Q-functions, synchronization, and Arnold tongues for coupled stochastic oscillators

Area of Science:

• Nonlinear Dynamics
• Statistical Physics
• Complex Systems

Background:

• Phase reduction is crucial for analyzing coupled deterministic oscillators.
• Stochastic oscillators necessitate new definitions for asymptotic phase and synchronization.
• The Q-function, the slowest decaying mode of the stochastic Koopman operator (SKO), was proposed for stochastic phase reduction.

Purpose of the Study:

• To define synchronization for coupled stochastic oscillators using the Q-function.
• To investigate the relationship between Q-functions of uncoupled and coupled systems.
• To propose a novel definition of synchronization based on eigenvalue spectra.

Main Methods:

• Utilizing the Q-function derived from the stochastic Koopman operator (SKO).
• Analyzing the eigenvalue spectrum of Kolmogorov's backward operator for coupled systems.
• Examining cross-spectral density to identify bifurcations.

Main Results:

• The Q-function approach provides a novel definition for stochastic oscillator synchronization.
• A new type of bifurcation is observed related to SKO eigenvalues and cross-spectral density.
• Synchronization domains analogous to Arnold tongues are identified for coupled stochastic oscillators.

Conclusions:

• The Q-function offers a robust framework for defining and analyzing synchronization in stochastic systems.
• The proposed definition advances the understanding of collective behavior in noisy oscillator networks.
• This work bridges the gap between deterministic and stochastic synchronization phenomena.