In the 1980s, most researchers approached empirical analysis of low-dimensional dynamical systems through the famous Takens embedding theorem, which guarantees that the attractor of any dynamical system can be reconstructed from samples of the values of key variables. By the 1990s, however, as researchers turned to more complex systems of higher dimension, many came to believe that the embedding approach was no longer useful, and instead turned to analysis through so-called proper orthogonal decomposition. This technique sought to find an orthogonal transformation that would turn a dataset of correlated variables to a simpler one of linearly uncorrelated variables, essentially by systematically identifying the most important components
However, other theoretical developments at the time suggested an alternative – that the embedding approach might still work, and involve only a small number of variables, if one lumped the small scale contributions into stochastic noise terms, and focussed the embedding procedure on global quantities tracking symmetry properties of the flow. Applied to turbulent flows such as the von Karman swirling flow, or for atmospheric dynamics, this revised approach produces low-dimensional models which capture the essential features of the dynamics.
In a new paper, LML External Fellow Davide Faranda and colleagues examine how information provided by this embedding procedure compares with that obtained by a more extensive analysis using proper orthogonal decomposition. They consider the specific case of 2D turbulent Rayleigh-Bénard convection, a flow characterized by a series of random transitions from one quasi-steady state to another. This system offers an interesting test, they note, because its turbulent character would naively seem to prevent successful projection of the dynamics onto a low dimensional space. As they show, however, this is indeed possible – low-dimensional descriptions of the turbulent flow do exist, providing that the right observable is embedded. In particular, the approach works if the embedding uses the time series of a global observable tracing the symmetry of the flow, such as the angular momentum in the Rayleigh-Bénard convection. Overall, the study establishes that, even for complex high-dimensional systems, significant information on the structure of the stationary states and their stability can be recovered by the embedding technique, consistent with the characteristics of reversals provided by POD analysis.
The paper is available at https://doi.org/10.1063/1.5081031