Discontinuous Attractor Dimension at the Synchronization Transition of Time-Delayed Chaotic Systems

TL;DR

This paper studies how the attractor dimension changes abruptly at the synchronization transition in time-delayed chaotic systems, using Lyapunov exponents and applying to various models including semiconductor lasers.

Contribution

It demonstrates that the Kaplan-Yorke dimension exhibits a discontinuity at the synchronization transition and analyzes its dependence on system size and delay.

Findings

Discontinuous jump in Kaplan-Yorke dimension at transition

Correlation dimension also shows a jump in Bernoulli maps

Scaling laws for dimension and entropy with system size and delay

Abstract

The attractor dimension at the transition to complete synchronization in a network of chaotic units with time-delayed couplings is investigated. In particular, we determine the Kaplan-Yorke dimension from the spectrum of Lyapunov exponents for iterated maps and for two coupled semiconductor lasers. We argue that the Kaplan-Yorke dimension must be discontinuous at the transition and compare it to the correlation dimension. For a system of Bernoulli maps we indeed find a jump in the correlation dimension. The magnitude of the discontinuity in the Kaplan-Yorke dimension is calculated for networks of Bernoulli units as a function of the network size. Furthermore the scaling of the Kaplan-Yorke dimension as well as of the Kolmogorov entropy with system size and time delay is investigated.