Heterogeneous delays making parents synchronized: A coupled maps on Cayley tree model

TL;DR

This paper investigates how heterogeneous delays in coupled map networks on Cayley trees influence the formation of synchronized clusters, revealing new cluster states and lag synchronization phenomena with potential applications in biology and social sciences.

Contribution

It introduces the effects of heterogeneous delays on cluster formation and synchronization in Cayley tree networks, a novel aspect not extensively studied before.

Findings

Heterogeneous delays produce distinct cluster states including inner-boundary and boundary-only clusters.

Delay heterogeneity causes lag synchronization among sibling nodes on the boundary.

Exact synchronization among last-generation nodes is disrupted by delay heterogeneity.

Abstract

We study the phase synchronized clusters in the diffusively coupled maps on the Cayley tree networks for heterogeneous delay values. Cayley tree networks comprise of two parts: the inner nodes and the boundary nodes. We find that heterogeneous delays lead to various cluster states, such as; (a) cluster state consisting of inner nodes and boundary nodes, and (b) cluster state consisting of only boundary nodes. The former state may comprise of nodes from all the generations forming self-organized cluster or nodes from few generations yielding driven clusters depending upon on the parity of heterogeneous delay values. Furthermore, heterogeneity in delays leads to the lag synchronization between the siblings lying on the boundary by destroying the exact synchronization among them. The time lag being equal to the difference in the delay values. The Lyapunov function analysis sheds light on the destruction of the exact synchrony among the last generation nodes. To the end we discuss the relevance of our results with respect to their applications in the family business as well as in understanding the occurrence of genetic diseases.