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Chaos / Volume 22 / Issue 1 / REGULAR ARTICLES

Chaos 22, 013114 (2012); http://dx.doi.org/10.1063/1.3677365 (10 pages)

Multistability of twisted states in non-locally coupled Kuramoto-type models

Taras Girnyk1, Martin Hasler2, and Yuriy Maistrenko3,4

1Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany
2Laboratory of Nonlinear Systems, School of Computer and Communication Sciences, Ecole Polytechnique Federale de Lausanne, CH 1015 Lausanne, Switzerland
3Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska St. 3, 01601 Kyiv, Ukraine
4National Centre for Medical and Biotechnical Research, NAS of Ukraine, Volodymyrska St. 54, 01030 Kyiv, Ukraine

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(Received 9 October 2011; accepted 27 December 2011; published online 2 February 2012)

A ring of N identical phase oscillators with interactions between L-nearest neighbors is considered, where L ranges from 1 (local coupling) to N/2 (global coupling). The coupling function is a simple sinusoid, as in the Kuramoto model, but with a minus sign which has a profound influence on its behavior. Without the limitation of the generality, the frequency of the free-running oscillators can be set to zero. The resulting system is of gradient type, and therefore, all its solutions converge to an equilibrium point. All so-called q-twisted states, where the phase difference between neighboring oscillators on the ring is 2πq/N, are equilibrium points, where q is an integer. Their stability in the limit N → ∞ is discussed along the line of Wiley et al. [Chaos 16, 015103 (2006)] In addition, we prove that when a twisted state is asymptotically stable for the infinite system, it is also asymptotically stable for sufficiently large N. Note that for smaller N, the same q-twisted states may become unstable and other q-twisted states may become stable. Finally, the existence of additional equilibrium states, called here multi-twisted states, is shown by numerical simulation. The phase difference between neighboring oscillators is approximately 2πq/N in one sector of the ring, −2πq/N in another sector, and it has intermediate values between the two sectors. Our numerical investigation suggests that the number of different stable multi-twisted states grows exponentially as N → ∞. It is possible to interpret the equilibrium points of the coupled phase oscillator network as trajectories of a discrete-time translational dynamical system where the space-variable (position on the ring) plays the role of time. The q-twisted states are then fixed points, and the multi-twisted states are periodic solutions of period N that are close to a heteroclinic cycle. Due to the apparently exponentially fast growing number of such stable periodic solutions, the system shows spatial chaos as N → ∞.

© 2012 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. MODEL
  3. STABILITY ANALYSIS OF THE TWISTED STATES FOR N → ∞
  4. STABILITY ANALYSIS OF THE TWISTED STATES FOR FINITE N
  5. COMPARISON OF ANALYTICAL AND NUMERICAL RESULTS
  6. MULTI-TWISTED STATES

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KEYWORDS and PACS

Keywords

chaos, nonlinear dynamical systems, numerical analysis, oscillators

PACS

  • 05.45.Xt

    Synchronization; coupled oscillators

  • 02.60.-x

    Numerical approximation and analysis

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3677365

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.
    D. A. Wiley, S. H. Strogatz, and M. Girvan, Chaos 16, 015103 (2006)CHAOEH000016000001015103000001.

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

    D. M. Abrams and S. H. Strogatz, Phys. Rev. Lett. 93, 174102 (2004).

    O. E. Omelchenko, M. Wolfrum, and Y. L. Maistrenko, Phys. Rev. E 81, 065201(R) (2010).

    P. Coullet, C. Elphick, and D. Repaux, Phys Rev. Lett. 58, 431 (1987).

    B. Fernandez, B. Luna, and E. Ugalde, Phys Rev. E 80, 025203 (2009).


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