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Breaking the symmetry of the parametrically excited pendulum

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  • Sofroniou, Anastasia
  • Bishop, Steven R.

Abstract

This paper considers a parametrically excited pendulum whose symmetry is destroyed by a bias term. This study investigates the effect of this symmetry-breaking by comparing the control parameter space of frequency and amplitude of the forcing with its symmetric counterpart. Approximate bifurcation analysis is used to predict the new escape boundary using a harmonic balance scheme. The bifurcations exhibited in the asymmetrical model are echoed in the case of the tilted pendulum, whose drive is not quite vertical. The experimental importance of this alteration is furthermore discussed with a view of predicting the onset of rotating motions. More specifically, an easily viewed drop in amplitude experienced by the asymmetric oscillatory solution may be considered as the trigger of escape and can therefore be regarded as a precursor of imminent danger or operational difficulties. The paper goes on to examine the impact that a variation in the bias term causes in terms of the changes to the region of safe, oscillatory motion. Applications in ship dynamics give a physical significance of this papers findings.

Suggested Citation

  • Sofroniou, Anastasia & Bishop, Steven R., 2006. "Breaking the symmetry of the parametrically excited pendulum," Chaos, Solitons & Fractals, Elsevier, vol. 28(3), pages 673-681.
    • Handle: RePEc:eee:chsofr:v:28:y:2006:i:3:p:673-681
    DOI: 10.1016/j.chaos.2005.07.014
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    References listed on IDEAS

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    1. 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:

    1. 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.
    2. Feng, Jingjing & Zhang, Qichang & Wang, Wei, 2012. "Chaos of several typical asymmetric systems," Chaos, Solitons & Fractals, Elsevier, vol. 45(7), pages 950-958.
    3. Liu, Yachong & Hu, Ankang & Han, Fenglei & Lu, Yu, 2015. "Stability analysis of nonlinear ship-roll dynamics under wind and wave," Chaos, Solitons & Fractals, Elsevier, vol. 76(C), pages 32-39.
    4. Anastasia Sofroniou & Steven Bishop, 2014. "Dynamics of a Parametrically Excited System with Two Forcing Terms," Mathematics, MDPI, vol. 2(3), pages 1-24, September.
    5. 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).
    6. 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.
    7. 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.

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