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Symmetry-breaking analysis for the general Helmholtz–Duffing oscillator

Author

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  • Cao, Hongjun
  • Seoane, Jesús M.
  • Sanjuán, Miguel A.F.

Abstract

The symmetry breaking phenomenon for a general Helmholtz–Duffing oscillator as a function of a symmetric parameter in the nonlinear force is investigated. Different values of this parameter convert the general oscillator into either the Helmholtz or the Duffing oscillator. Due to the variation of the symmetric parameter, the phase space patterns of the unperturbed Helmholtz–Duffing oscillator will cause a huge difference between the left-hand homoclinic orbit and the right-hand one. In particular, the area of the left-hand homoclinic orbits is a strictly monotonously decreasing function, while the area of the right-hand homoclinic orbit varies only in a very small range. There exist distinct local supercritical and subcritical saddle-node bifurcations at two different centers. The left-hand and the right-hand existing regions of the harmonic solutions of the Helmholtz–Duffing oscillator created by the left-hand and the right-hand saddle-node bifurcation curves will lead to different transition in the amplitude–frequency plane. There exists also a critical frequency which has the effect that the left-hand homoclinic bifurcation value is equal to the right-hand homoclinic bifurcation value. And, if the amplitude coefficient of the Helmholtz–Duffing oscillator is used as the control parameter, and it is larger than the same left-hand and right-hand homoclinic bifurcation, then the global stability of the system will be destroyed at a lowest cost. Besides this critical frequency, the left-hand and the right-hand homoclinic bifurcations are not only unequal, but also their effects for the system’s stability are different. Among them, the effect resulting from the small homoclinic bifurcation for the system’s stability is local and negligible, while the effect from the large homoclinic bifurcation is global but this is accomplished at a quite larger cost.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:chsofr:v:34:y:2007:i:2:p:197-212
    DOI: 10.1016/j.chaos.2006.04.010
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    References listed on IDEAS

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    1. Sofroniou, Anastasia & Bishop, Steven R., 2006. "Breaking the symmetry of the parametrically excited pendulum," Chaos, Solitons & Fractals, Elsevier, vol. 28(3), pages 673-681.
    2. Bishop, S.R. & Sofroniou, A. & Shi, P., 2005. "Symmetry-breaking in the response of the parametrically excited pendulum model," Chaos, Solitons & Fractals, Elsevier, vol. 25(2), pages 257-264.
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    Cited by:

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    2. Siewe, M. Siewe & Cao, Hongjun & Sanjuán, Miguel A.F., 2009. "On the occurrence of chaos in a parametrically driven extended Rayleigh oscillator with three-well potential," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 772-782.
    3. Girault, Jean-Marc, 2015. "Recurrence and symmetry of time series: Application to transition detection," Chaos, Solitons & Fractals, Elsevier, vol. 77(C), pages 11-28.
    4. 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.
    5. Ramadoss, Janarthanan & Kengne, Jacques & Kengnou Telem, Adélaïde Nicole & Rajagopal, Karthikeyan, 2022. "Broken symmetry and dynamics of a memristive diodes bridge-based Shinriki oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 588(C).
    6. Wu, H. & Zhou, J. & Chen, M. & Xu, Q. & Bao, B., 2022. "DC-offset induced asymmetry in memristive diode-bridge-based Shinriki oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 154(C).
    7. Attili, Basem S., 2009. "A direct method for the numerical computation of bifurcation points underlying symmetries," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1545-1551.
    8. Ramadoss, Janarthanan & Kengne, Jacques & Tanekou, Sosthene Tsamene & Rajagopal, Karthikeyan & Kenmoe, Germaine Djuidje, 2022. "Reversal of period doubling, multistability and symmetry breaking aspects for a system composed of a van der pol oscillator coupled to a duffing oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).
    9. Zhou, Peipei & Cao, Hongjun, 2008. "The effect of symmetry-breaking on the parameterically excited pendulum," Chaos, Solitons & Fractals, Elsevier, vol. 38(2), pages 590-597.
    10. Feng, Jingjing & Zhang, Qichang & Wang, Wei, 2012. "Chaos of several typical asymmetric systems," Chaos, Solitons & Fractals, Elsevier, vol. 45(7), pages 950-958.

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