Scalar variable method draws analogies between some systems dealing with space, time and memory

Whether it’s for physics, biology or economics, many mathematical systems rely on large numbers of equations to adequately describe real-world phenomena. As these systems evolve dynamically over time, the many variables involved in a working system often lead to what chaoticians refer to as a high-dimensional phase space. One group of researchers, however, found that it is possible to describe the dynamical evolution of some types of high-dimensional systems in three-dimensional phase spaces.

These scientists demonstrated a method showing that some one-dimensional spatially extended systems (1D SES) and time-delayed systems (TDS) can be governed by the same polynomial potential in a three-dimensional pseudo phase space. Publishing their work in Chaos, the group reconstructed the pseudo phase space of these systems from the time series of a scalar variable. They compared a bistable 1D SES with a bistable scalar system with delayed feedback, as well as with a model system of two lasers that has two time-delay terms.

Using appropriated definitions for the pseudo-variables, the team constructed a three-dimensional pseudo space in which both systems were governed by a polynomial potential with three variables. In the laser model they found that the evolution in the pseudo space was only qualitatively described by the polynomial potential, likely because the laser system contained multiple variables and delay terms.

The authors note that a main limitation for successfully reconstructing the phase space of a SES or TDS is reliably estimating a large number of parameters.

Source: “State space reconstruction of spatially extended systems and of time delayed systems from the time series of a scalar variable,” by C. Quintero-Quiroz, M. C. Torrent, and C. Masoller, Chaos (2018). The article can be accessed at https://doi.org/10.1063/1.5023485 .