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Kuramoto dynamics in Hamiltonian systems.

Dirk Witthaut1, Marc Timme2

  • 1Network Dynamics, Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Göttingen, Germany.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
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Summary
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Researchers reveal a classical Hamiltonian system that precisely replicates Kuramoto model dynamics, linking dissipative and conservative systems. This discovery offers new insights into oscillator synchronization and collective dynamics.

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Area of Science:

  • Physics
  • Nonlinear Dynamics
  • Complex Systems

Background:

  • The Kuramoto model is a key framework for understanding how coupled oscillators synchronize (phase locking).
  • Existing models often focus on dissipative systems, limiting the scope of collective dynamics analysis.

Purpose of the Study:

  • To present a conservative Hamiltonian system that exactly reproduces Kuramoto dynamics.
  • To establish a direct link between dissipative and conservative dynamics in coupled oscillator systems.
  • To identify new indicators for synchronization transitions in oscillator networks.

Main Methods:

  • Formulation of a classical Hamiltonian system with 2N state variables.
  • Analysis of the system in its action-angle representation.
  • Investigation of invariant manifolds and transverse Hamiltonian action dynamics.
  • Utilizing the inverse participation ratio to analyze dynamics off invariant manifolds.

Main Results:

  • The Hamiltonian system exactly yields Kuramoto dynamics on N-dimensional invariant manifolds.
  • Phase locking in the Kuramoto model emerges when the transverse Hamiltonian action dynamics become unstable.
  • The inverse participation ratio provides a more precise indicator of global synchronization transition points for finite N compared to the standard Kuramoto order parameter.

Conclusions:

  • This work bridges the gap between dissipative (Kuramoto) and conservative (Hamiltonian) dynamics.
  • The findings offer a novel perspective on collective phenomena in physical systems.
  • The identified synchronization indicators enhance the analysis of complex oscillator networks.