Synchronization of chaotic networks with time-delayed couplings: An analytic study

TL;DR

This paper provides an analytical study of how networks of nonlinear units with time-delayed couplings can achieve complete synchronization of chaotic trajectories, even with large delays, using mathematical analysis of different network models.

Contribution

It derives analytic stability conditions for chaotic synchronization in networks with single and multiple delay times, extending understanding of synchronization mechanisms.

Findings

Synchronization stability relates to the spectral gap of the coupling matrix.

Analytic results are obtained for Bernoulli maps and polynomial theory.

Comparisons with tent maps and laser equations validate the analysis.

Abstract

Networks of nonlinear units with time-delayed couplings can synchronize to a common chaotic trajectory. Although the delay time may be very large, the units can synchronize completely without time shift. For networks of coupled Bernoulli maps, analytic results are derived for the stability of the chaotic synchronization manifold. For a single delay time, chaos synchronization is related to the spectral gap of the coupling matrix. For networks with multiple delay times, analytic results are obtained from the theory of polynomials. Finally, the analytic results are compared with networks of iterated tent maps and Lang-Kobayashi equations which imitate the behaviour of networks of semiconductor lasers.