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Codimension 3 nonresonant bifurcations of homoclinic orbits with two inclination flips

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Abstract

Homoclinic bifurcations in four-dimensional vector fields are investigated by setting up a local coordinate near a homoclinic orbit. This homoclinic orbit is principal but its stable and unstable foliations take inclination flip. The existence, nonexistence, and uniqueness of the 1-homoclinic orbit and 1-periodic orbit are studied. The existence of the two-fold 1-periodic orbit and three-fold 1-periodic orbit are also obtained. It is indicated that the number of periodic orbits bifurcated from this kind of homoclinic orbits depends heavily on the strength of the inclination flip.

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Authors and Affiliations

  1. College of Mathematics and Physics, Zhejiang Normal University, 321004, Jinhua, China

    Shuliang Shui

  2. Department of Mathematics, East China Normal University, 200062, Shanghai, China

    Deming Zhu

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  1. Shuliang Shui
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  2. Deming Zhu
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Correspondence to Shuliang Shui.

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Shui, S., Zhu, D. Codimension 3 nonresonant bifurcations of homoclinic orbits with two inclination flips. Sci. China Ser. A-Math. 48, 248–260 (2005). https://doi.org/10.1360/03ys0201

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  • DOI: https://doi.org/10.1360/03ys0201

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