Chaos. 2025 Jul 1;35(7):073150. doi: 10.1063/5.0276714.
ABSTRACT
Phase reduction is an effective theoretical and numerical tool for studying the synchronization of coupled deterministic oscillators. Stochastic oscillators require new definitions of the asymptotic phase. The Q-function, i.e., the slowest decaying complex mode of the stochastic Koopman operator (SKO), was proposed as a means of phase reduction for stochastic oscillators. In this paper, we show that the Q-function approach also leads to a novel definition of “synchronization” for coupled stochastic oscillators. A system of coupled oscillators in the synchronous regime may be viewed as a single (higher-dimensional) oscillator. Therefore, we investigate the relation between the Q-functions of the uncoupled oscillators and the higher-dimensional Q-function for the coupled system. We propose a definition of synchronization between coupled stochastic oscillators in terms of the eigenvalue spectrum of Kolmogorov’s backward operator (the generator of the Markov process, or the SKO) of the higher-dimensional coupled system. We observe a novel type of bifurcation reflecting (i) the relationship between the leading eigenvalues of the SKO for the coupled system and (ii) qualitative changes in the cross-spectral density of the coupled oscillators. Using our proposed definition, we observe synchronization domains for symmetrically coupled stochastic oscillators that are analogous to Arnold tongues for coupled deterministic oscillators.
PMID:40737692 | DOI:10.1063/5.0276714
