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
References:
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L. Y. Adrianova, Introduction to Linear Systems of Differential Equations, vol. 146 of Translations of Mathematical Monographs,, American Mathematical Society, (1995). Google Scholar

[2]

M. Allshouse and J.-L. Thiffeault, Detecting coherent structures using braids,, Physica D: Nonlinear Phenomena, 241 (2012), 95. doi: 10.1016/j.physd.2011.10.002. Google Scholar

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S. Balasuriya, Explicit invariant manifolds and specialised trajectories in a class of unsteady flows,, Physics of Fluids (1994-present), 24 (2012). doi: 10.1063/1.4769979. Google Scholar

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D. Blazevski and G. Haller, Hyperbolic and elliptic transport barriers in three-dimensional unsteady flows,, Physica D: Nonlinear Phenomena, 273/274 (2014), 46. doi: 10.1016/j.physd.2014.01.007. Google Scholar

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S. L. Brunton and C. W. Rowley, Fast computation of finite-time Lyapunov exponent fields for unsteady flows,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 20 (2010). doi: 10.1063/1.3270044. Google Scholar

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M. Budišić and I. Mezić, Geometry of the ergodic quotient reveals coherent structures in flows,, Physica D. Nonlinear Phenomena, 241 (2012), 1255. doi: 10.1016/j.physd.2012.04.006. Google Scholar

[9]

M. S. Chong, A. E. Perry and B. J. Cantwell, A general classification of three-dimensional flow fields,, Physics of Fluids A: Fluid Dynamics (1989-1993), 2 (1990), 1989. doi: 10.1063/1.857730. Google Scholar

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W. A. Coppel, Dichotomies in Stability Theory,, Lecture Notes in Mathematics, (1978). Google Scholar

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M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior,, SIAM Journal on Numerical Analysis, 36 (1999), 491. doi: 10.1137/S0036142996313002. Google Scholar

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M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems,, in Handbook of dynamical systems, 2 (2002), 221. doi: 10.1016/S1874-575X(02)80026-1. Google Scholar

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S. Gouëzel and I. Melbourne, Moment bounds and concentration inequalities for slowly mixing dynamical systems,, Electronic Journal of Probability, 19 (2014). doi: 10.1214/EJP.v19-3427. Google Scholar

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J. M. Greene, Two-dimensional measure-preserving mappings,, Journal of Mathematical Physics, 9 (1968), 760. doi: 10.1063/1.1664639. Google Scholar

[26]

J. M. Greene, Method for Determining a Stochastic Transition,, Journal of Mathematical Physics, 20 (1979), 1183. doi: 10.1063/1.524170. Google Scholar

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G. Haller, Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence,, Physics of Fluids, 13 (2001), 3365. doi: 10.1063/1.1403336. Google Scholar

[28]

G. Haller, A variational theory of hyperbolic Lagrangian Coherent Structures,, Physica D. Nonlinear Phenomena, 240 (2011), 574. doi: 10.1016/j.physd.2010.11.010. Google Scholar

[29]

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[31]

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[33]

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[34]

B. O. Koopman, Hamiltonian systems and transformations in Hilbert space,, Proceedings of National Academy of Sciences, 17 (1931), 315. doi: 10.1073/pnas.17.5.315. Google Scholar

[35]

Z. Levnajić and I. Mezić, Ergodic theory and visualization. I. Mesochronic plots for visualization of ergodic partition and invariant sets,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 20 (2010). doi: 10.1063/1.3458896. Google Scholar

[36]

T. Ma and E. M. Bollt, Differential geometry perspective of shape coherence and curvature evolution by finite-time nonhyperbolic splitting,, SIAM Journal on Applied Dynamical Systems, 13 (2014), 1106. doi: 10.1137/130940633. Google Scholar

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T. Ma and E. M. Bollt, Shape coherence and finite-time curvature evolution,, International Journal of Bifurcation and Chaos, 25 (2015). doi: 10.1142/S0218127415500765. Google Scholar

[38]

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J. A. J. Madrid and A. M. Mancho, Distinguished trajectories in time dependent vector fields,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 19 (2009). doi: 10.1063/1.3056050. Google Scholar

[40]

