Reducing the number of time delays in coupled dynamical systems

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.