IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v33y2007i3p795-812.html
   My bibliography  Save this article

Complex dynamics in three-well duffing system with two external forcings

Author

Listed:
  • Jing, Zhujun
  • Huang, Jicai
  • Deng, Jin

Abstract

Three-well duffing system with two external forcing terms is investigated. The criterion of existence of chaos under the periodic perturbation is given by using Melnikov’s method. By using second-order averaging method and Melnikov’s method we proved the criterion of existence of chaos in averaged systems under quasi-periodic perturbation for ω2=nω1+εν, n=1, 3, 5, and cannot prove the criterion of existence of chaos in second-order averaged system under quasi-periodic perturbation for ω2=nω1+εν, n=2, 4, 6, 7, 8, 9, 10, 11, 12, where ν is not rational to ω1, but can show the occurrence of chaos in original system by numerical simulation. Numerical simulations including heteroclinic and homoclinic bifurcation surfaces, bifurcation diagrams, maximum Lyapunov exponents and Poincaré map are given to illustrate the theoretical analysis, and to expose the more new complex dynamical behaviors. We show that cascades of period-doubling bifurcations from period-one to four orbits, cascades of interlocking period-doubling bifurcations from period-two orbits of two sets, from quasi-periodicity leading to chaos, onset of chaos which occurs more than one, interleaving occurrences of chaotic behavior and invariant torus, transient chaos with complex period windows and interior crisis, chaos converting to torus, different kind of chaotic attractors. Our results shows that the dynamical behaviors are different from the dynamics of duffing equation with two-well and two external forcings.

Suggested Citation

  • Jing, Zhujun & Huang, Jicai & Deng, Jin, 2007. "Complex dynamics in three-well duffing system with two external forcings," Chaos, Solitons & Fractals, Elsevier, vol. 33(3), pages 795-812.
    • Handle: RePEc:eee:chsofr:v:33:y:2007:i:3:p:795-812
    DOI: 10.1016/j.chaos.2006.03.071
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077906003079
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2006.03.071?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. de Souza, S.L.T. & Caldas, I.L. & Viana, R.L. & Balthazar, J.M. & Brasil, R.M.L.R.F., 2005. "Basins of attraction changes by amplitude constraining of oscillators with limited power supply," Chaos, Solitons & Fractals, Elsevier, vol. 26(4), pages 1211-1220.
    2. Li, Wei & Xu, Wei & Zhao, Junfeng & Ma, Shaojuan, 2005. "Stochastic optimal control of first-passage failure for coupled Duffing–van der Pol system under Gaussian white noise excitations," Chaos, Solitons & Fractals, Elsevier, vol. 25(5), pages 1221-1228.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Huang, Jicai & Jing, Zhujun, 2009. "Bifurcations and chaos in three-well Duffing system with one external forcing," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1449-1466.
    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. Ario, Ichiro, 2013. "Multiple duffing problem in a folding structure with hill-top bifurcation," Chaos, Solitons & Fractals, Elsevier, vol. 51(C), pages 52-63.
    4. Caneco, Acilina & Grácio, Clara & Leonel Rocha, J., 2009. "Kneading theory analysis of the Duffing equation," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1529-1538.

    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. Li, Wei & Xu, Wei & Zhao, Junfeng & Jin, Yanfei, 2006. "First-passage problem for strong nonlinear stochastic dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 28(2), pages 414-421.
    2. Blazejczyk-Okolewska, Barbara & Czolczynski, Krzysztof & Kapitaniak, Tomasz, 2009. "Dynamics of a two-degree-of-freedom cantilever beam with impacts," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1991-2006.
    3. Li, Wei & Li, Jiaorui & Chen, Weisheng, 2012. "The reliability of a stochastically complex dynamical system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(13), pages 3556-3565.
    4. Figueiredo, Annibal & Rocha Filho, Tarcisio M., 2009. "Basins of attraction of invariant regular manifolds," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1877-1889.
    5. Lyu, Xiaohong & Zhu, Xifeng & Gao, Quanfu & Luo, Guanwei, 2020. "Two-parameter bifurcations of an impact system under different damping conditions," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).

    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:33:y:2007:i:3:p:795-812. 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.