# American Institute of Mathematical Sciences

September  2012, 32(9): 3029-3042. doi: 10.3934/dcds.2012.32.3029

## The efficient approximation of coherent pairs in non-autonomous dynamical systems

 1 University of Paderborn, Warburger Str. 100, Paderborn, 33098, Germany, Germany

Received  February 2012 Revised  March 2012 Published  April 2012

The aim of this paper is the construction of numerical tools for the efficient approximation of transport phenomena in non-autonomous dynamical systems. We focus on transfer operator methods which have been developed in the last years for the treatment of non-autonomous dynamical systems. For instance Froyland et al. [11] proposed a method for the approximation of so-called coherent pairs -- these pairs of sets represent time-dependent slowly mixing structures -- by thresholding singular vectors of a normalized transfer operator over a fixed time-interval. In principle such transfer operator methods involve long term simulations of trajectories on the whole state space. In our main result we show that transport phenomena over a fixed (long) time horizon imply the existence of almost invariant sets over shorter time intervals if the transport process is slow enough. This fact is used to formulate an algorithm that preselects part of state space as a candidate for containing one of the sets of a coherent pair. By this we significantly reduce the related numerical effort.
Citation: Michael Dellnitz, Christian Horenkamp. The efficient approximation of coherent pairs in non-autonomous dynamical systems. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3029-3042. doi: 10.3934/dcds.2012.32.3029
##### References:
 [1] Michael Dellnitz, Gary Froyland, Christian Horenkamp, Kathrin Padberg-Gehle and Alex Sen Gupta, Seasonal variability of the subpolar gyres in the southern ocean: A numerical investigation based on transfer operators, Nonlinear Processes in Geophysics, 16 (2009), 655-664. doi: 10.5194/npg-16-655-2009. [2] Michael Dellnitz, Gary Froyland and Oliver Junge, The algorithms behind GAIO-set oriented numerical methods for dynamical systems, in "Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems" (ed. Bernold Fiedler), Springer, Berlin, (2001), 145-174, 805-807. [3] Michael Dellnitz and Oliver Junge, On the approximation of complicated dynamical behavior, SIAM Journal for Numerical Analysis, 36 (1999), 491-515. doi: 10.1137/S0036142996313002. [4] Michael Dellnitz, Oliver Junge, Wang Sang Koon, Francois Lekien, Martin W. Lo, Jerrold E. Marsden, Kathrin Padberg, Robert Preis, Shane D. Ross and Bianca Thiere, Transport in dynamical astronomy and multibody problems, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 15 (2005), 699-727. doi: 10.1142/S0218127405012545. [5] Michael Dellnitz, Oliver Junge, Martin W. Lo, Jerrold E. Marsden, Kathrin Padberg, Robert Preis, Shane D. Ross and Bianca Thiere, Transport of Mars-crossing asteroids from the quasi-Hilda region, Physical Review Letters, 94 (2005). [6] Gary Froyland and Michael Dellnitz, Detecting and locating near-optimal almost-invariant sets and cycles, SIAM Journal on Scientific Computing, 24 (2003), 1839-1863. [7] Gary Froyland, Christian Horenkamp, Vincent Rossi, Naratip Santitissadeekorn and Alex Sen Gupta, Three-dimensional characterization and tracking of an Agulhas ring, submitted to Ocean Modelling, 2011. [8] Gary Froyland, Simon Lloyd and Anthony Quas, Coherent structures and