Averaging of globally coupled oscillators

James W Swift, Steven H. Strogatz, Kurt Wiesenfeld

Research output: Contribution to journalArticle

81 Citations (Scopus)

Abstract

We study a specific system of symmetrically coupled oscillators using the method of averaging. The equations describe a series array of Josephson junctions. We concentrate on the dynamics near the splay-phase state (also known as the antiphase state, ponies on a merry-go-round, or rotating wave). We calculate the Floquet exponents of the splay-phase periodic orbit in the weak-coupling limit, and find that all of the Floquet exponents are purely imaginary; in fact, all the Floquet exponents are zero except for a single complex conjugate pair. Thus, nested two-tori of doubly periodic solutions surround the splay-phase state in the linearized averaged equations. We numerically integrate the original system, and find startling agreement with the averaging results on two counts: The observed ratio of frequencies is very close to the prediction, and the solutions of the full equations appear to be either periodic or doubly periodic, as they are in the averaged equations. Such behavior is quite surprising from the point of view of generic dynamical systems theory-one expects higher-dimensional tori and chaotic solutions. We show that the functional form of the equations, and not just their symmetry, is responsible for this nongeneric behavior.

Original languageEnglish (US)
Pages (from-to)239-250
Number of pages12
JournalPhysica D: Nonlinear Phenomena
Volume55
Issue number3-4
DOIs
StatePublished - 1992

Fingerprint

System theory
Coupled Oscillators
Averaging
Dynamical systems
Orbits
oscillators
Exponent
exponents
Torus
Complex conjugate
Josephson Junction
Weak Coupling
Systems Theory
Josephson junctions
dynamical systems
Periodic Orbits
Periodic Solution
Rotating
Count
High-dimensional

ASJC Scopus subject areas

  • Applied Mathematics
  • Statistical and Nonlinear Physics

Cite this

Averaging of globally coupled oscillators. / Swift, James W; Strogatz, Steven H.; Wiesenfeld, Kurt.

In: Physica D: Nonlinear Phenomena, Vol. 55, No. 3-4, 1992, p. 239-250.

Research output: Contribution to journalArticle

Swift, James W ; Strogatz, Steven H. ; Wiesenfeld, Kurt. / Averaging of globally coupled oscillators. In: Physica D: Nonlinear Phenomena. 1992 ; Vol. 55, No. 3-4. pp. 239-250.
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