Self-averaging of an order parameter in randomly coupled limit-cycle oscillators

Area of Science:

• Physics
• Nonlinear Dynamics
• Statistical Mechanics

Background:

• Previous studies claimed power-law decay for magnetization in coupled phase oscillators.
• Discrepancies were noted between reported power-law decay and observed deviations.

Purpose of the Study:

• To investigate deviations from power-law decay in the magnetization of coupled phase oscillators.
• To clarify the reasons for discrepancies with prior research, particularly concerning numerical methods and model assumptions.

Main Methods:

• Analysis of magnetization decay in a system of coupled phase oscillators with random interactions.
• Comparison of results obtained for individual system magnetization (Z) versus sample-averaged magnetization ([Z]).
• Utilizing Gaussian random variables for intrinsic oscillator frequencies, differing from deterministic approximations.

Main Results:

• Observed notable deviations from power-law decay for the magnetization (Z) of the system.
• Identified strong time discretization effects in numerical procedures for Lyaponov exponent calculations.
• Found that prior claims of power-law decay applied to sample-averaged magnetization ([Z]), not individual magnetization (Z).

Conclusions:

• The observed deviations from power-law decay are consistent with the specific quantity analyzed (Z vs. [Z]).
• Differences in numerical methods (time discretization) and model assumptions (Gaussian vs. deterministic frequencies) contribute to the discrepancies.
• The results suggest that the findings may be compatible when accounting for these methodological and assumption variations.