium point of the fast subsystem. This class of bursting has been referred to as “source/sink bursting of node/focus type”. In addition, an analog circuit is designed and implemented to realize experimentally the periodic and chaotic bursting oscillations. Good agreement is shown between the numerical and the experimental results. Further, the commensurate fractional-order version of the chaotic slow–fast system is studied using stability theorem of fractional-order systems and numerical simulations. By tuning the fractional-order, the slow–fast system displays a wide variety of dynamical behaviors ranging from chaotic bursting to fixed point dynamics via continuous chaotic spiking. Because of the asymmetric distribution of the system equilibrium points, this transition includes peculiar phenomena such as transient chaos and the coexistence of point and chaotic attractors.

Suggested Citation

  • Kingni, S.T. & Nana, B. & Mbouna Ngueuteu, G.S. & Woafo, P. & Danckaert, J., 2015. "Bursting oscillations in a 3D system with asymmetrically distributed equilibria: Mechanism, electronic implementation and fractional derivation effect," Chaos, Solitons & Fractals, Elsevier, vol. 71(C), pages 29-40.
    • Handle: RePEc:eee:chsofr:v:71:y:2015:i:c:p:29-40
    DOI: 10.1016/j.chaos.2014.11.011
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    References listed on IDEAS

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    1. Vladimir K. Vanag & Lingfa Yang & Milos Dolnik & Anatol M. Zhabotinsky & Irving R. Epstein, 2000. "Oscillatory cluster patterns in a homogeneous chemical system with global feedback," Nature, Nature, vol. 406(6794), pages 389-391, July.
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    Cited by:

    1. Bao, B.C. & Wu, P.Y. & Bao, H. & Xu, Q. & Chen, M., 2018. "Numerical and experimental confirmations of quasi-periodic behavior and chaotic bursting in third-order autonomous memristive oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 106(C), pages 161-170.
    2. Kingni, Sifeu Takougang & Pham, Viet-Thanh & Jafari, Sajad & Woafo, Paul, 2017. "A chaotic system with an infinite number of equilibrium points located on a line and on a hyperbola and its fractional-order form," Chaos, Solitons & Fractals, Elsevier, vol. 99(C), pages 209-218.
    3. Anague Tabejieu, L.M. & Nana Nbendjo, B.R. & Woafo, P., 2016. "On the dynamics of Rayleigh beams resting on fractional-order viscoelastic Pasternak foundations subjected to moving loads," Chaos, Solitons & Fractals, Elsevier, vol. 93(C), pages 39-47.
    4. Zhou, Chengyi & Xie, Fei & Li, Zhijun, 2020. "Complex bursting patterns and fast-slow analysis in a smallest chemical reaction system with two slow parametric excitations," Chaos, Solitons & Fractals, Elsevier, vol. 137(C).
    5. Lin, Y. & Liu, W.B. & Bao, H. & Shen, Q., 2020. "Bifurcation mechanism of periodic bursting in a simple three-element-based memristive circuit with fast-slow effect," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    6. Kengne, J. & Njikam, S.M. & Signing, V.R. Folifack, 2018. "A plethora of coexisting strange attractors in a simple jerk system with hyperbolic tangent nonlinearity," Chaos, Solitons & Fractals, Elsevier, vol. 106(C), pages 201-213.
    7. Zhang, Shaohua & Zhang, Hongli & Wang, Cong & Ma, Ping, 2020. "Bursting oscillations and bifurcation mechanism in a permanent magnet synchronous motor system with external load perturbation," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    8. Kengne, Jacques & Mogue, Ruth Line Tagne & Fozin, Theophile Fonzin & Telem, Adelaide Nicole Kengnou, 2019. "Effects of symmetric and asymmetric nonlinearity on the dynamics of a novel chaotic jerk circuit: Coexisting multiple attractors, period doubling reversals, crisis, and offset boosting," Chaos, Solitons & Fractals, Elsevier, vol. 121(C), pages 63-84.
    9. Bao, B.C. & Wu, P.Y. & Bao, H. & Wu, H.G. & Zhang, X. & Chen, M., 2018. "Symmetric periodic bursting behavior and bifurcation mechanism in a third-order memristive diode bridge-based oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 109(C), pages 146-153.
    10. Tchitnga, R. & Mezatio, B.A. & Fozin, T. Fonzin & Kengne, R. & Louodop Fotso, P.H. & Fomethe, A., 2019. "A novel hyperchaotic three-component oscillator operating at high frequency," Chaos, Solitons & Fractals, Elsevier, vol. 118(C), pages 166-180.
    11. Kingni, Sifeu Takougang & Jafari, Sajad & Pham, Viet-Thanh & Woafo, Paul, 2017. "Constructing and analyzing of a unique three-dimensional chaotic autonomous system exhibiting three families of hidden attractors," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 132(C), pages 172-182.
    12. Lin, Yi & Liu, Wenbo & Hang, Cheng, 2023. "Revelation and experimental verification of quasi-periodic bursting, periodic bursting, periodic oscillation in third-order non-autonomous memristive FitzHugh-Nagumo neuron circuit," Chaos, Solitons & Fractals, Elsevier, vol. 167(C).

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