Shilnikov Saddle-Focus Homoclinic Orbits from Numerics: Higher Dimensions

Brian A. Coomes, Hüseyin Koçak, Kenneth J. Palmer

Research output: Contribution to journalArticlepeer-review


In a previous paper we studied parametrized autonomous systems and gave a computable criterion that an approximate orbit connecting hyperbolic equilibria is shadowed by a true connecting orbit. This criterion was used to give rigorously verified examples of Shilnikov saddle-focus homoclinic orbits in three dimensions. This involved verifying a condition on the eigenvalues of the linearization at the equilibrium. In dimensions greater than three, there are three more conditions which must be established: general position, asymptotic tangency and a transversality condition. In this paper we give computable criteria for verifying these three conditions. An example in four dimensions, in which detailed rigorous computations are carried out, is given.

Original languageEnglish (US)
JournalJournal of Dynamics and Differential Equations
StateAccepted/In press - 2021
Externally publishedYes


  • Homoclinic
  • Rigorous numerics
  • Shadowing
  • Shilnikov chaos

ASJC Scopus subject areas

  • Analysis


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