Acta Mathematica Sinica, English Series

, Volume 34, Issue 5, pp 901–910 | Cite as

Analysis of a Shil’nikov Type Homoclinic Bifurcation

  • Yan Cong Xu
  • Xing Bo Liu


The bifurcation associated with a homoclinic orbit to saddle-focus including a pair of pure imaginary eigenvalues is investigated by using related homoclinic bifurcation theory. It is proved that, in a neighborhood of the homoclinic bifurcation value, there are countably infinite saddle-node bifurcation values, period-doubling bifurcation values and double-pulse homoclinic bifurcation values. Also, accompanied by the Hopf bifurcation, the existence of certain homoclinic connections to the periodic orbit is proved.


Homoclinic bifurcation Hopf bifurcation Poincaré map 

MR(2010) Subject Classification

34C23 34C37 37C29 


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The authors would like to thank the referees for their helpful comments and suggestions.


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© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsHangzhou Normal UniversityHangzhouP. R. China
  2. 2.Department of Mathematics, Shanghai Key Laboratory of PMMPEast China Normal UniversityShanghaiP. R. China

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