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Non-resonance 3D homoclinic bifurcation with an inclination flip

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  • Lu, Qiuying

Abstract

Local active coordinates approach is employed to study the bifurcation of a non-resonance three-dimensional smooth system which has a homoclinic orbit to a hyperbolic equilibrium point with three real eigenvalues -α,-β,1 satisfying α>β>0. A homoclinic orbit is called an inclination-flip homoclinic orbit if the strong inclination property of the stable manifold is violated. In this paper, we show the existence of 1-homoclinic orbit, 1-periodic orbit, 2n-homoclinic orbit and 2n-periodic orbit in the unfolding of an inclination-flip homoclinic orbit. And we figure out the bifurcation diagram based on the existence region of the corresponding bifurcation.

Suggested Citation

  • Lu, Qiuying, 2009. "Non-resonance 3D homoclinic bifurcation with an inclination flip," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 2597-2605.
    • Handle: RePEc:eee:chsofr:v:42:y:2009:i:5:p:2597-2605
    DOI: 10.1016/j.chaos.2009.03.112
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    References listed on IDEAS

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    1. He, Ji-Huan, 2005. "Limit cycle and bifurcation of nonlinear problems," Chaos, Solitons & Fractals, Elsevier, vol. 26(3), pages 827-833.
    2. Geng, Fengjie & Zhu, Deming & Xu, Yancong, 2009. "Bifurcations of heterodimensional cycles with two saddle points," Chaos, Solitons & Fractals, Elsevier, vol. 39(5), pages 2063-2075.
    3. Zhang, Weipeng & Zhu, Deming, 2009. "Codimension 2 bifurcations of double homoclinic loops," Chaos, Solitons & Fractals, Elsevier, vol. 39(1), pages 295-303.
    4. Zhang, Wei & Wang, Feng-Xia & Zu, Jean W., 2005. "Local bifurcations and codimension-3 degenerate bifurcations of a quintic nonlinear beam under parametric excitation," Chaos, Solitons & Fractals, Elsevier, vol. 24(4), pages 977-998.
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