Back
 JAMP  Vol.3 No.7 , July 2015
Contrast of Perspectives of Coherency
Abstract: Mixing and coherence are fundamental issues at the heart of understanding fluid dynamics and other non-autonomous dynamical systems. Recently the notion of coherence has come to a more rigorous footing, in particular, within the studies of finite-time nonautonomous dynamical systems. Here we recall “shape coherent sets” which is proven to correspond to slowly evolving curvature, for which tangency of finite time stable foliations (related to a “forward time” perspective) and finite time unstable foliations (related to a “backwards time” perspective) serve a central role. We compare and contrast this perspective to both the variational method of geodesics [17], as well as the coherent pairs perspective [12] from transfer operators.
Cite this paper: Ma, T. and Bollt, E. (2015) Contrast of Perspectives of Coherency. Journal of Applied Mathematics and Physics, 3, 781-791. doi: 10.4236/jamp.2015.37095.
References

[1]   Brown, L.G. (1992) A Survey of Image Registration Techniques. ACM Computing Surveys (CSUR), 24, 325-376. http://dx.doi.org/10.1145/146370.146374

[2]   Bollt, E. and Santitissadeekorn, N. (2013) Applied and Computational Measurable Dynamics. SIAM. http://dx.doi.org/10.1137/1.9781611972641

[3]   Barnea, D.I. and Silverman, H.F. (1972) A Class of Algo-rithms for Fast Digital Registration. IEEE Trans. Comput., C-21, 179-186. http://dx.doi.org/10.1109/TC.1972.5008923

[4]   De Castro, E. and Morandi, C. (1987) Registration of Translated and Rotated Images Using Finite Fourier Transforms. IEEE Thans. Pattern Anal. Machine Intell., PAMI-95, 700-703. http://dx.doi.org/10.1109/TPAMI.1987.4767966

[5]   Ding, J., Li, T.Y. and Zhou, A. (2002) Finite Approximations of Markov Operators. Journal of Computational and Applied Mathematics, 147-1, 137-152. http://dx.doi.org/10.1016/s0377-0427(02)00429-6

[6]   Froyland, G. (2005) Statistically Optimal Almost-Invariant Sets. Physica D: Nonlinear Phenomena, 200, 205-219. http://dx.doi.org/10.1016/j.physd.2004.11.008

[7]   Farazmand, M. and Haller, G. (2012) Computing La-grangian Coherent Structures from Their Variational Theory. Chaos, 22, 013128. http://dx.doi.org/10.1063/1.3690153

[8]   Farazmand, M. and Haller, G. (2013) Attracting and Repelling Lagrangian Coherent Structures from a Single Computation. Chaos, 23, 023101. http://dx.doi.org/10.1063/1.4800210

[9]   Farazmand, M. and Haller, G. (2012) Erratum and Addendum to “A Variational Theory of Hyperbolic Lagrangian Coherent Structures” [Physica D 240 (2011) 574–598]. Physica D, 241 439-441. http://dx.doi.org/10.1016/j.physd.2011.09.013

[10]   Froyland, G. and Padberg, K. (2009) Almost-Invariant Sets and Invariant Manifolds-Connecting Probabilistic and Geometric Descriptions of Coherent Structures in Flows. Physica D, 238, 1507-1523. http://dx.doi.org/10.1016/j.physd.2009.03.002

[11]   Froyland, G. and Padberg, K. (2014) Almost-Invariant and Finite-Time Coherent Sets: Directionality, Duration, and Diffusion. To appear in Ergodic Theory, Open Dynamics, and Coherent Structures, 70, 171-216. http://dx.doi.org/10.1007/978-1-4939-0419-8_9

[12]   Froyland, G., Santitissadeekorn, N. and Monahan, A. (2010) Transport in Time-Dependent Dynamical Systems: Finite- Time Coherent Sets. Chaos, 20, 043116. http://dx.doi.org/10.1063/1.3502450

[13]   Froyland, G. (2013) An Analytic Framework for Identifying Fi-nite-Time Coherent Sets in Time-Dependent Dynamical Systems. Physica D, 250, 1-19. http://dx.doi.org/10.1016/j.physd.2013.01.013

[14]   Golub, G.H. and Van Loan, C.F. (1989) Matrix Computa-tions. 2nd Edition, The Johns Hopkins University Press.

[15]   Haller, G. (2000) Finding Finite-Time Invariant Manifolds in Two-Dimensional Velocity Fields. Chaos, 10, 99. http://dx.doi.org/10.1063/1.166479

[16]   Haller, G. (2002) Lagrangian Coherent Structures from Approximate Velocity Data. Physics of Fluids, 14, 1851. http://dx.doi.org/10.1063/1.1477449

[17]   Haller, G. and Beron-Vera, F.J. (2012) Geodesic Theory of Transport Barriers in Two-Dimensional Flows. Physica D, 241, 1680-1702. http://dx.doi.org/10.1016/j.physd.2012.06.012

[18]   Haller, G. and Beron-Vera, F.J. (2013) Coherent La-grangian Vortices: The Black Holes of Turbulence. J. Fluid Mech. 731. http://dx.doi.org/10.1017/jfm.2013.391

[19]   NDSG at ETH Zurich, Led by Prof. Haller, G., LCS Tool. https://github.com/jeixav/LCS-Tool

[20]   Lasota, A. and Mackey, M.C. (1985) Chaos, Fractals, and Noise Sto-chastic Aspects of Dynamics. Springer.

[21]   Ma, T. and Bollt, E. (2013) Relatively Coherent Sets as a Hierarchical Partition Method. International Journal of Bifurcation and Chaos, 23, 1330026. http://dx.doi.org/10.1142/S0218127413300267

[22]   Ma, T. and Bollt, E. (2014) Differential Geometry Perspective of Shape Coherence and Curvature Evolution by Finite- Time Nonhyperbolic Splitting. SIADS, 13, 1106-1136. http://dx.doi.org/10.1137/130940633

[23]   Ma, T. and Bollt, E. (2014) Shape Coherence and Fi-nite-Time Curvature Evolution. To appear in International Journal of Bifurcation and Chaos, 25.

[24]   Rypina, I.I., Brown, M.G., Beron-Vera, F.J., Kocak, H., Olascoaga, M.J. and Udovydchenkov, I.A. (2007) On the La-grangian Dynamics of Atmospheric Zonal Jets and the Permeability of the Stratospheric Polar Vortex. Journal of the Atmospheric Sciences, 64.

[25]   Shadden, S.C., Lekien, F. and Marsden, J.E. (2005) Definition and Properties of Lagrangian Coherent Structures from Finite-Time Lyapunov Exponents in Two-Dimensional Aperiodic Flows. Physica D, 212, 271-304. http://dx.doi.org/10.1016/j.physd.2005.10.007

[26]   Tallapragada, P. and Ross, S.D. (2013) A Set Oriented Definition of Finite-Time Lyapunov Exponents and Coherent Sets. Commun Nonlinear Sci Number Simulation, 18, 1106-1126. http://dx.doi.org/10.1016/j.cnsns.2012.09.017

 
 
Top