Acta mathematica scientia,Series A ›› 2021, Vol. 41 ›› Issue (5): 1516-1528.

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The Influence of Splitting Index on Heteroclinic Orbit Bifurcation Under Periodic Perturbation

Bin Long*(),Shanshan Xu,Hui Cao,Jianquan Li   

  1. Department of Mathematics, Shaanxi University of Science and Technology, Xi'an 710021
  • Received:2020-04-09 Online:2021-10-26 Published:2021-10-08
  • Contact: Bin Long
  • Supported by:
    the NSFC(11801343);the NSFC(12071268);the NSFC(11971281);the NSFC(11801342);the Natural Science Basic Research Plan in Shaanxi Province(2018JQ1031);the Scientific Research Initiation Foundation of Shaanxi University of Science and Technology(2017BJ-45)


By using the method of Lyapunov-Schmidt reduction and exponential dichotomies, we consider the degenerate heteroclinic orbit bifurcation with $m$ dimensional periodic perturbations. The variational equation along the heteroclinic orbit has $d (d\ge1)$ bounded solutions. The splitting index of the unperturbed heteroclinic orbit is $s$. The bifurcation equation has $d+m$ variables and $d-s$ equations. The zeros of bifurcation function correspond to the existence of heteroclinic orbits for perturbed equation. If the splitting index $s<0$, it needs at least $1-s$ dimensional periodic perturbation can break the unperturbed heteroclinic orbit. If the splitting index $s\geq0$, there is a small perturbation can break the unperturbed heteroclinic orbit.

Key words: Degenerate heteroclinic bifurcation, Lyapunov-Schmidt reduction, Exponential dichotomy

CLC Number: 

  • O175.1