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A closing scheme for finding almost-invariant sets in open dynamical systems
1. | School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052 |
2. | School of Mathematics and Physics, The University of Queensland, St Lucia QLD 4072, Australia |
3. | School of Mathematics and Statistics, The University of New South Wales, Sydney NSW 2052, Australia |
References:
[1] |
W. Bahsoun, Rigorous numerical approximation of escape rates, Nonlinearity, 19 (2006), 25-29.
doi: 10.1088/0951-7715/19/11/002. |
[2] |
M. S. Bartlett, Stochastic Population Models in Ecology and Epidemiology, Methuen, London, 1960. |
[3] |
L. Billings and I. B. Schwartz, Identifying almost invariant sets in stochastic dynamical systems, Chaos, 18 (2008), 023122.
doi: 10.1063/1.2929748. |
[4] |
C. Bose, G. Froyland, C. Gonzáles Tokman and R. Murray, Ulam's method for Lasota-Yorke maps with holes, To appear in SIAM J. Appl. Dynam. Syst. arXiv:1204.2329. |
[5] |
A. Boyarsky and P. Gora, Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension (Probability and its Applications), Springer, Berlin, 1997.
doi: 10.1007/978-1-4612-2024-4. |
[6] |
P. Brémaud, Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues, Springer-Verlag, 1999. |
[7] |
H. Bruin, M. Demers and I. Melbourne, Existence and convergence properties of physical measures for certain dynamical systems with holes, Ergodic Theory and Dynamical Systems, 30 (2010), 687-728.
doi: 10.1017/S0143385709000200. |
[8] |
D. Clancy and P. K. Pollett, A note on quasi-stationary distributions of birth-death processes and the SIS logistic epidemic, Journal of Applied Probability, 40 (2003), 821-825.
doi: 10.1239/jap/1059060909. |
[9] |
P. Collet, S. Martínez and V. Maume-Deschamps, On the existence of conditionally invariant probability measures in dynamical systems, Nonlinearity, 13 (2000), 1263-1274.
doi: 10.1088/0951-7715/13/4/315. |
[10] |
P. Collet, S. Martínez and B. Schmitt, The Lasota-Yorke measure and the asymptotic law in the limit Cantor set of expanding systems, Nonlinearity, 7 (1996), 1437-1443.
doi: 10.1088/0951-7715/7/5/010. |
[11] |
P. Collet, S. Martínez and B. Schmitt, On the enhancement of diffusion by chaos, escape rates and stochastic stability, Transactions of the American Mathematical Society, 351 (1999), 2875-2897.
doi: 10.1090/S0002-9947-99-02023-1. |
[12] |
M. Dellnitz, G. Froyland, C. Horenkamp, K. Padberg-Gehle and A. 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), 655-663.
doi: 10.5194/npg-16-655-2009. |
[13] |
M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - set oriented numerical methods for dynamical systems, In Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems (ed. B. Fiedler), Springer (2001) 145-174. |
[14] |
M. Dellnitz and O. Junge, Almost-invariant sets in Chua's circuit, International Journal of Bifurcation and Chaos Appl. Sci. Engrg., 7 (1997), 2475-2485.
doi: 10.1142/S0218127497001655. |
[15] |
M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior, SIAM Journal on Numerical Analysis, 36 (1999), 491-515.
doi: 10.1137/S0036142996313002. |
[16] |
M. Demers, Markov extensions for dynamical systems with holes: An application to expanding maps of the interval, Israel J. Math., 146 (2005), 189-221.
doi: 10.1007/BF02773533. |
[17] |
M. Demers and L.-S. Young, Escape rates and conditionally invariant measures, Nonlinearity, 19 (2006), 377-397.
doi: 10.1088/0951-7715/19/2/008. |
[18] |
P. Deuflhard, M. Dellnitz, O. Junge and C. Schütte, Computation of essential molecular dynamics by subdivision techniques, in Computational Molecular Dynamics: Challenges, Methods, Ideas (eds. P. Deuflhard, J. Hermans, B. Leimkuhler, A.E. Mark, S. Reich, R.D. Skeel), Springer Berlin Heidelberg (1999), 98-115.
