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Effect of nonlinear dissipation on the basin boundaries of a driven two-well Rayleigh–Duffing oscillator

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  • Siewe, M. Siewe
  • Cao, Hongjun
  • Sanjuán, Miguel A.F.

Abstract

The Rayleigh oscillator is one canonical example of self-excited systems. However, simple generalizations of such systems, such as the Rayleigh–Duffing oscillator, have not received much attention. The presence of a cubic term makes the Rayleigh–Duffing oscillator a more complex and interesting case to analyze. In this work, we use analytical techniques such as the Melnikov theory, to obtain the threshold condition for the occurrence of Smale-horseshoe type chaos in the Rayleigh–Duffing oscillator. Moreover, we examine carefully the phase space of initial conditions in order to analyze the effect of the nonlinear damping, and in particular how the basin boundaries become fractalized.

Suggested Citation

  • Siewe, M. Siewe & Cao, Hongjun & Sanjuán, Miguel A.F., 2009. "Effect of nonlinear dissipation on the basin boundaries of a driven two-well Rayleigh–Duffing oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1092-1099.
    • Handle: RePEc:eee:chsofr:v:39:y:2009:i:3:p:1092-1099
    DOI: 10.1016/j.chaos.2007.05.007
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    References listed on IDEAS

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    1. Cao, Hongjun & Seoane, Jesús M. & Sanjuán, Miguel A.F., 2007. "Symmetry-breaking analysis for the general Helmholtz–Duffing oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 34(2), pages 197-212.
    2. Feudel, F. & Witt, A. & Gellert, M. & Kurths, J. & Grebogi, C. & Sanjuán, M.A.F., 2005. "Intersections of stable and unstable manifolds: the skeleton of Lagrangian chaos," Chaos, Solitons & Fractals, Elsevier, vol. 24(4), pages 947-956.
    3. Siewe, M. Siewe & Kakmeni, F.M. Moukam & Bowong, S. & Tchawoua, C., 2006. "Non-linear response of a self-sustained electromechanical seismographs to fifth resonance excitations and chaos control," Chaos, Solitons & Fractals, Elsevier, vol. 29(2), pages 431-445.
    4. Yamapi, René, 2006. "Synchronization dynamics in a ring of four mutually inertia coupled self-sustained electrical systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 366(C), pages 187-196.
    5. Lazzouni, Sihem A. & Siewe Siewe, M. & Moukam Kakmeni, F.M. & Bowong, Samuel, 2006. "Slow flow solutions and chaos control in an electromagnetic seismometer system," Chaos, Solitons & Fractals, Elsevier, vol. 29(4), pages 988-1001.
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    Cited by:

    1. Zhou, Liangqiang & Chen, Fangqi, 2022. "Chaos of the Rayleigh–Duffing oscillator with a non-smooth periodic perturbation and harmonic excitation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 192(C), pages 1-18.
    2. Yu, Yue & Zhang, Cong & Chen, Zhenyu & Lim, C.W., 2020. "Relaxation and mixed mode oscillations in a shape memory alloy oscillator driven by parametric and external excitations," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).

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