Time delay embeddings to uncover unstable periodic orbits and exact coherent structures in chaotic fluid systems

The data-driven modeling of dynamical systems is rapidly developing, especially for fluid systems. However, many leading algorithms require high-dimensional, full-state training data, while for many real-world systems only very limited or partial measurements are available. Time delay embeddings provide a principled approach to reconstruct an attractor from such limited time series measurements. In this work, we evaluate the use of long-time delay embeddings to capture progressively sophisticated unstable periodic orbits (UPOs) of a chaotic system; in fluid dynamics, these orbits correspond to exact coherent structures (ECSs). The behavior of a chaotic attractor can be understood as the combination of all its periodic orbits. This is particularly relevant when studying the turbulent behavior of fluid flows, which can be represented by an infinite number of these orbits. While an infinite number of orbits are needed to fully describe the system, the dominant dynamics can be captured by the shortest, or ‘fundamental’, orbits. By finding enough of these orbits, we can make predictions about the statistical behavior of turbulence.

In this talk, we evaluate numerical techniques for finding these UPOs for chaotic systems based on delay embedded data. The time delay embeddings obtained for these UPOs can be further used to predict the global characterizations of the dynamics of the attractor including natural measure, fractal dimensions, and Lyapunov exponents. Our results show that, for long time delay embeddings, UPOs disentangle and converge to a discrete Fourier decomposition. We present the analysis for the Lorenz attractor, Rössler attractor, Kuramoto-Shivashinsky equations, and plane Couette flow.