# Stabilization of direct numerical simulation for finite truncations of circular cumulant expansions

## TL;DR

This paper investigates numerical instabilities in direct simulations of truncated circular cumulant equations for coupled oscillators and proposes stabilization methods effective near the Ott-Antonsen manifold.

## Contribution

It introduces two stabilization approaches for direct numerical simulations of truncated circular cumulant equations and evaluates their effectiveness.

## Key findings

- Stabilization techniques work well near the Ott-Antonsen manifold.
- Effectiveness decreases as deviation from the Ott-Antonsen manifold increases.
- Methods enable stable simulations with up to 20 cumulants.

## Abstract

We study a numerical instability of direct simulations with truncated equation chains for the "circular cumulant" representation and two approaches to its suppression. The approaches are tested for a chimera-bearing hierarchical population of coupled oscillators. The stabilization techniques can be efficiently applied without significant effect on the natural system dynamics within a finite vicinity of the Ott-Antonsen manifold for direct numerical simulations with up to 20 cumulants; with increasing deviation from the Ott-Antonsen manifold the stabilization becomes more problematic.