Nonlinear Dynamics

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

Bifurcations of twisted heteroclinic loop with resonant eigenvalues

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.


  1. 1.
    Chow, S.N., Hale, J.K.: Methods of Bifurcation Theory. Springer, New York (1982)CrossRefMATHGoogle Scholar
  2. 2.
    Chow, S.N., Deng, B., Fiedler, B.: Homoclinic bifurcation at resonant eigenvalues. J. Dyn. Syst. Differ. Equ. 2(2), 177–244 (1990)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Han, M.A., Zhu, D.M.: Bifurcation Theory of Differential Equations. Coal Industry Press, Beijing (1994)Google Scholar
  4. 4.
    Luo, D.J., Wang, X., Zhu, D.M., Han, M.A.: Bifurcation Theory and Methods of Dynamical Systems. World Scientific, Singapore (1997)CrossRefMATHGoogle Scholar
  5. 5.
    Wiggins, S.: Global Bifurcations and Chaos-Analytical Methods. Springer, New York (1988)CrossRefMATHGoogle Scholar
  6. 6.
    Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edn. Springer, New York (2003)MATHGoogle Scholar
  7. 7.
    Zhang, J.Y., Feng, B.Y.: Geometrical Theory and Bifurcation Problems of Ordinary Differential Equations. Peking Press, Beijing (2000)Google Scholar
  8. 8.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. Springer, New York (1983)CrossRefMATHGoogle Scholar
  9. 9.
    Arnold, V.I.: Geometric Methods in the Theory of Ordinary Differential Equations, 2nd edn. Springer, New York (1983)CrossRefGoogle Scholar
  10. 10.
    Gruendler, J.: The existence of homoclinic orbits and the method of Melnikov for systems in \(R^n\). SIAM J. Math. Anal. 16(5), 907–931 (1985)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kovacic, G., Wiggins, S.: Orbits homoclinic to resonance with an application to chaos in a model of the forced and damped sine-Gordon equation. Phys. D 57, 185–225 (1992)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Li, J.B., Feng, B.: Stability, Bifurcations and Chaos. Yunnan Scientific & Technical Publishers, Kunming (1995)Google Scholar
  13. 13.
    Zhu, D.M.: Problems in homoclinic bifurcation with higher dimensions. Acta Math. Sin. New Ser. 14(3), 341–352 (1998)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Zhu, D.M., Xia, Z.H.: Bifurcations of heteroclinic loops. Sci. China Ser. A 41(8), 837–848 (1998)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Jin, Y.L., Zhu, D.M.: Degenerated homoclinic bifurcations with higher dimensions. Chin. Ann. Math. Ser. B 21(2), 201–210 (2000)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Liu, X.B., Zhu, D.M.: On the stability of homoclinic loop with higher dimension. Discrete Contin. Dyn. Syst. Ser. B 17(3), 915–932 (2012)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Tian, Q.P., Zhu, D.M.: Non-twisted bifurcations of heteroclinic loop. Sci. China Ser. A 43(8), 818–828 (2000)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Jin, Y.L., Zhu, D.M.: Bifurcations of rough heteroclinic loops with two saddle points. Sci. China Ser. A 46(4), 459–468 (2003)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Jin, Y.L., Zhu, X.W.: Bifurcations of non-twisted heteroclinic loop with resonant eigenvalues. Sci. World J. (2014). Google Scholar
  20. 20.
    Yamashita, M.: Melnikov vector in higher dimensions. Nonlinear Anal. Theory Methods Appl. 18(7), 657–670 (1992)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Battelli, F., Lazzari, C.: Exponential dichotomies, heteroclinic orbits and Melnikov functions. J. Differ. Equ. 86(2), 342–366 (1990)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Zhu, D.M.: Melnikov vector and heteroclinic manifold. Sci. China Ser. A 37(6), 673–682 (1994)MathSciNetMATHGoogle Scholar
  23. 23.
    Zhu, D.M.: Melnikov-type vectors and principal normals. Sci. China Ser. A 37(7), 814–822 (1994)MathSciNetMATHGoogle Scholar
  24. 24.
    Zhu, D.M.: Exponential trichotomy and heteroclinic bifurcation. Nonlinear Anal. Theory Methods Appl. 28(3), 547–557 (1997)MathSciNetCrossRefMATHGoogle Scholar

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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLinyi UniversityLinyiChina

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