Mesochronic classification of trajectories in incompressible 3D vector fields over finite times

The mesochronic velocity is the average of the velocity field along trajectories generated by the same velocity field over a time interval of finite duration. In this paper we classify initial conditions of trajectories evolving in incompressible vector fields according to the character of motion of material around the trajectory. In particular, we provide calculations that can be used to determine the number of expanding directions and the presence of rotation from the characteristic polynomial of the Jacobian matrix of mesochronic velocity. In doing so, we show that (a) the mesochronic velocity can be used to characterize dynamical deformation of three-dimensional volumes, (b) the resulting mesochronic analysis is a finite-time extension of the Okubo--Weiss--Chong analysis of incompressible velocity fields, (c) the two-dimensional mesochronic analysis from Mezic et al. emph{A New Mixing Diagnostic and Gulf Oil Spill Movement}, Science 330, (2010), 486-489, extends to three-dimensional state spaces. Theoretical considerations are further supported by numerical computations performed for a dynamical system arising in fluid mechanics, the unsteady Arnold--Beltrami--Childress (ABC) flow.

Applied Koopmanism

A majority of methods from dynamical systems analysis, especially those in applied settings, rely on Poincares geometric picture that focuses on dynamics of states. While this picture has fueled our field for a century, it has shown difficulties in handling high-dimensional, ill-described, and uncertain systems, which are more and more common in engineered systems design and analysis of big data measurements. This overview article presents an alternative framework for dynamical systems, based on the dynamics of observables picture. The central object is the Koopman operator: an infinite-dimensional, linear operator that is nonetheless capable of capturing the full nonlinear dynamics. The first goal of this paper is to make it clear how methods that appeared in different papers and contexts all relate to each other through spectral properties of the Koopman operator. The second goal is to present these methods in a concise manner in an effort to make the framework accessible to researchers who would like to apply them, but also, expand and improve them. Finally, we aim to provide a road map through the literature where each of the topics was described in detail. We describe three main concepts: Koopman mode analysis, Koopman eigenquotients, and continuous indicators of ergodicity. For each concept we provide a summary of theoretical concepts required to define and study them, numerical methods that have been developed for their analysis, and, when possible, applications that made use of them. The Koopman framework is showing potential for crossing over from academic and theoretical use to industrial practice. Therefore, the paper highlights its strengths, in applied and numerical contexts. Additionally, we point out areas where an additional research push is needed before the approach is adopted as an off-the-shelf framework for analysis and design.