Heteroclinic cycles and modulated travelling waves in systems with O(2) symmetry

Dieter Armbruster, John Guckenheimer, Philip Holmes

Research output: Contribution to journalArticlepeer-review

217 Scopus citations


We analyze unfoldings of a codimension two, steady-state/steady-state modal interaction possessing O(2) symmetry. At the degenerate bifurcation point there are two zero eigenvalues, each of multiplicity two. The spatial wavenumbers of the critical modes ki are assumed to satisfy k2 = 2k1. We base our analysis on a detailed study of the third order truncation of the resulting equivariant normal form, which is a four-dimensional vector field. We find that heteroclinic cycles and modulated travelling waves exist for open sets of parameter values near the codimension two bifurcation point. We provide conditions on parameters which guarantee existence and uniqueness of such solutions and we investigate their stability types. We argue that such motions will be prevalent in continuum systems having the symmetry of translation and reflection with respect to one (or more) spatial directions.

Original languageEnglish (US)
Pages (from-to)257-282
Number of pages26
JournalPhysica D: Nonlinear Phenomena
Issue number3
StatePublished - Jan 1988
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics


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