Can the state space of spatially extended systems and of time delayed systems be reconstructed from the time series of a scalar variable?

Carlos Quintero-Quiroz
Universitat Politècnica de Catalonia
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. By using the space-time representation many analogies between one-dimensional spatially extended systems (1D SESs) and time delayed systems(TDSs) have been found, including similar pattern formation and propagation of localized structures.An open question is whether such analogies are limited to the space-time representation, or it is also possible to recover similar evolution in a low-dimensional pseudo-space. To address this issue, we analyze a 1D SES (a bistable reaction-diffusion system), a scalar TDS (a bistable system with delayed feedback), and a non-scalar TDS (a model of two delay-coupled lasers). In these three examples, we show that we can reconstruct the dynamics in a three-dimensional phase space, where the evolution is governed by the same polynomial potential. We also discuss the limitations of the analogy between1D SESs and TDSs.