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Nonlinear Dynamics

, Volume 70, Issue 2, pp 1645–1655 | Cite as

Bifurcations of heteroclinic loop accompanied by pitchfork bifurcation

  • Fengjie Geng
  • Yancong Xu
Original Paper

Abstract

In this paper, using the local coordinate moving frame approach, we investigate bifurcations of generic heteroclinic loop with a hyperbolic equilibrium and a nonhyperbolic equilibrium which undergoes a pitchfork bifurcation. Under some generic hypotheses, the existence of homoclinic loop, heteroclinic loop, periodic orbit and three or four heteroclinic orbits is obtained. In addition, the non-coexistence conditions for homoclinic loop and periodic orbit are also given. Note that the results achieved here can be extended to higher dimensional systems.

Keywords

Local coordinate moving frame Pitchfork bifurcation Generic heteroclinic loop Nonhyperbolic equilibrium 

Notes

Acknowledgements

Project supported by the National NSF of China (No. 10926051, 11101385, 11171085), the Fundamental Research Funds for the Central Universities (2010ZY30, 2012093), and by the Science Foundation of Zhejiang Province through grant Y6100081.

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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.College of Mathematics and ScienceChina University of Geosciences (Beijing)BeijingChina
  2. 2.Department of MathematicsHangzhou Normal UniversityHangzhouChina
  3. 3.Division of Applied MathematicsBrown UniversityProvidenceUSA

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