Geometry of the ergodic quotient reveals coherent structures in flows

Dynamical systems that exhibit diverse behaviors can rarely be completely understood using a single approach. However, by identifying coherent structures in their state spaces, i.e., regions of uniform and simpler behavior, we could hope to study each of the structures separately and then form the understanding of the system as a whole. The method we present in this paper uses trajectory averages of scalar functions on the state space to: (a) identify invariant sets in the state space, and (b) to form coherent structures by aggregating invariant sets that are similar across multiple spatial scales. First, we construct the ergodic quotient, the object obtained by mapping trajectories to the space of the trajectory averages of a function basis on the state space. Second, we endow the ergodic quotient with a metric structure that successfully captures how similar the invariant sets are in the state space. Finally, we parametrize the ergodic quotient using intrinsic diffusion modes on it. By segmenting the ergodic quotient based on the diffusion modes, we extract coherent features in the state space of the dynamical system. The algorithm is validated by analyzing the Arnold–Beltrami–Childress flow, which was the test-bed for alternative approaches: the Ulam’s approximation of the transfer operator and the computation of Lagrangian Coherent Structures. Furthermore, we explain how the method extends the Poincaré map analysis for periodic flows. As a demonstration, we apply the method to a periodically-driven three-dimensional Hill’s vortex flow, discovering unknown coherent structures in its state space. Finally, we discuss differences between the ergodic quotient and alternatives, propose a generalization to analysis of (quasi-)periodic structures, and lay out future research directions.

► Averaging the Fourier basis along trajectories maps them into the ergodic quotient.
► A multi-scale Sobolev metric on the quotient captures similarity between trajectories.
► Trajectories are aggregated into coherent structures using diffusion on the quotient.
► The method was successfully validated on steady and time-periodic flows.
► The method can handle systems with costly or restricted trajectory initialization.