Complex transitions to synchronization in delay-coupled networks of logistic maps

TL;DR

This paper investigates how delay heterogeneity influences synchronization in networks of logistic maps, revealing a transition from time-dependent to steady-state synchronization and complex unsynchronized states.

Contribution

It characterizes the transition mechanisms to synchronization in delay-coupled logistic maps, highlighting the effects of delay distribution heterogeneity and coupling strength.

Findings

Homogeneous delays lead to time-dependent synchronization.

Heterogeneous delays induce steady-state synchronization.

Complex unsynchronized states emerge with increasing delay heterogeneity.

Abstract

A network of delay-coupled logistic maps exhibits two different synchronization regimes, depending on the distribution of the coupling delay times. When the delays are homogeneous throughout the network, the network synchronizes to a time-dependent state [Atay et al., Phys. Rev. Lett. 92, 144101 (2004)], which may be periodic or chaotic depending on the delay; when the delays are sufficiently heterogeneous, the synchronization proceeds to a steady-state, which is unstable for the uncoupled map [Masoller and Marti, Phys. Rev. Lett. 94, 134102 (2005)]. Here we characterize the transition from time-dependent to steady-state synchronization as the width of the delay distribution increases. We also compare the two transitions to synchronization as the coupling strength increases. We use transition probabilities calculated via symbolic analysis and ordinal patterns. We find that, as the coupling strength increases, before the onset of steady-state synchronization the network splits into two clusters which are in anti-phase relation with each other. On the other hand, with increasing delay heterogeneity, no cluster formation is seen at the onset of steady-state synchronization; however, a rather complex unsynchronized state is detected, revealed by a diversity of transition probabilities in the network nodes.