Synchronization time in a hyperbolic dynamical system with long-range interactions

TL;DR

This paper investigates how synchronization time in hyperbolic coupled map lattices is influenced by linear stability analysis, emphasizing the importance of accurate observation and providing both numerical and theoretical insights.

Contribution

It offers a rigorous mathematical framework for understanding synchronization thresholds and times in hyperbolic dynamical systems with long-range interactions.

Findings

Synchronization threshold is determined by transversal linear stability.

Inadequate observation can lead to incorrect conclusions about synchronization.

The results generalize to hyperbolic coupled map lattices.

Abstract

We show that the threshold of complete synchronization in a lattice of coupled non-smooth chaotic maps is determined by linear stability along the directions transversal to the synchronization subspace. We examine carefully the sychronization time and show that a inadequate observation of the system evolution leads to wrong results. We present both careful numerical experiments and a rigorous mathematical explanation confirming this fact, allowing for a generalization involving hyperbolic coupled map lattices.