Coherent sets for nonautonomous dynamical systems. (English) Zbl 1193.37032
Summary: We describe a mathematical formalism and numerical algorithms for identifying and tracking slowly mixing objects in nonautonomous dynamical systems. In the autonomous setting, such objects are variously known as almost-invariant sets, metastable sets, persistent patterns, or strange eigenmodes, and have proved to be important in a variety of applications. In this current work, we explain how to extend existing autonomous approaches to the nonautonomous setting. We call the new time-dependent slowly mixing objects coherent sets as they represent regions of phase space that disperse very slowly and remain coherent. The new methods are illustrated via detailed examples in both discrete and continuous time.
MSC:
| 37C60 | Nonautonomous smooth dynamical systems |
| 37A25 | Ergodicity, mixing, rates of mixing |
| 37B55 | Topological dynamics of nonautonomous systems |
Keywords:
Perron-Frobenius operator; coherent set; nonautonomous dynamical system; Oseledets subspace; Lyapunov exponent; almost-invariant set; metastable set; strange eigenmode; persistent patternReferences:
| [1] | Meiss, J. D., Symplectic maps, variational principles, and transport, Rev. Modern Phys., 64, 3, 795-848 (1992) · Zbl 1160.37302 |
| [2] | Wiggins, S., Chaotic Transport in Dynamical Systems (1992), Springer-Verlag: Springer-Verlag New York, NY · Zbl 0747.34028 |
| [3] | Aref, H., The development of chaotic advection, Phys. Fluids, 14, 4, 1315-1325 (2002) · Zbl 1185.76034 |
| [4] | Wiggins, S., The dynamical systems approach to Lagrangian tranport in oceanic flows, Annu. Rev. Fluid Mech., 37, 295-328 (2005) · Zbl 1117.76058 |
| [5] | Pikovsky, A.; Popovych, O., Persistent patterns in deterministic mixing flows, Europhys. Lett., 61, 5, 625-631 (2003) |
| [6] | Liu, Weijiu; Haller, George, Strange eigenmodes and decay of variance in the mixing of diffusive tracers, Physica D, 188, 1-39 (2004) · Zbl 1098.76550 |
| [7] | Popovych, O. V.; Pikovsky, A.; Eckhardt, B., Abnormal mixing of passive scalars in chaotic flows, Phys. Rev. E, 75, 036308 (2007) |
| [8] | Froyland, Gary; Padberg, Kathrin; England, Matthew H.; Treguier, Anne Marie, Detection of coherent oceanic structures via transfer operators, Phys. Rev. Lett., 98, 22, 224503 (2007) |
| [9] | Dellnitz, Michael; Froyland, Gary; Horenkamp, Christian; Padberg-Gehle, Kathrin; Gupta, Alex Sen, Seasonal variability of the subpolar gyres in the Southern Ocean: a numerical investigation based on transfer operators, Nonlinear Process. Geophys., 16, 655-664 (2009) |
| [10] | Dellnitz, M.; Junge, O.; Koon, W. S.; Lekien, F.; Lo, M. W.; Marsden, J. E.; Padberg, K.; Preis, R.; Ross, S. D.; Thiere, B., Transport in dynamical astronomy and multibody problems, Int. J. Bifur. Chaos, 15, 3, 699-727 (2005) · Zbl 1085.70012 |
| [11] | Deuflhard, P.; Huisinga, W.; Fischer, A.; Schütte, C., Identification of almost invariant aggregates in nearly uncoupled Markov chains, Linear Algebra Appl., 315, 39-59 (2000) · Zbl 0963.65008 |
| [12] | Schütte, Christof; Huisinga, Wilhelm; Deuflhard, Peter, Transfer operator approach to conformational dynamics in biomolecular systems, (Fiedler, Bernold, Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems (2001), Springer: Springer Berlin), 191-223 · Zbl 0996.92012 |
