State space reconstruction of spatially extended systems and of time delayed systems from the time series of a scalar variable

Abstract: The space-time representation of high-dimensional dynamical systems that have a well defined characteristic time scale has proven to be very useful to deepen the understanding of such systems and to uncover hidden features in their output signals. Genuine analogies between one-dimensional (1D) spatially extended systems (1D SESs) and time delayed systems (TDSs) have been observed, including similar pattern formation and propagation of localized structures. An open question is if such analogies are limited to the space-time representation, or, if it is possible to reconstruct similar attractors, from the time series of an observed variable. In this work we address this issue by considering a bistable 1D SES and two TDSs (a bistable system and a model of two lasers with time delayed coupling). In these three examples we find that we can reconstruct the underlying attractor in a three-dimensional pseudo-space, where the evolution is governed by a polynomial potential. We also discuss the limitations of the analogy between 1D SESs and TDSs.