N. Malhotra, I. Mezić and S. Wiggins, Patchiness: A new diagnostic for lagrangian trajectory analysis in time-dependent fluid flows,, International Journal of Bifurcation and Chaos, 8 (1998), 1053. doi: 10.1142/S0218127498000875. Google Scholar

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I. Mezić, S. Loire, V. A. Fonoberov and P. J. Hogan, A new mixing diagnostic and Gulf oil spill movement,, Science Magazine, 330 (2010), 486. Google Scholar

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[48]

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[51]

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show all references

References:
[1]

L. Y. Adrianova, Introduction to Linear Systems of Differential Equations, vol. 146 of Translations of Mathematical Monographs,, American Mathematical Society, (1995). Google Scholar

[2]

M. Allshouse and J.-L. Thiffeault, Detecting coherent structures using braids,, Physica D: Nonlinear Phenomena, 241 (2012), 95. doi: 10.1016/j.physd.2011.10.002. Google Scholar

[3]

H. Aref and E. P. Flinchem, Dynamics of a vortex filament in a shear-flow,, Journal of Fluid Mechanics, 148 (1984), 477. doi: 10.1017/S0022112084002457. Google Scholar

[4]

S. Balasuriya, Explicit invariant manifolds and specialised trajectories in a class of unsteady flows,, Physics of Fluids (1994-present), 24 (2012). doi: 10.1063/1.4769979. Google Scholar

[5]

D. Blazevski and G. Haller, Hyperbolic and elliptic transport barriers in three-dimensional unsteady flows,, Physica D: Nonlinear Phenomena, 273/274 (2014), 46. doi: 10.1016/j.physd.2014.01.007. Google Scholar

[6]

P. L. Boyland, H. Aref and M. A. Stremler, Topological fluid mechanics of stirring,, Journal of Fluid Mechanics, 403 (2000), 277. doi: 10.1017/S0022112099007107. Google Scholar

[7]

S. L. Brunton and C. W. Rowley, Fast computation of finite-time Lyapunov exponent fields for unsteady flows,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 20 (2010). doi: 10.1063/1.3270044. Google Scholar

[8]

M. Budišić and I. Mezić, Geometry of the ergodic quotient reveals coherent structures in flows,, Physica D. Nonlinear Phenomena, 241 (2012), 1255. doi: 10.1016/j.physd.2012.04.006. Google Scholar

[9]

M. S. Chong, A. E. Perry and B. J. Cantwell, A general classification of three-dimensional flow fields,, Physics of Fluids A: Fluid Dynamics (1989-1993), 2 (1990), 1989. doi: 10.1063/1.857730. Google Scholar

[10]

W. A. Coppel, Dichotomies in Stability Theory,, Lecture Notes in Mathematics, (1978). Google Scholar

[11]

M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior,, SIAM Journal on Numerical Analysis, 36 (1999), 491. doi: 10.1137/S0036142996313002. Google Scholar

[12]

M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems,, in Handbook of dynamical systems, 2 (2002), 221. doi: 10.1016/S1874-575X(02)80026-1. Google Scholar

[13]

T. Dombre, U. Frisch, J. M. Greene, M. Hénon, A. Mehr and A. M. Soward, Chaotic streamlines in the ABC flows,, Journal of Fluid Mechanics, 167 (1986), 353. doi: 10.1017/S0022112086002859. Google Scholar

[14]

R. Durrett, Probability: Theory and Examples,, 4th edition, (2010). doi: 10.1017/CBO9780511779398. Google Scholar

[15]

M. Farazmand and G. Haller, Polar rotation angle identifies elliptic islands in unsteady dynamical systems,, Physica D: Nonlinear Phenomena, 315 (2016), 1. doi: 10.1016/j.physd.2015.09.007. Google Scholar

[16]

A. M. Fox and J. D. Meiss, Greene's residue criterion for the breakup of invariant tori of volume-preserving maps,, Physica D: Nonlinear Phenomena, 243 (2013), 45. doi: 10.1016/j.physd.2012.09.005. Google Scholar

[17]

G. Froyland and M. Dellnitz, Detecting and locating near-optimal almost-invariant sets and cycles,, SIAM Journal on Scientific Computing, 24 (2003), 1839. doi: 10.1137/S106482750238911X. Google Scholar

[18]

