# American Institute of Mathematical Sciences

August  2016, 9(4): 923-958. doi: 10.3934/dcdss.2016035

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

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

Received  October 2015 Revised  April 2016 Published  August 2016

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.
Citation: Marko Budišić, Stefan Siegmund, Doan Thai Son, Igor Mezić. Mesochronic classification of trajectories in incompressible 3D vector fields over finite times. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 923-958. doi: 10.3934/dcdss.2016035
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