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 & Continuous Dynamical Systems - A, 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.  doi: 10.5194/npg-16-655-2009.  Google Scholar

[2]

Michael Dellnitz, Gary Froyland and Oliver Junge, The algorithms behind GAIO-set oriented numerical methods for dynamical systems,, in, (2001), 145.   Google Scholar

[3]

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

[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.  doi: 10.1142/S0218127405012545.  Google Scholar

[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).   Google Scholar

[6]

Gary Froyland and Michael Dellnitz, Detecting and locating near-optimal almost-invariant sets and cycles,, SIAM Journal on Scientific Computing, 24 (2003), 1839.   Google Scholar

[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).   Google Scholar

[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.  doi: 10.1017/S0143385709000339.  Google Scholar

[9]

Gary Froyland, Simon Lloyd and Naratip Santitissadeekorn, Coherent sets for nonautonomous dynamical systems,, Physica D, 239 (2010), 1527.  doi: 10.1016/j.physd.2010.03.009.  Google Scholar

[10]

Gary Froyland, Kathrin Padberg, Matthew England and Anne Marie Treguier, Detection of coherent oceanic structures via transfer operators,, Physical Review Letters, 98 (2007).  doi: 10.1103/PhysRevLett.98.224503.  Google Scholar

[11]

Gary Froyland, Naratip Santitissadeekorn and Adam Monahan., Transport in time-dependent dynamical systems: Finite-time coherent sets,, Chaos, 20 (2010).   Google Scholar

[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, (2008), 313.   Google Scholar

[13]

George Haller, Lagrangian coherent structures from approximate velocity data,, Physics of Fluids, 14 (2002), 1851.  doi: 10.1063/1.1477449.  Google Scholar

[14]

Wilhelm Huisinga, Sean Meyn and Christof Schütte, Phase transitions and metastability in Markovian and molecular systems,, Annals of Applied Probability, 14 (2004), 419.  doi: 10.1214/aoap/1075828057.  Google Scholar

[15]

Wilhelm Huisinga and Bernd Schmidt, Metastability and dominant eigenvalues of transfer operators,, In, 49 (2006).   Google Scholar

[16]

Christopher Jones and Sean Winkler, Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere,, in, (2002), 55.  doi: 10.1016/S1874-575X(02)80023-6.  Google Scholar

[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.   Google Scholar

[18]

Martin Rasmussen, "Attractivity and Bifurcation for Nonautonomous Dynamical Systems,", Lecture Notes in Mathematics, 1907 (2007).   Google Scholar

[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).  doi: 10.1103/PhysRevE.82.056311.  Google Scholar

[20]

Christof Schütte, Wilhelm Huisinga and Peter Deuflhard, Transfer operator approach to conformational dynamics in biomolecular systems,, in, (2001), 191.   Google Scholar

[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.  doi: 10.1016/j.physd.2005.10.007.  Google Scholar

[22]

Shawn C. Shadden and Charles Taylor, Characterization of coherent structures in the cardiovascular system,, Annals of Biomedical Engineering, 36 (2008), 1152.  doi: 10.1007/s10439-008-9502-3.  Google Scholar

[23]

Stanislaw Marcin Ulam, "A Collection of Mathematical Problems,", Interscience Tracts in Pure and Applied Mathematics, (1960).   Google Scholar

[24]

Stephen Wiggins, The dynamical systems approach to Lagrangian transport in oceanic flows,, in, 37 (2005), 295.   Google Scholar

show all references

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.  doi: 10.5194/npg-16-655-2009.  Google Scholar

[2]

Michael Dellnitz, Gary Froyland and Oliver Junge, The algorithms behind GAIO-set oriented numerical methods for dynamical systems,, in, (2001), 145.   Google Scholar

[3]

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

[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.  doi: 10.1142/S0218127405012545.  Google Scholar

[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).   Google Scholar

[6]

Gary Froyland and Michael Dellnitz, Detecting and locating near-optimal almost-invariant sets and cycles,, SIAM Journal on Scientific Computing, 24 (2003), 1839.   Google Scholar

[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).   Google Scholar

[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.  doi: 10.1017/S0143385709000339.  Google Scholar

[9]

Gary Froyland, Simon Lloyd and Naratip Santitissadeekorn, Coherent sets for nonautonomous dynamical systems,, Physica D, 239 (2010), 1527.  doi: 10.1016/j.physd.2010.03.009.  Google Scholar

[10]

Gary Froyland, Kathrin Padberg, Matthew England and Anne Marie Treguier, Detection of coherent oceanic structures via transfer operators,, Physical Review Letters, 98 (2007).  doi: 10.1103/PhysRevLett.98.224503.  Google Scholar

[11]

Gary Froyland, Naratip Santitissadeekorn and Adam Monahan., Transport in time-dependent dynamical systems: Finite-time coherent sets,, Chaos, 20 (2010).   Google Scholar

[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, (2008), 313.   Google Scholar

[13]

George Haller, Lagrangian coherent structures from approximate velocity data,, Physics of Fluids, 14 (2002), 1851.  doi: 10.1063/1.1477449.  Google Scholar

[14]

Wilhelm Huisinga, Sean Meyn and Christof Schütte, Phase transitions and metastability in Markovian and molecular systems,, Annals of Applied Probability, 14 (2004), 419.  doi: 10.1214/aoap/1075828057.  Google Scholar

[15]

Wilhelm Huisinga and Bernd Schmidt, Metastability and dominant eigenvalues of transfer operators,, In, 49 (2006).   Google Scholar

[16]

Christopher Jones and Sean Winkler, Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere,, in, (2002), 55.  doi: 10.1016/S1874-575X(02)80023-6.  Google Scholar

[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.   Google Scholar

[18]

Martin Rasmussen, "Attractivity and Bifurcation for Nonautonomous Dynamical Systems,", Lecture Notes in Mathematics, 1907 (2007).   Google Scholar

[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).  doi: 10.1103/PhysRevE.82.056311.  Google Scholar

[20]

Christof Schütte, Wilhelm Huisinga and Peter Deuflhard, Transfer operator approach to conformational dynamics in biomolecular systems,, in, (2001), 191.   Google Scholar

[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.  doi: 10.1016/j.physd.2005.10.007.  Google Scholar

[22]

Shawn C. Shadden and Charles Taylor, Characterization of coherent structures in the cardiovascular system,, Annals of Biomedical Engineering, 36 (2008), 1152.  doi: 10.1007/s10439-008-9502-3.  Google Scholar

[23]

Stanislaw Marcin Ulam, "A Collection of Mathematical Problems,", Interscience Tracts in Pure and Applied Mathematics, (1960).   Google Scholar

[24]

Stephen Wiggins, The dynamical systems approach to Lagrangian transport in oceanic flows,, in, 37 (2005), 295.   Google Scholar

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