Almost-invariant sets and invariant manifolds - Connecting probabilistic and geometric descriptions of coherent structures in flows

Physica D: Nonlinear Phenomena

Volumn 238, Issue 16, 2009, Pages 1507-1523

  Froyland, Gary ^a   Padberg, Kathrin ^b  

^a UNIVERSITY OF NEW SOUTH WALES   (Australia)

^b DRESDEN UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Almost invariant set; Coherent structure; Finite time Lyapunov exponent; Invariant manifold; Transfer operator; Transport

Indexed keywords

EIGENVALUES AND EIGENFUNCTIONS; LYAPUNOV METHODS; MIXING;

COHERENT STRUCTURE; FINITE-TIME LYAPUNOV EXPONENT; INVARIANT MANIFOLDS; INVARIANT SET; TRANSFER OPERATOR; TRANSPORT;

FLOW OF FLUIDS;

EID: 67650066712     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2009.03.002     Document Type: Article

References (59)

1. Aref H. The development of chaotic advection. Phys. Fluids 14 4 (2002) 1315-1325
2. Meiss J.D. Symplectic maps variational principles transport. Rev. Modern Phys. 64 3 (1992) 795-848
3. Wiggins S. Chaotic Transport in Dynamical Systems (1992), Springer-Verlag, New York, NY
4. Wiggins S. The dynamical systems approach to Lagrangian transport in oceanic flows. Annu. Rev. Fluid Mech. 37 (2005) 295-328
5. Rom-Kedar V., Leonard A., and Wiggins S. An analytical study of transport mixing and chaos in an unsteady vortical flow. J. Fluid Mech. 214 (1990) 347-394
6. Rom-Kedar V., and Wiggins S. Transport in two-dimensional maps. Arch. Ration. Mech. Anal. 109 (1990) 239-298
7. Haller G. Finding finite-time invariant manifolds in two-dimensional velocity fields. Chaos 10 (2000) 99-108
8. Haller G. Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Physica D 149 (2001) 248-277
9. Haller G., and Poje A. Finite-time transport in aperiodic flows. Physica D 119 (1998) 352-380
10. Haller G., and Yuan G. Lagrangian coherent structures and mixing in two-dimensional turbulence. Physica D 147 (2000) 352-370
11. Shadden S.C., Lekien F., and Marsden J.E. Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Physica D 212 (2005) 271-304
12. Lekien F., Shadden S.C., and Marsden J.E. Lagrangian coherent structures inn-dimensional systems. J. Math. Phys. 48 (2007) 065404
13. Lekien F., and Coulliette C. Chaotic stirring in quasi-turbulent flows. Roy. Soc. London Philos. Trans. Ser. A 365 (2007) 3061-3084
14. Dellnitz M., and Junge O. On the approximation of complicated dynamical behaviour. SIAM J. Numer. Anal. 36 2 (1999) 491-515
15. Dellnitz M., and Junge O. Almost invariant sets in Chua's circuit. Internat. J. Bifur. Chaos 7 11 (1997) 2475-2485
16. Froyland G., and Dellnitz M. Detecting and locating near-optimal almost-invariant sets and cycles. SIAM J. Sci. Comput. 24 6 (2003) 1839-1863
17. Froyland G. Statistically optimal almost-invariant sets. Physica D 200 (2005) 205-219
18. Froyland G. Unwrapping eigenfunctions to discover the geometry of almost-invariant sets in hyperbolic maps. Physica D 237 6 (2008) 840-853
19. Deuflhard P., Huisinga W., Fischer A., and Schütte C. Identification of almost invariant aggregates in nearly uncoupled Markov chains. Linear Algebra Appl. 315 (2000) 39-59
20. Schütte C., Huisinga W., and Deuflhard P. Transfer operator approach to conformational dynamics in biomolecular systems. In: Fiedler B. (Ed). Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems (2001), Springer, Berlin 191-223
21. Gaveau B., and Schulman L. Theory of nonequilibrium first-order phase transitions for stochastic dynamics. J. Math. Phys. 39 3 (1998) 1517-1533
22. Gaveau B., and Schulman L. Multiple phases in stochastic dynamics: Geometry and probabilities. Phys. Rev. E 73 (2006) 036124
23. Liu W., and Haller G. Strange eigenmodes and decay of variance in the mixing of diffusive tracers. Physica D 188 (2004) 1-39
24. Pikovsky A., and Popovych O. Persistent patterns in deterministic mixing flows. Europhys. Lett. 61 5 (2003) 625-631
25. Popovych O., Pikovsky A., and Eckhardt B. Abnormal mixing of passive scalars in chaotic flows. Phys. Rev. E 75 (2007) 036308
26. Mezić I., and Wiggins S. A method of visualization of invariant sets of dynamical systems based on the ergodic partition. Chaos 9 1 (1999) 213-218
27. Malhotra N., Mezic I., and Wiggins S. Patchiness: A new diagnostic for Lagrangian trajectory analysis in fluid flows. Internat. J. Bifur. Chaos 8 (1998) 1053-1094
28. Poje A., Haller G., and Mezić I. The geometry and statistics of mixing in aperiodic flows. Phys. Fluids 11 (1999) 2963-2968
