# American Institute of Mathematical Sciences

January  2014, 1(1): 135-162. doi: 10.3934/jcd.2014.1.135

## 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

Received  September 2011 Revised  June 2012 Published  April 2014

We explore the concept of metastability or almost-invariance in open dynamical systems. In such systems, the loss of mass through a hole'' occurs in the presence of metastability. We extend existing techniques for finding almost-invariant sets in closed systems to open systems by introducing a closing operation that has a small impact on the system's metastability.
Citation: Gary Froyland, Philip K. Pollett, Robyn M. Stuart. A closing scheme for finding almost-invariant sets in open dynamical systems. Journal of Computational Dynamics, 2014, 1 (1) : 135-162. doi: 10.3934/jcd.2014.1.135
##### 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. , (). [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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##### 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. , (). [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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