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Chimera states in the Kuramoto system become short-lived when rotators are three-dimensional. Transverse-stability analysis reveals these states are vulnerable to perturbations, explaining their reduced lifetime.

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Area of Science:

  • Complex systems
  • Nonlinear dynamics
  • Statistical physics

Background:

  • Chimera states, characterized by coexisting synchronous and asynchronous oscillators, are typically long-lived in the 2D Kuramoto system.
  • Their lifetime in the 2D system scales exponentially with system size.
  • Recent findings suggest 3D rotators lead to short-lived chimera states.

Purpose of the Study:

  • To investigate the mechanism behind the short-lived nature of chimera states in the 3D Kuramoto system.
  • To introduce and apply transverse-stability analysis to understand this phenomenon.
  • To derive an exact formula for the transient lifetime of chimera states.

Main Methods:

  • Utilized transverse-stability analysis on the unit sphere for 3D rotations.
  • Calculated the largest transverse Lyapunov exponent for chimera states.
  • Analyzed the stability of states on the equator under latitudinal perturbations.

Main Results:

  • Identified that long-lived 2D chimera states reside on the equator.
  • Demonstrated that latitudinal perturbations (3D rotations) are transverse to these states.
  • Found a typically positive largest transverse Lyapunov exponent, indicating instability and short-lived states.
  • Transformed a numerical scaling law into an exact formula for transient lifetime.

Conclusions:

  • Transverse-stability analysis provides a theoretical explanation for short-lived chimera states in 3D Kuramoto systems.
  • The largest transverse Lyapunov exponent precisely determines the lifetime scaling.
  • Chimera states in physical systems are likely short-lived due to vulnerability to transverse perturbations.