G. Froyland, S. Lloyd and N. Santitissadeekorn, Coherent sets for nonautonomous dynamical systems,, Physica D: Nonlinear Phenomena, 239 (2010), 1527. doi: 10.1016/j.physd.2010.03.009. Google Scholar

[19]

G. Froyland and K. Padberg, Almost-invariant sets and invariant manifolds-connecting probabilistic and geometric descriptions of coherent structures in flows,, Physica D. Nonlinear Phenomena, 238 (2009), 1507. doi: 10.1016/j.physd.2009.03.002. Google Scholar

[20]

G. Froyland, N. Santitissadeekorn and A. Monahan, Transport in time-dependent dynamical systems: Finite-time coherent sets,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 20 (2010). doi: 10.1063/1.3502450. Google Scholar

[21]

I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants,, Modern Birkhäuser Classics, (2008). Google Scholar

[22]

I. Goldhirsch, P.-L. Sulem and S. A. Orszag, Stability and Lyapunov stability of dynamical systems: A differential approach and a numerical method,, Physica D: Nonlinear Phenomena, 27 (1987), 311. doi: 10.1016/0167-2789(87)90034-0. Google Scholar

[23]

S. Gouëzel, Central limit theorem and stable laws for intermittent maps,, Probability Theory and Related Fields, 128 (2004), 82. doi: 10.1007/s00440-003-0300-4. Google Scholar

[24]

S. Gouëzel and I. Melbourne, Moment bounds and concentration inequalities for slowly mixing dynamical systems,, Electronic Journal of Probability, 19 (2014). doi: 10.1214/EJP.v19-3427. Google Scholar

[25]

J. M. Greene, Two-dimensional measure-preserving mappings,, Journal of Mathematical Physics, 9 (1968), 760. doi: 10.1063/1.1664639. Google Scholar

[26]

J. M. Greene, Method for Determining a Stochastic Transition,, Journal of Mathematical Physics, 20 (1979), 1183. doi: 10.1063/1.524170. Google Scholar

[27]

G. Haller, Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence,, Physics of Fluids, 13 (2001), 3365. doi: 10.1063/1.1403336. Google Scholar

[28]

G. Haller, A variational theory of hyperbolic Lagrangian Coherent Structures,, Physica D. Nonlinear Phenomena, 240 (2011), 574. doi: 10.1016/j.physd.2010.11.010. Google Scholar

[29]

G. Haller, Lagrangian coherent structures,, Annual Review of Fluid Mechanics, 47 (2015), 137. doi: 10.1146/annurev-fluid-010313-141322. Google Scholar

[30]

G. Haller and F. J. Beron-Vera, Geodesic theory of transport barriers in two-dimensional flows,, Physica D: Nonlinear Phenomena, 241 (2012), 1680. doi: 10.1016/j.physd.2012.06.012. Google Scholar

[31]

G. Haller and A. C. Poje, Finite time transport in aperiodic flows,, Physica D. Nonlinear Phenomena, 119 (1998), 352. doi: 10.1016/S0167-2789(98)00091-8. Google Scholar

[32]

G. Haller and G. Yuan, Lagrangian coherent structures and mixing in two-dimensional turbulence,, Physica D. Nonlinear Phenomena, 147 (2000), 352. doi: 10.1016/S0167-2789(00)00142-1. Google Scholar

[33]

R. S. Irving, Integers, Polynomials, and Rings,, Undergraduate Texts in Mathematics, (2004). Google Scholar

[34]

B. O. Koopman, Hamiltonian systems and transformations in Hilbert space,, Proceedings of National Academy of Sciences, 17 (1931), 315. doi: 10.1073/pnas.17.5.315. Google Scholar

[35]

Z. Levnajić and I. Mezić, Ergodic theory and visualization. I. Mesochronic plots for visualization of ergodic partition and invariant sets,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 20 (2010). doi: 10.1063/1.3458896. Google Scholar

[36]

T. Ma and E. M. Bollt, Differential geometry perspective of shape coherence and curvature evolution by finite-time nonhyperbolic splitting,, SIAM Journal on Applied Dynamical Systems, 13 (2014), 1106. doi: 10.1137/130940633. Google Scholar

[37]

T. Ma and E. M. Bollt, Shape coherence and finite-time curvature evolution,, International Journal of Bifurcation and Chaos, 25 (2015). doi: 10.1142/S0218127415500765. Google Scholar