doi: 10.1007/978-3-642-58360-5. |
[19] |
P. Deuflhard, W. Huisinga, A. Fischer and C. Schütte, Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains, Linear Algebra and its Applications, 315 (2000), 39-59.
doi: 10.1016/S0024-3795(00)00095-1. |
[20] |
J. Ding and A. Zhou, Finite element approximations of Frobenius-Perron operators - a solution to Ulam's conjecture for multi-dimensional transformations, Physica D, 92 (1996), 61-68. |
[21] |
J. Ding and A. Zhou, Statistical Properties of Deterministic Systems, Springer, Berlin, 2009.
doi: 10.1007/978-3-540-85367-1. |
[22] |
P.A. Ferrari, H. Kesten, S. Martínez and S. Picco, Existence of quasistationary distributions. a renewal dynamical approach, Annals of Probability, 23 (1995), 501-521.
doi: 10.1214/aop/1176988277. |
[23] |
G. Froyland, Finite approximation of Sinai-Bowen-Ruelle measures of Anosov systems in two dimensions, Random Comput. Dynamics, 3 (1995), 251-264. |
[24] |
G. Froyland, Ulam's method for random interval maps, Nonlinearity, 12 (1999), 1029-1052.
doi: 10.1088/0951-7715/12/4/318. |
[25] |
G. Froyland, Extracting dynamical behaviour via Markov models, in Nonlinear Dynamics and Statistics: Proceedings, Newton Institute, Cambridge, 1998, (ed. Alistair Mees), Birkhauser (2001), 283-324. |
[26] |
G. Froyland, Statistically optimal almost-invariant sets, Physica D, 200 (2005), 205-219.
doi: 10.1016/j.physd.2004.11.008. |
[27] |
G. Froyland and M. Dellnitz, Detecting and locating near-optimal almost-invariant sets and cycles, SIAM Journal Sci. Comput., 24 (2003), 1839-1863.
doi: 10.1137/S106482750238911X. |
[28] |
G. Froyland and K. Padberg, Almost-invariant sets and invariant manifolds: Connecting probabilistic and geometric descriptions of coherent structures in flows, Physica D, 238 (2009), 1507-1523.
doi: 10.1016/j.physd.2009.03.002. |
[29] |
G. Froyland, K. Padberg, M. England and A.-M. Treguier, Detection of coherent oceanic structures via transfer operators, Physics Review Letters, 98 (2007), 224503. |
[30] |
G. Froyland, N. Santitissadeekorn and A. Monahan, Transport in time-dependent dynamical systems: Finite-time coherent sets, Chaos, 20 (2010), 043116.
doi: 10.1063/1.3502450. |
[31] |
G. Froyland and O. Stancevic, Escape rates and Perron-Frobenius operators: Open and closed dynamical systems, Disc. Cont. Dynam. Sys. B, 14 (2010), 457-472.
doi: 10.3934/dcdsb.2010.14.457. |
[32] |
C. González Tokman, B.R. Hunt and P. Wright, Approximating invariant densities of metastable systems, Ergodic Theory and Dynamical Systems, 31 (2010), 1345-1361. |
[33] |
G. Keller and C. Liverani, Rare events, escape rates and quasistationarity: Some exact formulae, Journal of Statistical Physics, 135 (2009), 519-534.
doi: 10.1007/s10955-009-9747-8. |
[34] |
T.-Y. Li, Finite approximation for the Perron-Frobenius operator: a solution to Ulam's conjecture, Journal of Approximation Theory, 17 (1976), 177-186.
doi: 10.1016/0021-9045(76)90037-X. |
[35] |
C. Liverani and V. Maume-Deschamps, Lasota-Yorke maps with holes: conditionally invariant probability measures and invariant probability measures on the survivor set, Ann. Inst. H. Poinc. Probab. Statist., 39 (2003), 385-412.
