A new method to reduce the number of time delays in a network

Abstract

Time delays may cause dramatic changes to the dynamics of interacting oscillators. Coupled networks of interacting dynamical systems can have unexpected behaviours when the signal between the vertices are time delayed. It has been shown for a very general class of systems that the time delays can be rearranged as long as the total time delay over the constitutive loops of the network is conserved. This fact allows to reduce the number of time delays of the problem without loss of information. There is a theoretical lower bound for this number that can be numerically improved if the time delays are commensurable. Here we propose a formulation of the problem and a numerical method to even further reduce the number of time delays in a network.

Reducing the number of time delays in coupled dynamical systems

“…Now the problem is to find the time-shifts η i associated to each vertex for a desired configuration of time delays τk . It is possible to find a vector η based on the topology of the graph and the time delay requirements on each edge [10]. Notice that the initial history of the delay differential equation deserves a special treatment if we want to match the trajectory in the phase space for both sets of equations [9].…”

“…A new formulation of this method [10] has been developed to improve the number of zero time delays. With the application of this new method, we find a number of zero time delays n z ≥ n v − 1, and the total sum of the time delays is also considerably reduced.…”