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Journal of Dynamics and Differential Equations

, Volume 7, Issue 4, pp 521–548 | Cite as

Exponential dichotomies and transversal homoclinic orbits in degenerate cases

  • Weiyao Zeng
Article

Abstract

In this paper, we first give a sufficient condition which assures that a linear differential equation depending on a small parameter admits an exponential dichotomy onR, then we use the result obtained here on exponential dichotomies to investigate the existence of transversal homoclinic orbits of perturbed differential systems in two degenerate cases and obtain a Melnikov-type vector. The results on exponential dichotomies of this paper provide us a tool of proving the transversality of homoclinic orbits in studying degenerate bifurcations.

Key words

Exponential dichotomies homoclinic orbit Melnikov functions degenerate bifurcation 

AMS (MOS) classification numbers

34A34 34C25 34C28 

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Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • Weiyao Zeng
    • 1
  1. 1.Department of MathematicsHunan Light Industrial CollegeChangsha, HunanPeople's Republic of China

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