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Phase chaos in coupled oscillators.

Oleksandr V Popovych1, Yuri L Maistrenko, Peter A Tass

  • 1Institute of Medicine and Virtual Institute of Neuromodulation, Research Centre Jülich, 52425 Jülich, Germany.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|August 11, 2005
PubMed
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Researchers discovered phase chaos, a complex behavior, in coupled phase oscillators. This chaotic dynamics, strongest in medium-sized networks, is common across various oscillator types.

Area of Science:

  • Complex systems
  • Nonlinear dynamics
  • Network science

Background:

  • Coupled oscillator systems are fundamental in various scientific fields.
  • Understanding chaotic behavior in networks is crucial for predicting complex phenomena.
  • The Kuramoto model is a widely used framework for studying synchronization and dynamics in coupled oscillators.

Purpose of the Study:

  • To investigate and characterize a complex high-dimensional chaotic behavior, termed phase chaos, within the finite-dimensional Kuramoto model.
  • To analyze the conditions under which phase chaos emerges and its prevalence in different network configurations.

Main Methods:

  • Utilized the finite-dimensional Kuramoto model of coupled phase oscillators.
  • Analyzed the spectrum of Lyapunov exponents to quantify chaotic behavior.

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  • Calculated the Lyapunov dimension to determine the dimensionality of the chaotic attractor.

    • Main Results:

    • Identified and confirmed the existence of phase chaos in the Kuramoto model.
    • Observed that approximately half of the Lyapunov exponents are positive, indicating high-dimensional chaos.
    • Found that the Lyapunov dimension closely approaches the total system dimension.
    • Determined that the strongest phase chaos occurs in intermediate-sized ensembles.

    Conclusions:

    • Phase chaos is a significant dynamical property of the Kuramoto model.
    • This chaotic behavior is characterized by its high dimensionality.
    • The phenomenon of phase chaos is not limited to phase oscillators but is a common property observed in diverse networks, including limit-cycle and chaotic oscillators (e.g., Rössler systems).