29. Dellnitz M., Junge O., Koon W., Lekien F., Lo M., Marsden J., Padberg K., Preis R., Ross S., and Thiere B. Transport in dynamical astronomy and multibody problems. Internat. J. Bifur. Chaos 15 3 (2005) 699-727
30. Froyland G., Padberg K., England M.H., and Treguier A.M. Detection of coherent oceanic structures via transfer operators. Phys. Rev. Lett. 98 22 (2007) 224503
31. Lorenz E.N. Deterministic nonperiodic flow. J. Atmos. Sci. 20 2 (1963) 130-141
32. Matsumoto T. A chaotic attractor from Chua's circuit. IEEE Trans. CAS 31 (1984) 1055-1058
33. Lasota A., and Mackey M.C. Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics. 2nd edition. Applied Mathematical Sciences vol. 97 (1994), Springer-Verlag, New York
34. Baladi V. Positive Transfer Operators and Decay of Correlations (2000), World Scientific, Singapore
35. Froyland G. Extracting dynamical behaviour via Markov models. In: Mees A.I. (Ed). Nonlinear Dynamics and Statistics: Proceedings, Newton Institute, Cambridge, 1998 (2001), Birkhäuser 283-324
36. Ulam S. Problems in Modern Mathematics (1964), Interscience
37. Horn R.A., and Johnson C.R. Matrix Analysis (1985), Cambridge University Press, Cambridge
38. G. Froyland, Estimating physical invariant measures and space averages of dynamical systems indicators, Ph.D. Thesis, The University of Western Australia, Perth, 1996
39. Baladi V., Isola S., and Schmitt B. Transfer operator for piecewise affine approximations of interval maps. Ann. l'Inst. Henri Poincaré-Phys. théor. 62 3 (1995) 251-265
40. Froyland G. Computer-assisted bounds for the rate of decay of correlations. Comm. Math. Phys. 189 (1997) 237-257
41. Blank M., Keller G., and Liverani C. Ruelle-Perron-Frobenius spectrum for Anosov maps. Nonlinearity 15 (2002) 1905-1973
42. Brémaud P. Markov chains. Gibbs Fields Monte Carlo Simulation Queues. Texts in Applied Mathematics vol. 31 (1999), Springer, New York
43. Alpert C.J., Kahng A.B., and Yao S.-Z. Spectral partitioning with multiple eigenvectors. Discrete Appl. Math. 90 (1999) 3-26
44. W. Huisinga, B. Schmidt, Metastability and dominant eigenvalues of transfer operators, Free University Berlin, 2004. Preprint
45. Deuflhard P., and Weber M. Robust Perron cluster analysis in conformation dynamics. Linear Algebra Appl. 398 (2005) 161-184
46. Guckenheimer J., and Holmes P. Nonlinear Oscillations Dynamical Systems and Bifurcations of Vector Fields. Applied Mathematical Sciences vol. 42 (1983), Springer-Verlag, New York, NY
47. Krauskopf B., Osinga H., Doedel E., Henderson M., Guckenheimer J., Vladimirsky A., Dellnitz M., and Junge O. A survey of methods for computing (un)stable manifolds of vector fields. Internat. J. Bifur. Chaos 15 (2005) 763-792
48. Dellnitz M., and Hohmann A. A subdivision algorithm for the computation of unstable manifolds and global attractors. Numer. Math. 75 (1997) 293-317
49. Dellnitz M., Froyland G., and Junge O. The algorithms behind GAIO-Set oriented numerical methods for dynamical systems. In: Fiedler B. (Ed). Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems (2001), Springer 145-174
50. Dellnitz M., and Junge O. Set Oriented Numerical Methods for Dynamical Systems. In: Fiedler B., Iooss G., and Kopell N. (Eds). Handbook of Dynamical Systems II: Towards Applications (2002), World Scientific 221-264
51. K. Padberg, Numerical analysis of transport in dynamical systems, Ph.D. Thesis, Universität Paderborn, Paderborn, 2005
52. Huang D.-B., Zhao X.-H., and Dai H.-H. Invariant tori and chaotic streamlines in the ABC flow. Phys. Lett. A 237 (1998) 136-140
53. Dombre T., Frisch U., Greene J., Hénon M., Mehr A., and Soward A. Chaotic streamlines in the ABC flows. J. Fluid Mech. 167 (1986) 353-391
54. Tucker W. The Lorenz attractor exists. C. R. Acad. Sci. Paris, Sér. I (1999) 1197-1202
55. Mehta P., von Molo M.H., and Dellnitz M. Symmetry of attractors and the Perron-Frobenius operator. J. Difference Equ. Appl. 12 11 (2006) 1147-1178
56. Doedel E., Krauskopf B., and Osinga H. Global bifurcations of the Lorenz manifold. Nonlinearity 19 (2006) 2947-2972
57. Franceschini V., Gilberti C., and Zheng Z. Characterization of the Lorenz attractor by unstable periodic orbits. Nonlinearity 6 (1993) 251-258
58. G. Froyland, S. Lloyd, A. Quas, Coherent structures and isolated spectrum for Perron-Frobenius cocycles, Ergodic Theory and Dynamical Systems (in press)
59. G. Froyland, S. Lloyd, N. Santitissadeekorn, Coherent sets for nonautonomous dynamical systems (submitted for publication)