# Concurrent formation of nearly synchronous clusters in each intertwined cluster set with parameter mismatches

## TL;DR

This paper investigates how nearly synchronized clusters form concurrently in networks with intertwined structures when oscillators are nearly identical but have parameter mismatches, revealing stability conditions and deviation behaviors.

## Contribution

It extends the understanding of cluster synchronization to nearly identical oscillators with parameter mismatches, focusing on intertwined cluster sets and their stability.

## Key findings

- Nearly synchronized clusters form concurrently in intertwined sets with stable states.
- Deviation from synchronization increases linearly with parameter mismatch in nearly identical clusters.
- Numerical simulations confirm the theoretical stability and deviation results.

## Abstract

Cluster synchronization is a phenomenon in which oscillators in a given network are partitioned into synchronous clusters. As recently shown, diverse cluster synchronization patterns can be found using network symmetry when the oscillators are identical. For such symmetry-induced cluster synchronization patterns, subsets called intertwined clusters can exist, in which every cluster in the same subset should synchronize or desynchronize concurrently. In this work, to reflect the existence of noise in real systems, we consider networks composed of nearly identical oscillators. We show that every cluster in the same intertwined cluster set is nearly synchronized concurrently when the nearly synchronous state of the set is stable. We also consider an extreme case where only one cluster of an intertwined cluster set is composed of nearly identical oscillators while every other cluster in the set is composed of identical oscillators. In this case, deviation from the synchronous state of every cluster in the same set increases linearly with the magnitude of parameter mismatch within the cluster of nearly identical oscillators. We confirm these results by numerical simulation.