doi: 10.1016/S0246-0203(02)00005-5. |
[36] |
R. Murray, Discrete Approximation of Invariant Densities, Ph.D thesis, University of Cambridge, 1997. |
[37] |
R. Murray, Ulam's method for some non-uniformly expanding maps, Discrete and continuous dynamical systems, 26 (2010),1007-1018.
doi: 10.3934/dcds.2010.26.1007. |
[38] |
G. Pianigiani and J. Yorke, Expanding maps on sets which are almost invariant: Decay and chaos, Transactions of the American Mathematical Society, 252 (1979), 351-366.
doi: 10.2307/1998093. |
[39] |
V. Rom-Kedar, A. Leonard and S. Wiggins, An analytical study of transport, mixing and chaos in an unsteady vortical flow, Journal of Fluid Mechanics, 214 (1990), 347-394.
doi: 10.1017/S0022112090000167. |
[40] |
V. Rom-Kedar and S. Wiggins, Transport in two-dimensional maps, Archive for Rational Mechanics and Analysis, 109 (1990), 239-298.
doi: 10.1007/BF00375090. |
[41] |
C. Schütte, Conformational Dynamics: Modelling, Theory, Algorithm, and Application to Biomolecules, Ph.D thesis, Freie Universität Berlin, Department of Mathematics and Computer Science, 1999. |
[42] |
S. C. Shadden, F. Lekien and J. E. Marsden, Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Physica D, 212 (2005), 217-304.
doi: 10.1016/j.physd.2005.10.007. |
[43] |
A. Sinclair and M. Jerrum, Approximate counting, uniform generation and rapidly mixing Markov chains, Information and Computation, 82 (1989), 93-133.
doi: 10.1016/0890-5401(89)90067-9. |
[44] |
M. A. Stremler, S. D. Ross, P. Grover and P. Kumar, Topological chaos and periodic braiding of almost-cyclic sets, Phys. Rev. Lett., 106 (2011), 114101.
doi: 10.1103/PhysRevLett.106.114101. |
[45] |
S. Ulam, A Collection of Mathematical Problems, Interscience, 1979. |
[46] |
P. Walters, An introduction to Ergodic Theory, Springer-Verlag, 1982. |
[47] |
S. Wiggins, Chaotic Transport in Dynamical Systems, Springer, New York, 1992. |
show all references
References:
[1] |
W. Bahsoun, Rigorous numerical approximation of escape rates, Nonlinearity, 19 (2006), 25-29.
doi: 10.1088/0951-7715/19/11/002. |
[2] |
M. S. Bartlett, Stochastic Population Models in Ecology and Epidemiology, Methuen, London, 1960. |
[3] |
L. Billings and I. B. Schwartz, Identifying almost invariant sets in stochastic dynamical systems, Chaos, 18 (2008), 023122.
doi: 10.1063/1.2929748. |
[4] |
C. Bose, G. Froyland, C. Gonzáles Tokman and R. Murray, Ulam's method for Lasota-Yorke maps with holes, To appear in SIAM J. Appl. Dynam. Syst. arXiv:1204.2329. |
[5] |
A. Boyarsky and P. Gora, Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension (Probability and its Applications), Springer, Berlin, 1997.
doi: 10.1007/978-1-4612-2024-4. |
[6] |
P. Brémaud, Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues, Springer-Verlag, 1999. |
[7] |
H. Bruin, M. Demers and I. Melbourne, Existence and convergence properties of physical measures for certain dynamical systems with holes, Ergodic Theory and Dynamical Systems, 30 (2010), 687-728.
doi: 10.1017/S0143385709000200. |
[8] |
D. Clancy and P. K. Pollett, A note on quasi-stationary distributions of birth-death processes and the SIS logistic epidemic, Journal of Applied Probability, 40 (2003), 821-825.
doi: 10.1239/jap/1059060909. |
[9] |
P. Collet, S. Martínez and V. Maume-Deschamps, On the existence of conditionally invariant probability measures in dynamical systems, Nonlinearity, 13 (2000), 1263-1274.
doi: 10.1088/0951-7715/13/4/315. |
[10] |
P. Collet, S. Martínez and B. Schmitt, The Lasota-Yorke measure and the asymptotic law in the limit Cantor set of expanding systems, Nonlinearity, 7 (1996), 1437-1443.
