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

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  • 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
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    References listed on IDEAS

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    1. 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.
    2. 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.
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    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.

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