American Institute of Mathematical Sciences

Advanced
January  2015, 2(1): 51-64. doi: 10.3934/jcd.2015.2.51

An elementary way to rigorously estimate convergence to equilibrium and escape rates

Stefano Galatolo 1, , Isaia Nisoli 2, and  Benoît Saussol 3,

1. 

Dipartimento di Matematica Applicata, Università di Pisa, Via Bonanno Pisano
2. 

Instituto de Matemática, UFRJ Av. Athos da Silveira Ramos 149, Centro de Tecnologia, Bloco C Cidade Universitária, Ilha do Fundão, Caixa Postal 68530 21941-909 Rio de Janeiro, RJ, Brazil
3. 

Laboratoire de Mathématiques, CNRS UMR 6205, Université de Bretagne Occidentale, 6 av. Victor Le Gorgeu, CS 93837, 29238 BREST Cedex 3

Received  April 2014 Revised  January 2015 Published  August 2015

We show an elementary method to obtain (finite time and asymptotic) computer assisted explicit upper bounds on convergence to equilibrium (decay of correlations) and escape rates for systems satisfying a Lasota Yorke inequality. The bounds are deduced from the ones of suitable approximations of the system's transfer operator. We also present some rigorous experiments on some nontrivial example.
Keywords: interval arithmetics, Ulam method., Rigorous computation, decay of correlation, convergence to equilibrium.
Mathematics Subject Classification: Primary: 37M25, 65G3.
Citation: Stefano Galatolo, Isaia Nisoli, Benoît Saussol. An elementary way to rigorously estimate convergence to equilibrium and escape rates. Journal of Computational Dynamics, 2015, 2 (1) : 51-64. doi: 10.3934/jcd.2015.2.51
References:
[1]

V. Araujo, S. Galatolo and M. J. Pacifico, Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors,, Mathematiche Zeitcrift, 276 (2014), 1001. doi: 10.1007/s00209-013-1231-0. Google Scholar

[2]

W. Bahsoun, C. Bose and G. Froyland, (Eds.), Ergodic Theory, Open Dynamics, and Coherent Structures,, Springer Proceedings in Mathematics & Statistics, (2014). doi: 10.1007/978-1-4939-0419-8. Google Scholar

[3]

W. Bahsoun, Rigorous numerical approximation of escape rates,, Nonlinearity, 19 (2006), 2529. doi: 10.1088/0951-7715/19/11/002. Google Scholar

[4]

W. Bahsoun and C. Bose, Invariant densities and escape rates: Rigorous and computable approximations in the $L^{\infty }$,, Nonlinear Analysis, 74 (2011), 4481. doi: 10.1016/j.na.2011.04.012. Google Scholar

[5]

V. Baladi and M. Holschneider, Approximation of nonessential spectrum of transfer operators,, Nonlinearity Nonlinearity, 12 (1999), 525. doi: 10.1088/0951-7715/12/3/006. Google Scholar

[6]

L. Barreira and B. Saussol, Hausdorff dimension of measures via Poincaré recurrence,, Comm. Math. Phys., 219 (2001), 443. doi: 10.1007/s002200100427. Google Scholar

[7]

C. Bose, G. Froyland, C. Gonzales-Tokman and R. Murray, Ulam's Method for Lasota Yorke maps with holes,, , (). Google Scholar

[8]

M. D. Boshernitzan, Quantitative recurrence results,, Inv. Math., 113 (1993), 617. doi: 10.1007/BF01244320. Google Scholar

[9]

M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems,, Handbook of dynamical systems, 2 (2002), 221. doi: 10.1016/S1874-575X(02)80026-1. Google Scholar

[10]

G. Froyland, Extracting dynamical behaviour via Markov models,, in Alistair Mees, (1998), 281. Google Scholar

[11]

G. Froyland, Computer-assisted bounds for the rate of decay of correlations,, Comm. Math. Phys., 189 (1997), 237. doi: 10.1007/s002200050198. Google Scholar

[12]

S. Galatolo and I. Nisoli, An elementary approach to rigorous approximation of invariant measures,, SIAM J. Appl Dyn Sys., 13 (2014), 958. doi: 10.1137/130911044. Google Scholar

[13]

S. Galatolo, Dimension and hitting time in rapidly mixing systems,, Math. Res. Lett., 14 (2007), 797. doi: 10.4310/MRL.2007.v14.n5.a8. Google Scholar

[14]

S. Galatolo and I. Nisoli, Rigorous computation of invariant measures and fractal dimension for piecewise hyperbolic maps: 2D Lorenz like maps,, , (). Google Scholar