| [13] | Rom-Kedar, V.; Leonard, A.; Wiggins, S., An analytical study of transport, mixing and chaos in an unsteady vortical flow, J. Fluid Mech., 214, 347-394 (1990) · Zbl 0698.76028 |
| [14] | Rom-Kedar, V.; Wiggins, S., Transport in two-dimensional maps, Arch. Ration. Mech. Anal., 109, 239-298 (1990) · Zbl 0755.58070 |
| [15] | Haller, G., Finding finite-time invariant manifolds in two-dimensional velocity fields, Chaos, 10, 99-108 (2000) · Zbl 0979.37012 |
| [16] | Haller, George, Distinguished material surfaces and coherent structures in three-dimensional fluid flows, Physica D, 149, 248-277 (2001) · Zbl 1015.76077 |
| [17] | Shadden, Shawn C.; Lekien, Francois; Marsden, Jerrold E., Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Physica D, 212, 271-304 (2005) · Zbl 1161.76487 |
| [18] | Dellnitz, Michael; Junge, Oliver, On the approximation of complicated dynamical behaviour, SIAM J. Numer. Anal., 36, 2, 491-515 (1999) · Zbl 0916.58021 |
| [19] | Dellnitz, Michael; Junge, Oliver, Almost invariant sets in Chua’s circuit, Int. J. Bifur. Chaos, 7, 11, 2475-2485 (1997) · Zbl 0899.58029 |
| [20] | Froyland, Gary; Dellnitz, Michael, Detecting and locating near-optimal almost-invariant sets and cycles, SIAM J. Sci. Comput., 24, 6, 1839-1863 (2003) · Zbl 1042.37063 |
| [21] | Froyland, Gary, Statistically optimal almost-invariant sets, Physica D, 200, 205-219 (2005) · Zbl 1062.37103 |
| [22] | Froyland, Gary, Unwrapping eigenfunctions to discover the geometry of almost-invariant sets in hyperbolic maps, Physica D, 237, 6, 840-853 (2008) · Zbl 1157.37011 |
| [23] | Schütte, Christof, Conformational Dynamics: Modelling, Theory, Algorithm, and Application to Biomolecules (1999), Habilitation: Habilitation ZIB, Berlin |
| [24] | Deuflhard, P.; Dellnitz, M.; Junge, O.; Schütte, C., Computation of essential molecular dynamics by subdivision techniques I: basic concept, (Computational Molecular Dynamics: Challenges, Methods, Ideas (1998), Springer-Verlag), 98-115 · Zbl 0966.81067 |
| [25] | Arnold, Ludwig, Random Dynamical Systems (1998), Springer: Springer Berlin · Zbl 0906.34001 |
| [26] | Tucker, Warwick, The Lorenz attractor exists, C. R. Acad. Sci. Paris, Ser. I, 1197-1202 (1999) · Zbl 0935.34050 |
| [27] | Ulam, S., Problems in Modern Mathematics (1964), Interscience · Zbl 0137.24201 |
| [28] | Li, Tien-Yien, Finite approximation for the Frobenius-Perron operator. A solution to Ulam’s conjecture, J. Approx. Theory, 17, 177-186 (1976) · Zbl 0357.41011 |
| [29] | Froyland, Gary, Finite approximation of Sinai-Bowen-Ruelle measures for Anosov systems in two dimensions, Random Comput. Dyn., 3, 4, 251-263 (1995) · Zbl 0868.58052 |
| [30] | Ding, Jiu; Zhou, Aihui, Finite approximations of Frobenius-Perron operators. A solution of Ulam’s conjecture to multi-dimensional transformations, Physica D, 92, 1-2, 61-68 (1996) · Zbl 0890.58035 |
| [31] | Rua Murray, Discrete approximation of invariant densities, Ph.D. Thesis, University of Cambridge, 1997.; Rua Murray, Discrete approximation of invariant densities, Ph.D. Thesis, University of Cambridge, 1997. · Zbl 1193.37029 |
| [32] | Froyland, Gary, Ulam’s method for random interval maps, Nonlinearity, 12, 1029-1052 (1999) · Zbl 0989.37049 |
| [33] | Blank, Michael; Keller, Gerhard; Liverani, Carlangelo, Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity, 15, 1905-1973 (2002) · Zbl 1021.37015 |
| [34] | Murray, Rua, Ulam’s method for some non-uniformly expanding maps, Discrete Contin. Dyn. Syst., 26, 3, 1007-1018 (2010) · Zbl 1183.37139 |
