Chaos Nameplat
  • Volume/Page
  • Keyword
  • DOI
  • Citation
  • Advanced
   
 
 
 

You Tube Flickr Twitter iResearch App Facebook

Chaos / Volume 19 / Issue 4 / REGULAR ARTICLES

Chaos 19, 043104 (2009); http://dx.doi.org/10.1063/1.3247089 (11 pages)

Identical phase oscillators with global sinusoidal coupling evolve by Möbius group action

Seth A. Marvel1, Renato E. Mirollo2, and Steven H. Strogatz1

1Center for Applied Mathematics, Cornell University, Ithaca, New York 14853, USA
2Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02167, USA

View MapView Map

(Received 23 July 2009; accepted 21 September 2009; published online 15 October 2009)

Systems of N identical phase oscillators with global sinusoidal coupling are known to display low-dimensional dynamics. Although this phenomenon was first observed about 20 years ago, its underlying cause has remained a puzzle. Here we expose the structure working behind the scenes of these systems by proving that the governing equations are generated by the action of the Möbius group, a three-parameter subgroup of fractional linear transformations that map the unit disk to itself. When there are no auxiliary state variables, the group action partitions the N-dimensional state space into three-dimensional invariant manifolds (the group orbits). The N−3 constants of motion associated with this foliation are the N−3 functionally independent cross ratios of the oscillator phases. No further reduction is possible, in general; numerical experiments on models of Josephson junction arrays suggest that the invariant manifolds often contain three-dimensional regions of neutrally stable chaos.

© 2009 American Institute of Physics

Lead Paragraph

Large arrays of coupled limit-cycle oscillators have been used to model diverse systems in physics, biology, chemistry, engineering, and social science. The special case of phase oscillators coupled all to all through sinusoidal interactions has attracted mathematical interest because of its analytical tractability. About 20 years ago, numerical experiments revealed that when the oscillators are all identical, these systems display an exceptionally simple form of collective behavior: For all N ≥ 3, where N is the number of oscillators, all trajectories are confined to manifolds with N−3 fewer dimensions than the state space itself. Several insights have been obtained over the past two decades, but it has remained an open problem to pinpoint the symmetry or other structure that causes this nongeneric behavior. Here we show that group theory provides the explanation: The governing equations for these systems arise naturally from the action of the group of conformal mappings of the unit disk to itself. This link unifies and explains the previous numerical and analytical results, and yields new constants of motion for this class of dynamical systems.

Article Outline

  1. INTRODUCTION
  2. BACKGROUND
    1. Reducible systems with sinusoidal coupling
    2. Ott–Antonsen ansatz
    3. Möbius group
  3. MÖBIUS GROUP REDUCTION
    1. Algebraic method
    2. Geometric method of finding math
    3. Geometric method of finding math
  4. CONNECTIONS TO PREVIOUS RESULTS
    1. Relation to the Watanabe–Strogatz transformation
    2. Invariant manifold of Poisson kernels
  5. CHARACTERISTICS OF THE MOTION
    1. Cross ratios as constants of motion
    2. Fourier coefficients of the phase distribution
  6. CHAOS IN JOSEPHSON ARRAYS

RELATED DATABASES

To view database links for this article, you need to log in.

KEYWORDS and PACS

Keywords

chaos, group theory, nonlinear dynamical systems

PACS

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3247089

PUBLICATION DATA

ISSN

1054-1500 (print)  
1089-7682 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

For access to fully linked references, you need to log in.

    References

    N. J. Zabusky, Chaos 15, 015102 (2005)CHAOEH000015000001015102000001.

    N. J. Zabusky and M. D. Kruskal, Phys. Rev. Lett. 15, 240 (1965).

    C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, Phys. Rev. Lett. 19, 1095 (1967).

    K. Y. Tsang and I. B. Schwartz, Phys. Rev. Lett. 68, 2265 (1992).

    D. Golomb, D. Hansel, B. Shraiman, and H. Somopolinsky, Phys. Rev. A 45, 3516 (1992).

    S. Nichols and K. Wiesenfeld, Phys. Rev. A 45, 8430 (1992).

    T. M. Antonsen, Jr., R. T. Faghih, M. Girvan, E. Ott, and J. Platig, Chaos 18, 037112 (2008)CHAOEH000018000003037112000001.

    J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, Rev. Mod. Phys. 77, 137 (2005).

    S. Watanabe and S. H. Strogatz, Phys. Rev. Lett. 70, 2391 (1993).

    E. Ott and T. M. Antonsen, Chaos 18, 037113 (2008)CHAOEH000018000003037113000001.

    A. Pikovsky and M. Rosenblum, Phys. Rev. Lett. 101, 264103 (2008).

    E. Ott and T. M. Antonsen, Chaos 19, 023117 (2009)CHAOEH000019000002023117000001.

    S. A. Marvel and S. H. Strogatz, Chaos 19, 013132 (2009)CHAOEH000019000001013132000001.

    S. H. Strogatz and R. E. Mirollo, Phys. Rev. E 47, 220 (1993).


For access to citing articles, you need to log in.


Figures (3)

Access to article objects (figures, tables, multimedia) requires a subscription; log in to view available files.
(Access to supplementary files, where available, is free for this journal.)



Close

Google Calendar

ADVERTISEMENT

Your Recent History Recent History Help

Recently Viewed

  1. Identical phase oscillators with global sinusoidal coupling evolve by Möbius group action
Go to AIP Home   © AIP Publishing LLC
close