A new spatio-temporal 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 spatio-temporal representation, uncovering the role of multiple time-scales 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 new 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 depend on the vicinity of a super-critical bifurcation as required in previous approaches. We discuss the general validity and limit of this method and consider the exemplary cases of long-delayed excitable, bistable and Landau systems.