isolated spectrum for Perron-Frobenius cocycles, Ergodic Theory and Dynamical Systems, 30 (2010), 729-756. doi: 10.1017/S0143385709000339. [9] Gary Froyland, Simon Lloyd and Naratip Santitissadeekorn, Coherent sets for nonautonomous dynamical systems, Physica D, 239 (2010), 1527-1541. doi: 10.1016/j.physd.2010.03.009. [10] Gary Froyland, Kathrin Padberg, Matthew England and Anne Marie Treguier, Detection of coherent oceanic structures via transfer operators, Physical Review Letters, 98 (2007), 224503. doi: 10.1103/PhysRevLett.98.224503. [11] Gary Froyland, Naratip Santitissadeekorn and Adam Monahan., Transport in time-dependent dynamical systems: Finite-time coherent sets, Chaos, 20 (2010), 043116, 10 pp. [12] Gary Froyland, Marcel Schwalb, Kathrin Padberg and Michael Dellnitz, A transfer operator based numerical investigation of coherent structures in three-dimensional Southern Ocean circulation, in "Proceedings of the 2008 International Symposium on Nonlinear Theory and its Applications," Budapest, September, (2008), 313-316. [13] George Haller, Lagrangian coherent structures from approximate velocity data, Physics of Fluids, 14 (2002), 1851-1861. doi: 10.1063/1.1477449. [14] Wilhelm Huisinga, Sean Meyn and Christof Schütte, Phase transitions and metastability in Markovian and molecular systems, Annals of Applied Probability, 14 (2004), 419-458. doi: 10.1214/aoap/1075828057. [15] Wilhelm Huisinga and Bernd Schmidt, Metastability and dominant eigenvalues of transfer operators, In "Advances in Algorithms for Macromolecular Simulation," Lect. Notes Comput. Sci. Eng., 49, Springer, Berlin, 2006. [16] Christopher Jones and Sean Winkler, Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere, in "Handbook of Dynamical Systems" (ed. Bernold Fiedler), Vol. 2, North Holland, Amsterdam, (2002), 55-92. doi: 10.1016/S1874-575X(02)80023-6. [17] Francois Lekien, Chad Coulliette and Jerrold E. Marsden, Lagrangian structures in very high frequency radar data and optimal pollution timing, American Institute of Physics: 7th Experimental Chaos Conference, 676 (2003), 162-168. [18] Martin Rasmussen, "Attractivity and Bifurcation for Nonautonomous Dynamical Systems," Lecture Notes in Mathematics, 1907, Springer, Berlin, 2007. [19] Naratip Santitissadeekorn, Gary Froyland and Adam Monahan, Optimally coherent sets in geophysical flows: A transfer-operator approach to delimiting the stratospheric polar vortex, Physical Review E, 82 (2010), 056311. doi: 10.1103/PhysRevE.82.056311. [20] Christof Schütte, Wilhelm Huisinga and Peter Deuflhard, Transfer operator approach to conformational dynamics in biomolecular systems, in "Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems," Springer, Berlin, (2001), 191-223. [21] Shawn C. Shadden, Francois Lekien and Jerrold E. Marsden, Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Physica D, 212 (2005), 271-304. doi: 10.1016/j.physd.2005.10.007. [22] Shawn C. Shadden and Charles Taylor, Characterization of coherent structures in the cardiovascular system, Annals of Biomedical Engineering, 36 (2008), 1152-1162. doi: 10.1007/s10439-008-9502-3. [23] Stanislaw Marcin Ulam, "A Collection of Mathematical Problems," Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York-London, 1960. [24] Stephen Wiggins, The dynamical systems approach to Lagrangian transport in oceanic flows, in "Annual Review of Fluid Mechanics," 37, Annual Reviews, Palo Alto, CA, (2005), 295-328.

