×

Local expansion concepts for detecting transport barriers in dynamical systems. (English) Zbl 1221.37163

Summary: In the last two decades, the mathematical analysis of material transport has received considerable interest in many scientific fields such as ocean dynamics and astrodynamics. In this contribution we focus on the numerical detection and approximation of transport barriers in dynamical systems. Starting from a set-oriented approximation of the dynamics we combine discrete concepts from graph theory with established geometric ideas from dynamical systems theory. We derive the global transport barriers by computing the local expansion properties of the system. For the demonstration of our results we consider two different systems. First we explore a simple flow map inspired by the dynamics of the global ocean. The second example is the planar circular restricted three body problem with Sun and Jupiter as primaries, which allows us to analyze particle transport in the solar system.

MSC:

37M05 Simulation of dynamical systems
37B10 Symbolic dynamics
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
70F07 Three-body problems
86A05 Hydrology, hydrography, oceanography

Software:

GAIO; JOSTLE; Scotch; GADS
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Wiggins, S., Chaotic transport in dynamical systems (1992), Springer: Springer New York (NY) · Zbl 0747.34028
[2] MacKay, R.; Meiss, J.; Percival, I., Transport in Hamiltonian systems, Physica D, 13, 55-81 (1984) · Zbl 0585.58039
[3] Rom-Kedar, V.; Wiggins, S., Transport in two-dimensional maps, Arch Ration Mech Anal, 109, 239-298 (1990) · Zbl 0755.58070
[4] Haller, G., Finding finite-time invariant manifolds in two-dimensional velocity fields, Chaos, 10, 99-108 (2000) · Zbl 0979.37012
[5] Meiss, J., Symplectic maps, variational principles, and transport, Rev Mod Phys, 64, 3, 795-848 (1992) · Zbl 1160.37302
[6] Shadden, S.; Lekien, F.; Marsden, J., 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
[7] Dellnitz, M.; Junge, O.; Koon, W.; Lekien, F.; Lo, M.; Marsden, J., Transport in dynamical astronomy and multibody problems, Int J Bifurcat Chaos, 15, 3, 699-727 (2005) · Zbl 1085.70012
[8] Broer H, Hanßmann H. Hamiltonian perturbation theory (and transition to chaos). In: Meyers R, editor. Encyclopaedia of complexity & system science. Springer, in press.; Broer H, Hanßmann H. Hamiltonian perturbation theory (and transition to chaos). In: Meyers R, editor. Encyclopaedia of complexity & system science. Springer, in press.
[9] Aref, H., The development of chaotic advection, Phys Fluids, 14, 4, 1315-1325 (2002) · Zbl 1185.76034
[10] Wiggins, S., The dynamical systems approach to Lagrangian transport in oceanic flows, Annu Rev Fluid Mech, 37, 295-328 (2005) · Zbl 1117.76058
[11] Gladman, B.; Burns, J.; Duncan, M.; Lee, P.; Levison, H., The exchange of impact ejecta between terrestrial planets, Sciences, 271, 1387-1392 (1996)
[12] Froyland G, Padberg K. Almost-invariant sets and invariant manifolds – connecting probabilistic and geometric descriptions of coherent structures in flows. Physica D 2009 (in press).; Froyland G, Padberg K. Almost-invariant sets and invariant manifolds – connecting probabilistic and geometric descriptions of coherent structures in flows. Physica D 2009 (in press). · Zbl 1178.37119
[13] Haller, G., Distinguished material surfaces and coherent structures in three-dimensional fluid flows, Physica D, 149, 248-277 (2001) · Zbl 1015.76077
[14] Haller, G.; Poje, A., Finite-time transport in aperiodic flows, Physica D, 119, 352-380 (1998) · Zbl 1194.76089
[15] Haller, G.; Yuan, G., Lagrangian coherent structures and mixing in two-dimensional turbulence, Physica D, 147, 352-370 (2000) · Zbl 0970.76043
[16] Dellnitz, M.; Junge, O., On the approximation of complicated dynamical behavior, SIAM J Numer Anal, 36, 2, 491-515 (1999) · Zbl 0916.58021
[17] Deuflhard P, Dellnitz M, Junge O, Schütte C. Computation of essential molecular dynamics by subdivision techniques. In: Deuflhard P et al. editor. Computational molecular dynamics: challenges, methods, ideas, LNCSE, vol. 4. Springer-Verlag; 1998. p. 98-115.; Deuflhard P, Dellnitz M, Junge O, Schütte C. Computation of essential molecular dynamics by subdivision techniques. In: Deuflhard P et al. editor. Computational molecular dynamics: challenges, methods, ideas, LNCSE, vol. 4. Springer-Verlag; 1998. p. 98-115.
[18] Deuflhard, P.; Huisinga, W.; Fischer, A.; Schütte, C., Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains, Lin Alg Appl, 315, 39-59 (2000) · Zbl 0963.65008
