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

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 evolutions 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 between 1D SESs and TDSs. Real-world systems in physics, chemistry, biology, economy, etc. are typically described by a large number of equations, involving many variables, and therefore, their dynamical evolution occurs in a high dimensional phase space. One of the most exciting discoveries in the field of dynamical systems in the last decades is that, in spite of their high dimensionality, these systems can be described by low-dimensional attractors, which can be reconstructed even if one can only observe one variable, during a finite time interval, with finite resolution and with large measurement noise. Examples of such high dimensional systems are one-dimensional spatially extended systems (1D SESs), and time delayed systems (TDSs). In a space-time representation, these systems show similar phenomena (e.g., wave propagation, pattern formation, defects and dislocations, turbulence, etc.). In this work we study the state space reconstruction of these systems, from the time series of one scalar “observed” variable. We analyze a bistable reaction-diffusion 1D SES and two TDSs: a bistable scalar system with delayed feedback, and a system composed by two lasers with delayed mutual cross coupling (the system has several variables and two time-delay terms). We find that their dynamics can be reconstructed in a three-dimensional pseudo-space, where the evolution is governed by the same polynomial potential.Peer Reviewe