Hopf bifurcation with the symmetry of the square

Research output: Contribution to journalArticle

87 Citations (Scopus)

Abstract

Four identical nonlinear oscillators, coupled with the symmetry of a square, can undergo a symmetric version of the standard Hopf bifurcation. Golubitsky and Stewart (1986) have studied the case of N oscillators coupled in a ring with nearest-neighbour coupling. Their results are incomplete for the square case (N=4) because they only considered periodic solutions which have 'maximal' symmetry. Here the author studies the dynamics of all possible square-symmetric Hopf bifurcations; these codimension-one bifurcations are parametrised by the three complex cubic coefficients in the normal form. He finds that invariant tori (quasiperiodic solutions with two frequencies) and periodic solutions with 'minimal' symmetry bifurcate from the origin for open regions of the parameter space of cubic coefficients. The coefficients can be chosen so that the invariant tori are the only asymptotically stable solutions near the origin. Thus a direct transition from a stable fixed point to flow on a stable invariant torus is expected in certain laboratory experiments with only one adjustable parameter (provided the square symmetry is accurate enough). Furthermore, it is conjectured that there are attracting chaotic solutions arbitrarily close to the bifurcation in a certain codimension-two case.

Original languageEnglish (US)
Article number003
Pages (from-to)333-377
Number of pages45
JournalNonlinearity
Volume1
Issue number2
DOIs
StatePublished - 1988
Externally publishedYes

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Hopf bifurcation
Hopf Bifurcation
Invariant Tori
Symmetry
Bifurcation (mathematics)
symmetry
Codimension
Periodic Solution
Coefficient
coefficients
Bifurcation
oscillators
Quasi-periodic Solutions
Stable Solution
Nonlinear Oscillator
Coupled Oscillators
Asymptotically Stable
Normal Form
Parameter Space
Nearest Neighbor

ASJC Scopus subject areas

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

Cite this

Hopf bifurcation with the symmetry of the square. / Swift, James W.

In: Nonlinearity, Vol. 1, No. 2, 003, 1988, p. 333-377.

Research output: Contribution to journalArticle

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