# The dynamics of critical fluctuations in asymmetric Curie-Weiss models

## TL;DR

This paper investigates how fluctuations behave at critical points in asymmetric Curie-Weiss spin models undergoing Hopf bifurcations, revealing different time-scale dynamics and deriving their limiting behavior.

## Contribution

It introduces a study of fluctuation dynamics at criticality in asymmetric Curie-Weiss models with Hopf bifurcations, highlighting the impact of asymmetry and bifurcation type.

## Key findings

- Fluctuations exhibit different time scales depending on the observable.
- Averaging principle is used to derive the limiting dynamics of slow observables.
- Fluctuation behavior reflects the nature of the bifurcation at criticality.

## Abstract

We study the dynamics of fluctuations at the critical point for two time-asymmetric version of the Curie-Weiss model for spin systems that, in the macroscopic limit, undergo a Hopf bifurcation. The fluctuations around the macroscopic limit reflect the type of bifurcation, as they exhibit observables whose fluctuations evolve at different time scales. The limiting dynamics of fluctuations of slow observable is obtained via an averaging principle.