IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v38y2008i2p590-597.html
   My bibliography  Save this article

The effect of symmetry-breaking on the parameterically excited pendulum

Author

Listed:
  • Zhou, Peipei
  • Cao, Hongjun

Abstract

The effect of the symmetry-breaking on the parameterically excited pendulum including a bias term is investigated. At first, our numerical simulations show that the area of the safe region of the unexcited pendulum (without damping and without forcing) will decrease with the increasing of the bias term. Due to the variation, the critical homoclinic bifurcation of the excited pendulum will increase, and the region where the homoclinic transversal intersection occurs between the stable and unstable manifolds in the Poincaré map will be enlarged. Second, as the bias term increases, our analysis demonstrates that the number and the type of attractors of the Poincaré map, the phase portraits, the basins of attraction, and the bifurcation diagrams will produce a considerable variation. In particular, the stability of the parameterically excited pendulum will lose once the bias term exceeds a critical value. In this case there is no longer any steady state existing. These results suggest that much attention should be paid on controlling the increasing of bias term, especially when the parameterically excited pendulum as a main device is applied to some practical systems.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:chsofr:v:38:y:2008:i:2:p:590-597
    DOI: 10.1016/j.chaos.2007.06.073
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S096007790700464X
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.chaos.2007.06.073?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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. 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.
    3. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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).
    2. 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.
    3. Anastasia Sofroniou & Steven Bishop, 2014. "Dynamics of a Parametrically Excited System with Two Forcing Terms," Mathematics, MDPI, vol. 2(3), pages 1-24, September.
    4. 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.
    5. 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.
    6. Feng, Jingjing & Zhang, Qichang & Wang, Wei, 2012. "Chaos of several typical asymmetric systems," Chaos, Solitons & Fractals, Elsevier, vol. 45(7), pages 950-958.
    7. Liu, Zeyi & Rao, Xiaobo & Gao, Jianshe & Ding, Shunliang, 2023. "Non-quantum chirality and periodic islands in the driven double pendulum system," Chaos, Solitons & Fractals, Elsevier, vol. 177(C).
    8. Tiňo, Peter, 2009. "Bifurcation structure of equilibria of iterated softmax," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1804-1816.
    9. 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).
    10. Sofroniou, Anastasia & Bishop, Steven R., 2006. "Breaking the symmetry of the parametrically excited pendulum," Chaos, Solitons & Fractals, Elsevier, vol. 28(3), pages 673-681.
    11. 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).
    12. Girault, Jean-Marc, 2015. "Recurrence and symmetry of time series: Application to transition detection," Chaos, Solitons & Fractals, Elsevier, vol. 77(C), pages 11-28.
    13. 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.
    14. 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.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:38:y:2008:i:2:p:590-597. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.