[19] Huisinga W. Metastability of Markovian systems, Ph.D. thesis, Freie Universität Berlin; 2001.; Huisinga W. Metastability of Markovian systems, Ph.D. thesis, Freie Universität Berlin; 2001.
[20] Froyland, G.; Dellnitz, M., Detecting and locating near-optimal almost-invariant sets and cycles, SIAM J Sci Comput, 24, 1839-1863 (2003) · Zbl 1042.37063
[21] Deuflhard, P.; Weber, M., Robust Perron cluster analysis in conformation dynamics, Linear Algebra Appl, 398, 161-184 (2005) · Zbl 1070.15019
[22] Dellnitz M, Preis R. Congestion and almost invariant sets in dynamical systems. In: Winkler F, editor. Proceedings of SNSC’01, Springer; 2003. p. 183-209.; Dellnitz M, Preis R. Congestion and almost invariant sets in dynamical systems. In: Winkler F, editor. Proceedings of SNSC’01, Springer; 2003. p. 183-209. · Zbl 1027.65175
[23] Dellnitz, M.; Junge, O.; Lo, M. W.; Marsden, J. E.; Padberg, K.; Preis, R., Transport of Mars-crossers from the quasi-Hilda region, Phys Rev Lett, 94, 231102, 1-4 (2005)
[24] Padberg K. Numerical analysis of transport in dynamical systems, Ph.D. thesis, Universität Paderborn, Germany, June; 2005.; Padberg K. Numerical analysis of transport in dynamical systems, Ph.D. thesis, Universität Paderborn, Germany, June; 2005. · Zbl 1197.65207
[25] Dellnitz, M.; Hohmann, A., A subdivision algorithm for the computation of unstable manifolds and global attractors, Numer Math, 75, 293-317 (1997) · Zbl 0883.65060
[26] Dellnitz, M.; Froyland, G.; Junge, O., The algorithms behind GAIO - set oriented numerical methods for dynamical systems, (Fiedler, B., Ergodic theory, analysis, and efficient simulation of dynamical systems (2001), Springer), 145-174 · Zbl 0998.65126
[27] Ulam, S., A collection of mathematical problems (1960), Interscience Publishers · Zbl 0086.24101
[28] Hunt, F., A Monte Carlo approach to the approximation of invariant measures, Random Comput Dyn, 2, 111-133 (1994) · Zbl 0804.58033
[29] Froyland, G., Statistically optimal almost-invariant sets, Physica D, 200, 205-219 (2005) · Zbl 1062.37103
[30] Garey, M.; Johnson, D., Computers and intractability – a guide to the theory of NP-completeness (1979), Freeman · Zbl 0411.68039
[31] Preis R. Analyses and design of efficient graph partitioning methods, Ph.D. thesis, Universität Paderborn, Germany, November; 2000.; Preis R. Analyses and design of efficient graph partitioning methods, Ph.D. thesis, Universität Paderborn, Germany, November; 2000. · Zbl 1019.05055
[32] Hendrickson B, Leland R. A multilevel algorithm for partitioning graphs. In: Supercomputing’95: proceedings of the 1995 ACM/IEEE conference on supercomputing (CDROM). New York, NY, USA: ACM; 1995. p. 28.; Hendrickson B, Leland R. A multilevel algorithm for partitioning graphs. In: Supercomputing’95: proceedings of the 1995 ACM/IEEE conference on supercomputing (CDROM). New York, NY, USA: ACM; 1995. p. 28. · Zbl 0816.68093
[33] Walshaw C. The Jostle user manual: version 2.2, University of Greenwich; 2000.; Walshaw C. The Jostle user manual: version 2.2, University of Greenwich; 2000.
[34] Karypis, G.; Kumar, V., A fast and high quality multilevel scheme for partitioning irregular graphs, SIAM J Sci Comput, 20, 1, 359-392 (1998) · Zbl 0915.68129
[35] Pellegrini F. SCOTCH 3.1 user’s guide, Tech. Rep. 1137-96, LaBRI, University of Bordeaux; 1996.; Pellegrini F. SCOTCH 3.1 user’s guide, Tech. Rep. 1137-96, LaBRI, University of Bordeaux; 1996.
[36] Monien, B.; Preis, R.; Diekmann, R., Quality matching and local improvement for multilevel graph-partitioning, Parallel Comput, 26, 12, 1609-1634 (2000) · Zbl 0948.68131
[37] Preis R. GADS - graph algorithms for dynamical systems, Tech. Rep., Universität Paderborn; 2004.; Preis R. GADS - graph algorithms for dynamical systems, Tech. Rep., Universität Paderborn; 2004.
[38] Dellnitz M, Padberg K, Preis R. Integrating multilevel graph partitioning with hierarchical set oriented methods for the analysis of dynamical systems, Tech. Rep., Preprint 152, DFG priority program: analysis, modeling and simulation of multiscale problems; 2004.; Dellnitz M, Padberg K, Preis R. Integrating multilevel graph partitioning with hierarchical set oriented methods for the analysis of dynamical systems, Tech. Rep., Preprint 152, DFG priority program: analysis, modeling and simulation of multiscale problems; 2004.
[39] Szebehely, V., Theory of orbits (1967), Academic Press
[40] Koon, W.; Lo, M.; Marsden, J.; Ross, S., Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics, Chaos, 10, 2, 427-469 (2000) · Zbl 0987.70010
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.