Classification of coupled dynamical systems with multiple delays: Finding the minimal number of delays

TL;DR

This paper introduces a componentwise timeshift transformation (CTT) to classify coupled dynamical systems with delays, showing that systems with different delays can have equivalent dynamics and reducing the effective number of delays needed.

Contribution

The paper presents the CTT method to identify equivalent dynamics in delayed networks and demonstrates that the minimal number of delays can be bounded by the cycle space dimension.

Findings

Stability of attractors is invariant under CTT.

Equivalent systems can have different delay configurations.

The minimal number of delays is bounded by the cycle space dimension.

Abstract

In this article we study networks of coupled dynamical systems with time-delayed connections. If two such networks hold different delays on the connections it is in general possible that they exhibit different dynamical behavior as well. We prove that for particular sets of delays this is not the case. To this aim we introduce a componentwise timeshift transformation (CTT) which allows to classify systems which possess equivalent dynamics, though possibly different sets of connection delays. In particular, we show for a large class of semiflows (including the case of delay differential equations) that the stability of attractors is invariant under this transformation. Moreover we show that each equivalence class which is mediated by the CTT possesses a representative system in which the number of different delays is not larger than the cycle space dimension of the underlying graph. We conclude that the 'true' dimension of the corresponding parameter space of delays is in general smaller than it appears at first glance.