[38]

T. Ma, N. T. Ouellette and E. M. Bollt, Stretching and folding in finite time,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 26 (2016). doi: 10.1063/1.4941256. Google Scholar

[39]

J. A. J. Madrid and A. M. Mancho, Distinguished trajectories in time dependent vector fields,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 19 (2009). doi: 10.1063/1.3056050. Google Scholar

[40]

N. Malhotra, I. Mezić and S. Wiggins, Patchiness: A new diagnostic for lagrangian trajectory analysis in time-dependent fluid flows,, International Journal of Bifurcation and Chaos, 8 (1998), 1053. doi: 10.1142/S0218127498000875. Google Scholar

[41]

A. M. Mancho, S. Wiggins, J. Curbelo and C. Mendoza, Lagrangian descriptors: A method for revealing phase space structures of general time dependent dynamical systems,, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 3530. doi: 10.1016/j.cnsns.2013.05.002. Google Scholar

[42]

I. Mezić, On the Geometrical and Statistical Properties of Dynamical Systems: Theory and Applications,, Phd thesis, (1994). Google Scholar

[43]

I. Mezić, Spectral properties of dynamical systems, model reduction and decompositions,, Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, 41 (2005), 309. doi: 10.1007/s11071-005-2824-x. Google Scholar

[44]

I. Mezić and A. Banaszuk, Comparison of systems with complex behavior,, Physica D. Nonlinear Phenomena, 197 (2004), 101. doi: 10.1016/j.physd.2004.06.015. Google Scholar

[45]

I. Mezić, S. Loire, V. A. Fonoberov and P. J. Hogan, A new mixing diagnostic and Gulf oil spill movement,, Science Magazine, 330 (2010), 486. Google Scholar

[46]

I. Mezić and F. Sotiropoulos, Ergodic theory and experimental visualization of invariant sets in chaotically advected flows,, Physics of Fluids, 14 (2002), 2235. doi: 10.1063/1.1480266. Google Scholar

[47]

I. Mezić and S. Wiggins, A method for visualization of invariant sets of dynamical systems based on the ergodic partition,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 9 (1999), 213. doi: 10.1063/1.166399. Google Scholar

[48]

B. A. Mosovsky and J. D. Meiss, Transport in transitory dynamical systems,, SIAM Journal on Applied Dynamical Systems, 10 (2011), 35. doi: 10.1137/100794110. Google Scholar

[49]

J. Nocedal and S. J. Wright, Numerical Optimization,, 2nd edition, (2006). Google Scholar

[50]

A. Okubo, Horizontal dispersion of floatable particles in vicinity of velocity singularities such as convergences,, Deep-Sea Research, 17 (1970), 445. doi: 10.1016/0011-7471(70)90059-8. Google Scholar

[51]

J. M. Ottino, The Kinematics of Mixing: Stretching, Chaos, and Transport,, Cambridge Texts in Applied Mathematics, (1989). Google Scholar

[52]

K. J. Palmer, A finite-time condition for exponential dichotomy,, Journal of Difference Equations and Applications, 17 (2011), 221. doi: 10.1080/10236198.2010.549005. Google Scholar

[53]

A. D. Perry and S. Wiggins, KAM tori are very sticky: Rigorous lower bounds on the time to move away from an invariant Lagrangian torus with linear flow,, Physica D. Nonlinear Phenomena, 71 (1994), 102. doi: 10.1016/0167-2789(94)90184-8. Google Scholar

[54]

K. Petersen, Ergodic Theory, vol. 2 of Cambridge Studies in Advanced Mathematics,, Cambridge University Press, (1989). Google Scholar

[55]

A. Poje, G. Haller and I. Mezić, The geometry and statistics of mixing in aperiodic flows,, Physics of Fluids, 11 (1999), 2963. doi: 10.1063/1.870155. Google Scholar

[56]

L. Rey-Bellet and L.-S. Young, Large deviations in non-uniformly hyperbolic dynamical systems,, Ergodic Theory and Dynamical Systems, 28 (2008), 587. doi: 10.1017/S0143385707000478. Google Scholar

[57]

D. P. Ruelle, Rotation numbers for diffeomorphisms and flows,, Annales de l'Institut Henri Poincaré. Physique Théorique, 42 (1985), 109. Google Scholar

[58]

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