doi: 10.1088/0951-7715/7/5/010. |
[11] |
P. Collet, S. Martínez and B. Schmitt, On the enhancement of diffusion by chaos, escape rates and stochastic stability, Transactions of the American Mathematical Society, 351 (1999), 2875-2897.
doi: 10.1090/S0002-9947-99-02023-1. |
[12] |
M. Dellnitz, G. Froyland, C. Horenkamp, K. Padberg-Gehle and A. 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), 655-663.
doi: 10.5194/npg-16-655-2009. |
[13] |
M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - set oriented numerical methods for dynamical systems, In Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems (ed. B. Fiedler), Springer (2001) 145-174. |
[14] |
M. Dellnitz and O. Junge, Almost-invariant sets in Chua's circuit, International Journal of Bifurcation and Chaos Appl. Sci. Engrg., 7 (1997), 2475-2485.
doi: 10.1142/S0218127497001655. |
[15] |
M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior, SIAM Journal on Numerical Analysis, 36 (1999), 491-515.
doi: 10.1137/S0036142996313002. |
[16] |
M. Demers, Markov extensions for dynamical systems with holes: An application to expanding maps of the interval, Israel J. Math., 146 (2005), 189-221.
doi: 10.1007/BF02773533. |
[17] |
M. Demers and L.-S. Young, Escape rates and conditionally invariant measures, Nonlinearity, 19 (2006), 377-397.
doi: 10.1088/0951-7715/19/2/008. |
[18] |
P. Deuflhard, M. Dellnitz, O. Junge and C. Schütte, Computation of essential molecular dynamics by subdivision techniques, in Computational Molecular Dynamics: Challenges, Methods, Ideas (eds. P. Deuflhard, J. Hermans, B. Leimkuhler, A.E. Mark, S. Reich, R.D. Skeel), Springer Berlin Heidelberg (1999), 98-115.
doi: 10.1007/978-3-642-58360-5. |
[19] |
P. Deuflhard, W. Huisinga, A. Fischer and C. Schütte, Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains, Linear Algebra and its Applications, 315 (2000), 39-59.
doi: 10.1016/S0024-3795(00)00095-1. |
[20] |
J. Ding and A. Zhou, Finite element approximations of Frobenius-Perron operators - a solution to Ulam's conjecture for multi-dimensional transformations, Physica D, 92 (1996), 61-68. |
[21] |
J. Ding and A. Zhou, Statistical Properties of Deterministic Systems, Springer, Berlin, 2009.
doi: 10.1007/978-3-540-85367-1. |
[22] |
P.A. Ferrari, H. Kesten, S. Martínez and S. Picco, Existence of quasistationary distributions. a renewal dynamical approach, Annals of Probability, 23 (1995), 501-521.
doi: 10.1214/aop/1176988277. |
[23] |
G. Froyland, Finite approximation of Sinai-Bowen-Ruelle measures of Anosov systems in two dimensions, Random Comput. Dynamics, 3 (1995), 251-264. |
[24] |
G. Froyland, Ulam's method for random interval maps, Nonlinearity, 12 (1999), 1029-1052.
doi: 10.1088/0951-7715/12/4/318. |
[25] |
G. Froyland, Extracting dynamical behaviour via Markov models, in Nonlinear Dynamics and Statistics: Proceedings, Newton Institute, Cambridge, 1998, (ed. Alistair Mees), Birkhauser (2001), 283-324. |
[26] |
G. Froyland, Statistically optimal almost-invariant sets, Physica D, 200 (2005), 205-219.
doi: 10.1016/j.physd.2004.11.008. |
[27] |
G. Froyland and M. Dellnitz, Detecting and locating near-optimal almost-invariant sets and cycles, SIAM Journal Sci. Comput., 24 (2003), 1839-1863.
doi: 10.1137/S106482750238911X. |
[28] |
G. Froyland and K. Padberg, Almost-invariant sets and invariant manifolds: Connecting probabilistic and geometric descriptions of coherent structures in flows, Physica D, 238 (2009), 1507-1523.
