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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[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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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

[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.

[31]

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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.

[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.

[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.

[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.

[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.

[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.

[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.

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[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.

[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.

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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.

[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.

[48]

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

[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.

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

[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.

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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.

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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.

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R. M. Samelson, Lagrangian motion, coherent structures, and lines of persistent material strain,, Annual Review of Marine Science, 5 (2013), 137. doi: 10.1146/annurev-marine-120710-100819.

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S. C. Shadden, F. Lekien and J. E. Marsden, Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows,, Physica D. Nonlinear Phenomena, 212 (2005), 271. doi: 10.1016/j.physd.2005.10.007.

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J. D. Szezech Jr., A. B. Schelin, I. L. Caldas, S. R. Lopes, P. J. Morrison and R. L. Viana, Finite-time rotation number: A fast indicator for chaotic dynamical structures,, Physics Letters A, 377 (2013), 452.

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J.-L. Thiffeault, Braids of entangled particle trajectories,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 20 (2010). doi: 10.1063/1.3262494.

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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).

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[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.

[25]

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

[26]

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

[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.

[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.

[29]

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

[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.

[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.

[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.

[33]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[42]

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

[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.

[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.

[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.

[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.

[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.

[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.

[49]

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

[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.

[51]

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

[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.

[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.

[54]

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

[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.

[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.

[57]

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

[58]

R. M. Samelson, Lagrangian motion, coherent structures, and lines of persistent material strain,, Annual Review of Marine Science, 5 (2013), 137. doi: 10.1146/annurev-marine-120710-100819.

[59]

S. C. Shadden, F. Lekien and J. E. Marsden, Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows,, Physica D. Nonlinear Phenomena, 212 (2005), 271. doi: 10.1016/j.physd.2005.10.007.

[60]

J. D. Szezech Jr., A. B. Schelin, I. L. Caldas, S. R. Lopes, P. J. Morrison and R. L. Viana, Finite-time rotation number: A fast indicator for chaotic dynamical structures,, Physics Letters A, 377 (2013), 452.

[61]

J.-L. Thiffeault, Braids of entangled particle trajectories,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 20 (2010). doi: 10.1063/1.3262494.

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