Spatiotemporal description of long-delayed systems: Ruling the dynamics

The data generated by long-delayed dynamical systems can be organized in patterns by means of the so-called spatiotemporal representation, uncovering the role of multiple timescales as independent degrees of freedom. However, their identification as equivalent space and time variables does not lead to a correct dynamical rule. We introduce a framework for a proper description of the dynamics in the thermodynamic limit, providing a general avenue for the treatment of long-delayed systems in terms of partial differential equations. Such scheme is generic and does not rely on either the vicinity to bifurcations or on multiple-scale approaches. We discuss the validity of this method and consider the exemplary cases of long-delayed excitable, bistable, and Landau systems.