doi: 10.1016/j.physd.2009.03.002. |
[29] |
G. Froyland, K. Padberg, M. England and A.-M. Treguier, Detection of coherent oceanic structures via transfer operators, Physics Review Letters, 98 (2007), 224503. |
[30] |
G. Froyland, N. Santitissadeekorn and A. Monahan, Transport in time-dependent dynamical systems: Finite-time coherent sets, Chaos, 20 (2010), 043116.
doi: 10.1063/1.3502450. |
[31] |
G. Froyland and O. Stancevic, Escape rates and Perron-Frobenius operators: Open and closed dynamical systems, Disc. Cont. Dynam. Sys. B, 14 (2010), 457-472.
doi: 10.3934/dcdsb.2010.14.457. |
[32] |
C. González Tokman, B.R. Hunt and P. Wright, Approximating invariant densities of metastable systems, Ergodic Theory and Dynamical Systems, 31 (2010), 1345-1361. |
[33] |
G. Keller and C. Liverani, Rare events, escape rates and quasistationarity: Some exact formulae, Journal of Statistical Physics, 135 (2009), 519-534.
doi: 10.1007/s10955-009-9747-8. |
[34] |
T.-Y. Li, Finite approximation for the Perron-Frobenius operator: a solution to Ulam's conjecture, Journal of Approximation Theory, 17 (1976), 177-186.
doi: 10.1016/0021-9045(76)90037-X. |
[35] |
C. Liverani and V. Maume-Deschamps, Lasota-Yorke maps with holes: conditionally invariant probability measures and invariant probability measures on the survivor set, Ann. Inst. H. Poinc. Probab. Statist., 39 (2003), 385-412.
doi: 10.1016/S0246-0203(02)00005-5. |
[36] |
R. Murray, Discrete Approximation of Invariant Densities, Ph.D thesis, University of Cambridge, 1997. |
[37] |
R. Murray, Ulam's method for some non-uniformly expanding maps, Discrete and continuous dynamical systems, 26 (2010),1007-1018.
doi: 10.3934/dcds.2010.26.1007. |
[38] |
G. Pianigiani and J. Yorke, Expanding maps on sets which are almost invariant: Decay and chaos, Transactions of the American Mathematical Society, 252 (1979), 351-366.
doi: 10.2307/1998093. |
[39] |
V. Rom-Kedar, A. Leonard and S. Wiggins, An analytical study of transport, mixing and chaos in an unsteady vortical flow, Journal of Fluid Mechanics, 214 (1990), 347-394.
doi: 10.1017/S0022112090000167. |
[40] |
V. Rom-Kedar and S. Wiggins, Transport in two-dimensional maps, Archive for Rational Mechanics and Analysis, 109 (1990), 239-298.
doi: 10.1007/BF00375090. |
[41] |
C. Schütte, Conformational Dynamics: Modelling, Theory, Algorithm, and Application to Biomolecules, Ph.D thesis, Freie Universität Berlin, Department of Mathematics and Computer Science, 1999. |
[42] |
S. C. Shadden, F. Lekien and J. E. Marsden, Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Physica D, 212 (2005), 217-304.
doi: 10.1016/j.physd.2005.10.007. |
[43] |
A. Sinclair and M. Jerrum, Approximate counting, uniform generation and rapidly mixing Markov chains, Information and Computation, 82 (1989), 93-133.
doi: 10.1016/0890-5401(89)90067-9. |
[44] |
M. A. Stremler, S. D. Ross, P. Grover and P. Kumar, Topological chaos and periodic braiding of almost-cyclic sets, Phys. Rev. Lett., 106 (2011), 114101.
doi: 10.1103/PhysRevLett.106.114101. |
[45] |
S. Ulam, A Collection of Mathematical Problems, Interscience, 1979. |
[46] |
P. Walters, An introduction to Ergodic Theory, Springer-Verlag, 1982. |
[47] |
S. Wiggins, Chaotic Transport in Dynamical Systems, Springer, New York, 1992. |
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