| [35] | Gary Froyland, Simon Lloyd, Anthony Quas, Coherent Structures and Perron-Frobenius Cocycles, Published online by Cambridge University Press, 04 Sep. 2009, doi:10.1017/S0143385709000339.; Gary Froyland, Simon Lloyd, Anthony Quas, Coherent Structures and Perron-Frobenius Cocycles, Published online by Cambridge University Press, 04 Sep. 2009, doi:10.1017/S0143385709000339. |
| [36] | Dellnitz, Michael; Froyland, Gary; Sertl, Stefan, On the isolated spectrum of the Perron-Frobenius operator, Nonlinearity, 13, 1171-1188 (2000) · Zbl 0965.37008 |
| [37] | Keller, Gerhard, On the rate of convergence to equilibrium in one-dimensional systems, Comm. Math. Phys., 96, 181-193 (1984) · Zbl 0576.58016 |
| [38] | Froyland, Gary; Padberg, Kathrin, Almost-invariant sets and invariant manifolds—connecting probabilistic and geometric descriptions of coherent structures in flows, Physica D, 238, 16, 1507-1523 (2009) · Zbl 1178.37119 |
| [39] | Gary Froyland, Simon Lloyd, Anthony Quas, A semi-invertible Oseledets theorem with applications to transfer operator cocycles (submitted for publication). Available at: http://arxiv.org/abs/1001.5313v1.; Gary Froyland, Simon Lloyd, Anthony Quas, A semi-invertible Oseledets theorem with applications to transfer operator cocycles (submitted for publication). Available at: http://arxiv.org/abs/1001.5313v1. · Zbl 1405.37057 |
| [40] | Froyland, Gary; Judd, Kevin; Mees, Alistair, Estimation of Lyapunov exponents of dynamical systems using a spatial average, Phys. Rev. E, 51, 4, 2844-2855 (1995) |
| [41] | Trevisan, A.; Pancotti, F., Periodic orbits, Lyapunov vectors, and singular vectors in the Lorenz system, J. Atmospheric Sci., 55, 390-399 (1998) |
| [42] | Ershov, S.; Potapov, A., On the concept of stationary Lyapunov basis, Physica D, 118, 167-198 (1998) · Zbl 1194.34097 |
| [43] | Ginelli, F.; Poggi, P.; Turchi, A.; Chaté, H.; Livi, R.; Politi, A., Characterizing dynamics with covariant Lyapunov vectors, Phys. Rev. Lett., 99 (2007) |
| [44] | Lasota, Andrzej; Mackey, Michael C., Probabilistic Properties of Deterministic Systems (1985), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0606.58002 |
| [45] | Dunford, N.; Schwarz, J., Linear Operators, Part 1, General Theory (1988), Wiley: Wiley New York |
| [46] | Kato, Tosio, Perturbation Theory for Linear Operators (1980), Springer-Verlag: Springer-Verlag Berlin · Zbl 0435.47001 |
| [47] | Pierrehumbert, R. T., Chaotic mixing of tracers and vorticity by modulated travelling Rossby waves, Geophys. Astrophys. Fluid Dyn., 58, 285-320 (1991) |
| [48] | Samelson, R. M.; Wiggins, S., Lagrangian Transport in Geophysical Jets and Waves, The Dynamical System Approach (2006), Springer: Springer USA · Zbl 1132.37001 |
| [49] | Ruelle, David, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math., 115, 2, 243-290 (1982) · Zbl 0493.58015 |
| [50] | Naratip Santitissadeekorn, Gary Froyland, Adam Monahan, Optimally coherent sets in geophysical flows: a new approach to delimiting the stratospheric polar vortex (submitted for publication). Available at: http://arxiv.org/abs/1004.3596.; Naratip Santitissadeekorn, Gary Froyland, Adam Monahan, Optimally coherent sets in geophysical flows: a new approach to delimiting the stratospheric polar vortex (submitted for publication). Available at: http://arxiv.org/abs/1004.3596. · Zbl 1311.37008 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
