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A closing scheme for finding almost-invariant sets in open dynamical systems

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  • 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.
    Mathematics Subject Classification: Primary: 37M25; Secondary: 37A30.

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  • [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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