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

Marko  Budišić; Stefan  Siegmund; Doan Thai  Son; Igor  Mezić

doi:10.3934/dcdss.2016035

Article Contents

2016, Volume 9, Issue 4: 923-958. Doi: 10.3934/dcdss.2016035

This issue Previous Article Blue sky-like catastrophe for reversible nonlinear implicit ODEs Next Article Recurrent equations with sign and Fredholm alternative

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

1.
Department of Mathematics, Clarkson University, Potsdam, NY
2.
Center for Dynamics & Institute of Analysis, Department of Mathematics, TU Dresden, Dresden
3.
Department of Probability and Statistics, Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi
4.
Department of Mechanical Engineering, University of California, Santa Barbara, Santa Barbara, CA

Received: September 30, 2015

Revised: March 31, 2016

Published: July 2016

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Abstract

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 Mezić et al. ``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.

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Mathematics Subject Classification: Primary: 37N10, 37B55; Secondary: 37Exx.

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