Mesochronic classification of trajectories in incompressible 3D vector fields over finite times. Budišić, M.; Siegmund, S.; Son, D. T.; and Mezić, I.
Mesochronic classification of trajectories in incompressible 3D vector fields over finite times [link]Paper  abstract   bibtex   
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 unifies instantaneous, finite-time, and asymptotic analyses into a single approach, (c) the two-dimensional mesochronic analysis (Mezic, 2010) 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.
@article{Budisic2016,
  title = {Mesochronic classification of trajectories in incompressible {{3D}} vector fields over finite times},
  url = {http://arxiv.org/abs/1506.05333},
  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 unifies instantaneous, finite-time, and asymptotic analyses into a single approach, (c) the two-dimensional mesochronic analysis (Mezic, 2010) 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.},
  timestamp = {2016-05-27T00:38:27Z},
  journaltitle = {arXiv {[}nlin.CD]},
  author = {Budi\v{s}i{\'c}, Marko and Siegmund, Stefan and Son, Doan Thai and Mezi{\'c}, Igor},
  urldate = {2015-09-15},
  date = {2016-05},
  keywords = {Nonlinear Sciences - Chaotic Dynamics},
  eprinttype = {arxiv},
  eprint = {1506.05333}
}
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