Nonlinear Dynamics

, Volume 92, Issue 2, pp 557–573 | Cite as

Bifurcations of twisted heteroclinic loop with resonant eigenvalues

  • Yinlai Jin
  • Xiaowei Zhu
  • Yuanyuan Liu
  • Han Xu
  • Nana Zhang
Original Paper


In a small tubular neighborhood of the heteroclinic orbits, we establish a local coordinate system by using the foundational solutions of the linear variational equation of the unperturbed system along the heteroclinic orbits. We study the bifurcation problems of twisted heteroclinic loop with resonant eigenvalues. Under the twisted conditions and some transversal conditions, we obtain the existence, the number, the coexistence and non-coexistence problem of 1-heteroclinic loop, 1-homoclinic loop, 1-periodic orbit, double 1-periodic orbit, and 2-heteroclinic loop, 2-homoclinic loop, 2-periodic orbit. Moreover, the relative bifurcation surfaces and the existence regions are given, and the corresponding bifurcation graphs are drawn.


Bifurcation Poincar\(\acute{e}\) map Heteroclinic loop Resonant Twisted 

Mathematics Subject Classification

34C23 37C29 34C37 



This work is supported by the Shandong Province Natural Science Foundation (ZR2015AL005), the National Natural Science Foundation of China (No. 11601212), the Shandong Province Higher Educational Science and Technology Program (J16LI03), and the Applied Mathematics Enhancement Program of Linyi University.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLinyi UniversityLinyiChina

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