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##### References:
 [1] Michael Dellnitz, Gary Froyland, Christian Horenkamp, Kathrin Padberg-Gehle and Alex Sen Gupta, Seasonal variability of the subpolar gyres in the southern ocean: A numerical investigation based on transfer operators, Nonlinear Processes in Geophysics, 16 (2009), 655-664. doi: 10.5194/npg-16-655-2009. [2] Michael Dellnitz, Gary Froyland and Oliver Junge, The algorithms behind GAIO-set oriented numerical methods for dynamical systems, in "Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems" (ed. Bernold Fiedler), Springer, Berlin, (2001), 145-174, 805-807. [3] Michael Dellnitz and Oliver Junge, On the approximation of complicated dynamical behavior, SIAM Journal for Numerical Analysis, 36 (1999), 491-515. doi: 10.1137/S0036142996313002. [4] Michael Dellnitz, Oliver Junge, Wang Sang Koon, Francois Lekien, Martin W. Lo, Jerrold E. Marsden, Kathrin Padberg, Robert Preis, Shane D. Ross and Bianca Thiere, Transport in dynamical astronomy and multibody problems, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 15 (2005), 699-727. doi: 10.1142/S0218127405012545. [5] Michael Dellnitz, Oliver Junge, Martin W. Lo, Jerrold E. Marsden, Kathrin Padberg, Robert Preis, Shane D. Ross and Bianca Thiere, Transport of Mars-crossing asteroids from the quasi-Hilda region, Physical Review Letters, 94 (2005). [6] Gary Froyland and Michael Dellnitz, Detecting and locating near-optimal almost-invariant sets and cycles, SIAM Journal on Scientific Computing, 24 (2003), 1839-1863. [7] Gary Froyland, Christian Horenkamp, Vincent Rossi, Naratip Santitissadeekorn and Alex Sen Gupta, Three-dimensional characterization and tracking of an Agulhas ring, submitted to Ocean Modelling, 2011. [8] Gary Froyland, Simon Lloyd and Anthony Quas, Coherent structures and isolated spectrum for Perron-Frobenius cocycles, Ergodic Theory and Dynamical Systems, 30 (2010), 729-756. doi: 10.1017/S0143385709000339. [9] Gary Froyland, Simon Lloyd and Naratip Santitissadeekorn, Coherent sets for nonautonomous dynamical systems, Physica D, 239 (2010), 1527-1541. doi: 10.1016/j.physd.2010.03.009. [10] Gary Froyland, Kathrin Padberg, Matthew England and Anne Marie Treguier, Detection of coherent oceanic structures via transfer operators, Physical Review Letters, 98 (2007), 224503. doi: 10.1103/PhysRevLett.98.224503. [11] Gary Froyland, Naratip Santitissadeekorn and Adam Monahan., Transport in time-dependent dynamical systems: Finite-time coherent sets, Chaos, 20 (2010), 043116, 10 pp. [12] Gary Froyland, Marcel Schwalb, Kathrin Padberg and Michael Dellnitz, A transfer operator based numerical investigation of coherent structures in three-dimensional Southern Ocean circulation, in "Proceedings of the 2008 International Symposium on Nonlinear Theory and its Applications," Budapest, September, (2008), 313-316. [13] George Haller, Lagrangian coherent structures from approximate velocity data, Physics of Fluids, 14 (2002), 1851-1861. doi: 10.1063/1.1477449. [14] Wilhelm Huisinga, Sean Meyn and Christof Schütte, Phase transitions and metastability in Markovian and molecular systems, Annals of Applied Probability, 14 (2004), 419-458. doi: 10.1214/aoap/1075828057. [15] Wilhelm Huisinga and Bernd Schmidt, Metastability and dominant eigenvalues of transfer operators, In "Advances in Algorithms for Macromolecular Simulation," Lect. Notes Comput. Sci. Eng., 49, Springer, Berlin, 2006. [16] Christopher Jones and Sean Winkler, Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere, in "Handbook of Dynamical Systems" (ed. Bernold Fiedler), Vol. 2, North Holland, Amsterdam, (2002), 55-92. doi: 10.1016/S1874-575X(02)80023-6. [17] Francois Lekien, Chad Coulliette and Jerrold E. Marsden, Lagrangian structures in very high frequency radar data and optimal pollution timing, American Institute of Physics: 7th Experimental Chaos Conference, 676 (2003), 162-168. [18] Martin Rasmussen, "Attractivity and Bifurcation for Nonautonomous Dynamical Systems," Lecture Notes in Mathematics, 1907, Springer, Berlin, 2007. [19] Naratip Santitissadeekorn, Gary Froyland and Adam Monahan, Optimally coherent sets in geophysical flows: A transfer-operator approach to delimiting the stratospheric polar vortex, Physical Review E, 82 (2010), 056311. doi: 10.1103/PhysRevE.82.056311. [20] Christof Schütte, Wilhelm Huisinga and Peter Deuflhard, Transfer operator approach to conformational dynamics in biomolecular systems, in "Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems," Springer, Berlin, (2001), 191-223. [21] Shawn C. Shadden, Francois Lekien and Jerrold E. Marsden, Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Physica D, 212 (2005), 271-304. doi: 10.1016/j.physd.2005.10.007. [22] Shawn C. Shadden and Charles Taylor, Characterization of coherent structures in the cardiovascular system, Annals of Biomedical Engineering, 36 (2008), 1152-1162. doi: 10.1007/s10439-008-9502-3. [23] Stanislaw Marcin Ulam, "A Collection of Mathematical Problems," Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York-London, 1960. [24] Stephen Wiggins, The dynamical systems approach to Lagrangian transport in oceanic flows, in "Annual Review of Fluid Mechanics," 37, Annual Reviews, Palo Alto, CA, (2005), 295-328.
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