Nonlinear bias of collective oscillation frequency induced by asymmetric Cauchy noise

TL;DR

This paper investigates how asymmetric Cauchy noise causes nonlinear bias in the collective oscillation frequency of coupled phase oscillators, developing a circular cumulant formalism validated by exact solutions.

Contribution

It introduces a novel theoretical approach using circular cumulants to describe frequency bias under asymmetric non-Gaussian noise, surpassing traditional moment-based methods.

Findings

Asymmetric Cauchy noise induces nonlinear frequency bias.

Circular cumulant formalism accurately predicts the effect.

Validation with exact continued fraction solutions confirms the theory.

Abstract

We report the effect of nonlinear bias of the frequency of collective oscillations of sin-coupled phase oscillators subject to individual asymmetric Cauchy noises. The noise asymmetry makes the Ott-Antonsen Ansatz inapplicable. We argue that, for all stable non-Gaussian noises, the tail asymmetry is not only possible (in addition to the trivial shift of the distribution median) but also generic in many physical and biophysical set-ups. For the theoretical description of the effect, we develop a mathematical formalism based on the circular cumulants. The derivation of rigorous asymptotic results can be performed on this basis but seems infeasible in traditional terms of the circular moments (the Kuramoto-Daido order parameters). The effect of the entrainment of individual oscillator frequencies by the global oscillations is also reported in detail. The accuracy of theoretical results based on the low dimensional circular cumulant reductions is validated with the high-accuracy "exact" solutions calculated with the continued fraction method.