# Radial and circular synchronization clusters in extended starlike network of van der Pol oscillators

## TL;DR

This paper investigates complex synchronization patterns, including radial and circular clusters, in extended starlike networks of nonidentical van der Pol oscillators, revealing new forms of remote synchronization and their role in network synchronization.

## Contribution

It introduces the concept of circular and long-range remote synchronization in starlike networks, expanding understanding of synchronization phenomena in complex oscillator networks.

## Key findings

- Circular synchronization is a new form of remote synchronization.
- Long-range remote synchronization involves nodes interacting through multiple intermediates.
- Network synchronization progresses from the edges inward, with star symmetry affecting dynamics.

## Abstract

We consider extended starlike networks where the hub node is coupled with several chains of nodes representing star rays. Assuming that nodes of the network are occupied by nonidentical self-oscillators we study various forms of their cluster synchronization. Radial cluster emerges when the nodes are synchronized along a ray, while circular cluster is formed by nodes without immediate connections but located on identical distances to the hub. By its nature the circular synchronization is a new manifestation of so called remote synchronization [Phys. Rev. E 85 (2012), 026208]. We report its long-range form when the synchronized nodes interact through at least three intermediate nodes. Forms of long-range remote synchronization are elements of scenario of transition to the total synchronization of the network. We observe that the far ends of rays synchronize first. Then more circular clusters appear involving closer to hub nodes. Subsequently the clusters merge and, finally, all network become synchronous. Behavior of the extended starlike networks is found to be strongly determined by the ray length, while varying the number of rays basically affects fine details of a dynamical picture. Symmetry of the star also extensively influences the dynamics. In an asymmetric star circular cluster mainly vanish in favor of radial ones, however, long-range remote synchronization survives.