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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

Gary Froyland 1, , Philip K. Pollett 2, and  Robyn M. Stuart 3,

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.
Keywords: almostinvariant set, metastability, conditionally invariant measure, Open dynamical system, Perron-Frobenius operator, Ulam's method..
Mathematics Subject Classification: Primary: 37M25; Secondary: 37A3.
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.  Google Scholar

[2]

M. S. Bartlett, Stochastic Population Models in Ecology and Epidemiology, Methuen, London, 1960.  Google Scholar

[3]

L. Billings and I. B. Schwartz, Identifying almost invariant sets in stochastic dynamical systems, Chaos, 18 (2008), 023122. doi: 10.1063/1.2929748.  Google Scholar

[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. , ().   Google Scholar

[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.  Google Scholar

[6]

P. Brémaud, Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues, Springer-Verlag, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[21]

J. Ding and A. Zhou, Statistical Properties of Deterministic Systems, Springer, Berlin, 2009. doi: 10.1007/978-3-540-85367-1.  Google Scholar

[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.  Google Scholar

[23]

G. Froyland, Finite approximation of Sinai-Bowen-Ruelle measures of Anosov systems in two dimensions, Random Comput. Dynamics, 3 (1995), 251-264.  Google Scholar

[24]

G. Froyland, Ulam's method for random interval maps, Nonlinearity, 12 (1999), 1029-1052. doi: 10.1088/0951-7715/12/4/318.  Google Scholar

[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.  Google Scholar

[26]

G. Froyland, Statistically optimal almost-invariant sets, Physica D, 200 (2005), 205-219. doi: 10.1016/j.physd.2004.11.008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

R. Murray, Discrete Approximation of Invariant Densities, Ph.D thesis, University of Cambridge, 1997. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

S. Ulam, A Collection of Mathematical Problems, Interscience, 1979.  Google Scholar

[46]

P. Walters, An introduction to Ergodic Theory, Springer-Verlag, 1982.  Google Scholar

[47]

S. Wiggins, Chaotic Transport in Dynamical Systems, Springer, New York, 1992.  Google Scholar

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.  Google Scholar

[2]

M. S. Bartlett, Stochastic Population Models in Ecology and Epidemiology, Methuen, London, 1960.  Google Scholar

[3]

L. Billings and I. B. Schwartz, Identifying almost invariant sets in stochastic dynamical systems, Chaos, 18 (2008), 023122. doi: 10.1063/1.2929748.  Google Scholar

[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. , ().   Google Scholar

[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.  Google Scholar

[6]

P. Brémaud, Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues, Springer-Verlag, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[21]

J. Ding and A. Zhou, Statistical Properties of Deterministic Systems, Springer, Berlin, 2009. doi: 10.1007/978-3-540-85367-1.  Google Scholar

[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.  Google Scholar

[23]

G. Froyland, Finite approximation of Sinai-Bowen-Ruelle measures of Anosov systems in two dimensions, Random Comput. Dynamics, 3 (1995), 251-264.  Google Scholar

[24]

G. Froyland, Ulam's method for random interval maps, Nonlinearity, 12 (1999), 1029-1052. doi: 10.1088/0951-7715/12/4/318.  Google Scholar

[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.  Google Scholar

[26]

G. Froyland, Statistically optimal almost-invariant sets, Physica D, 200 (2005), 205-219. doi: 10.1016/j.physd.2004.11.008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

R. Murray, Discrete Approximation of Invariant Densities, Ph.D thesis, University of Cambridge, 1997. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

S. Ulam, A Collection of Mathematical Problems, Interscience, 1979.  Google Scholar

[46]

P. Walters, An introduction to Ergodic Theory, Springer-Verlag, 1982.  Google Scholar

[47]

S. Wiggins, Chaotic Transport in Dynamical Systems, Springer, New York, 1992.  Google Scholar

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