On the correspondence of time-delay and spatially extended systems

We establish a straightforward connection between spatially extended systems, the dynamics of which are modeled with the help of partial differential equations and time-delay systems. To this end, we give a linear partial differential equation with a nonlinear boundary condition whose solutions are equivalent to the solutions of a time-delay differential equation. We observe that the phase space of these systems exhibits a pronounced structure. In this paper, we express the structure of the phase space of time-delay systems and the corresponding spatially extended system by distinguishing between a 'linear subsystem' and a 'localized nonlinearity'. We find that the high-dimensional chaotic dynamics observed in time-delay systems is fundamentally different from the spatio-temporal chaos observed in homogeneous spatially extended systems, the dynamics of which is modeled with the help of nonlinear partial differential equations. To this end, we investigate the space-time correlation function and the 'thermodynamic limit'.