Hopf bifurcation and the Hopf fibration

M. Field, James W Swift

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

We present techniques for studying the local dynamics generated by an equivariant Hopf bifurcation. We show that under general hypotheses we can expect the formation of a branch of attracting invariant spheres which capture all the local dynamics. In addition, using the Hopf fibration, we show that the limit cycles generated by the Hopf bifurcation are determined by zeros of a vector field defined on complex projective space. We show how to compute these zeros and illustrate our methods with examples of Hopf bifurcations for the dihedral groups of order six and eight and the orthogonal groups.

Original languageEnglish (US)
Article number005
Pages (from-to)385-402
Number of pages18
JournalNonlinearity
Volume7
Issue number2
DOIs
StatePublished - 1994
Externally publishedYes

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Hopf Fibration
Hopf bifurcation
Hopf Bifurcation
Dihedral group
Complex Projective Space
Orthogonal Group
Zero
Equivariant
Limit Cycle
cycles
Vector Field
Branch
Invariant

ASJC Scopus subject areas

  • Mathematical Physics
  • Statistical and Nonlinear Physics
  • Applied Mathematics
  • Mathematics(all)

Cite this

Hopf bifurcation and the Hopf fibration. / Field, M.; Swift, James W.

In: Nonlinearity, Vol. 7, No. 2, 005, 1994, p. 385-402.

Research output: Contribution to journalArticle

Field, M. ; Swift, James W. / Hopf bifurcation and the Hopf fibration. In: Nonlinearity. 1994 ; Vol. 7, No. 2. pp. 385-402.
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