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Julio D da Fonseca1, Edson D Leonel1, Hugues Chaté2,3,4

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This study mathematically describes the instantaneous frequency distribution of coupled oscillators, extending Kuramoto theory. The findings reveal key differences between instantaneous and time-averaged frequencies, especially in synchronized states and distribution tails.

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Area of Science:

  • Mathematical Physics
  • Nonlinear Dynamics
  • Statistical Mechanics

Background:

  • The Kuramoto model is a fundamental framework for studying synchronization in systems of globally coupled phase oscillators.
  • Existing theory primarily describes the distribution of time-averaged frequencies, with less focus on instantaneous frequency dynamics.

Purpose of the Study:

  • To extend the Kuramoto theory by developing a mathematical description for the instantaneous frequency distribution.
  • To analyze the qualitative differences between instantaneous and time-averaged frequency distributions under varying coupling strengths.
  • To investigate the behavior of instantaneous frequencies for natural frequency distributions with power-law tails.

Main Methods:

  • Application of Kuramoto theory combined with probability theory and generalized function methods.
  • Geometric analysis to derive the mathematical description of instantaneous frequency.
  • Numerical simulations to validate theoretical results against systems with normal and Beta distributions.
  • Asymptotic analysis using power-series expansion for power-law tailed distributions.

Main Results:

  • A novel mathematical description for the instantaneous frequency (phase-velocity) distribution was derived.
  • Systematic comparison revealed qualitative differences between instantaneous and time-averaged frequency distributions, particularly near synchronization and in the distribution tails.
  • The instantaneous frequency distribution was analyzed for normal and Beta distributions, showing distinct behaviors based on coupling strength.
  • An asymptotic formula was obtained to analyze the tails of the instantaneous frequency distribution for power-law distributions like Cauchy-Lorentz.

Conclusions:

  • The extended Kuramoto theory provides a more complete picture of oscillator dynamics by characterizing instantaneous frequencies.
  • Understanding instantaneous frequency distributions is crucial for analyzing synchronization phenomena, especially in systems with non-trivial frequency distributions.
  • The derived formulas offer valuable tools for theoretical analysis and comparison with experimental or simulation data in complex oscillator systems.