Reducing the number of time delays in coupled dynamical systems

TL;DR

This paper demonstrates that under specific conditions, many time delays in coupled dynamical systems can be eliminated without affecting overall dynamics, simplifying the analysis of such systems.

Contribution

It introduces a method to reduce the number of time delays in coupled systems with identical delays and bidirectional links, preserving the global dynamics.

Findings

Significant number of delays can be removed without changing dynamics

The bounds for delay removal depend on the number of systems in the network

Applicable to systems with identical, bidirectional delays

Abstract

When several dynamical systems interact, the transmission of the information between them necessarily implies a time delay. When the time delay is not negligible, the study of the dynamics of these interactions deserve a special treatment. We will show here that under certain assumptions, it is possible to set to zero a significant amount of time-delayed connections without altering the global dynamics. We will focus on graphs of interactions with identical time delays and bidirectional connections. With these premises, it is possible to find a configuration where a number $n_z$ of time delays have been removed with $n_v-1 \leq n_z \leq n_v^2/4$, where $n_v$ is the number of dynamical systems on a connected graph.