Symbolic dynamics and synchronization of coupled map networks with multiple delays

TL;DR

This paper introduces a symbolic dynamics approach using avoiding sets to analyze coupled map networks with multiple delays, enabling detection of synchronization and delay estimation even with noise.

Contribution

It develops a novel method based on forbidden sequences derived from avoiding sets to identify synchronization and unknown delays in complex networks.

Findings

Successfully detects synchronization in coupled map lattices.

Accurately determines unknown transmission delays.

Applicable to noisy, directed, and time-varying networks.

Abstract

We use symbolic dynamics to study discrete-time dynamical systems with multiple time delays. We exploit the concept of avoiding sets, which arise from specific non-generating partitions of the phase space and restrict the occurrence of certain symbol sequences related to the characteristics of the dynamics. In particular, we show that the resulting forbidden sequences are closely related to the time delays in the system. We present two applications to coupled map lattices, namely (1) detecting synchronization and (2) determining unknown values of the transmission delays in networks with possibly directed and weighted connections and measurement noise. The method is applicable to multi-dimensional as well as set-valued maps, and to networks with time-varying delays and connection structure.