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

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