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

Hans Armbruster, John Guckenheimer, Philip Holmes

Research output: Contribution to journalArticle

204 Citations (Scopus)

Abstract

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
Volume29
Issue number3
DOIs
StatePublished - 1988
Externally publishedYes

Fingerprint

Heteroclinic Cycle
Bifurcation Point
traveling waves
Traveling Wave
Codimension
Symmetry
cycles
symmetry
uniqueness
Unfolding
Existence and Uniqueness of Solutions
Open set
Truncation
Equivariant
Normal Form
Vector Field
Multiplicity
Continuum
eigenvalues
continuums

ASJC Scopus subject areas

  • Applied Mathematics
  • Statistical and Nonlinear Physics

Cite this

Heteroclinic cycles and modulated travelling waves in systems with O(2) symmetry. / Armbruster, Hans; Guckenheimer, John; Holmes, Philip.

In: Physica D: Nonlinear Phenomena, Vol. 29, No. 3, 1988, p. 257-282.

Research output: Contribution to journalArticle

Armbruster, Hans ; Guckenheimer, John ; Holmes, Philip. / Heteroclinic cycles and modulated travelling waves in systems with O(2) symmetry. In: Physica D: Nonlinear Phenomena. 1988 ; Vol. 29, No. 3. pp. 257-282.
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