[15]

B. Hunt, Estimating invariant measures and Lyapunov exponents,, Erg. Th. Dyn. Sys., 16 (1996), 735. doi: 10.1017/S014338570000907X. Google Scholar

[16]

G. Keller and C. Liverani, Stability of the spectrum for transfer operators,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141. Google Scholar

[17]

O. Ippei, Computer-assisted verification method for invariant densities and rates of decay of correlations,, SIAM J. Applied Dynamical Systems, 10 (2011), 788. doi: 10.1137/09077864X. Google Scholar

[18]

O. E. Lanford III, Informal remarks on the orbit structure of discrete approximations to chaotic maps,, Exp. Math., 7 (1998), 317. doi: 10.1080/10586458.1998.10504377. Google Scholar

[19]

A. Lasota and J. Yorke, On the existence of invariant measures for piecewise monotonic transformations,, Trans. Amer. Math. Soc., 186 (1973), 481. doi: 10.1090/S0002-9947-1973-0335758-1. Google Scholar

[20]

C. Liverani, Rigorous numerical investigations of the statistical properties of piecewise expanding maps-A feasibility study,, Nonlinearity, 14 (2001), 463. doi: 10.1088/0951-7715/14/3/303. Google Scholar

[21]

C. Liverani, Invariant Measures and Their Properties. A Functional Analytic Point of View,, Dynamical Systems. Part II: Topological Geometrical and Ergodic Properties of Dynamics. Centro di Ricerca Matematica, (2004). Google Scholar

show all references

References:
[1]

V. Araujo, S. Galatolo and M. J. Pacifico, Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors,, Mathematiche Zeitcrift, 276 (2014), 1001. doi: 10.1007/s00209-013-1231-0. Google Scholar

[2]

W. Bahsoun, C. Bose and G. Froyland, (Eds.), Ergodic Theory, Open Dynamics, and Coherent Structures,, Springer Proceedings in Mathematics & Statistics, (2014). doi: 10.1007/978-1-4939-0419-8. Google Scholar

[3]

W. Bahsoun, Rigorous numerical approximation of escape rates,, Nonlinearity, 19 (2006), 2529. doi: 10.1088/0951-7715/19/11/002. Google Scholar

[4]

W. Bahsoun and C. Bose, Invariant densities and escape rates: Rigorous and computable approximations in the $L^{\infty }$,, Nonlinear Analysis, 74 (2011), 4481. doi: 10.1016/j.na.2011.04.012. Google Scholar

[5]

V. Baladi and M. Holschneider, Approximation of nonessential spectrum of transfer operators,, Nonlinearity Nonlinearity, 12 (1999), 525. doi: 10.1088/0951-7715/12/3/006. Google Scholar

[6]

L. Barreira and B. Saussol, Hausdorff dimension of measures via Poincaré recurrence,, Comm. Math. Phys., 219 (2001), 443. doi: 10.1007/s002200100427. Google Scholar

[7]

C. Bose, G. Froyland, C. Gonzales-Tokman and R. Murray, Ulam's Method for Lasota Yorke maps with holes,, , (). Google Scholar

[8]

M. D. Boshernitzan, Quantitative recurrence results,, Inv. Math., 113 (1993), 617. doi: 10.1007/BF01244320. Google Scholar

[9]

M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems,, Handbook of dynamical systems, 2 (2002), 221. doi: 10.1016/S1874-575X(02)80026-1. Google Scholar

[10]

G. Froyland, Extracting dynamical behaviour via Markov models,, in Alistair Mees, (1998), 281. Google Scholar

[11]

G. Froyland, Computer-assisted bounds for the rate of decay of correlations,, Comm. Math. Phys., 189 (1997), 237. doi: 10.1007/s002200050198. Google Scholar

[12]

S. Galatolo and I. Nisoli, An elementary approach to rigorous approximation of invariant measures,, SIAM J. Appl Dyn Sys., 13 (2014), 958. doi: 10.1137/130911044. Google Scholar

[13]

S. Galatolo, Dimension and hitting time in rapidly mixing systems,, Math. Res. Lett., 14 (2007), 797. doi: 10.4310/MRL.2007.v14.n5.a8. Google Scholar

[14]

S. Galatolo and I. Nisoli, Rigorous computation of invariant measures and fractal dimension for piecewise hyperbolic maps: 2D Lorenz like maps,, , (). Google Scholar

[15]

B. Hunt, Estimating invariant measures and Lyapunov exponents,, Erg. Th. Dyn. Sys., 16 (1996), 735. doi: 10.1017/S014338570000907X. Google Scholar

[16]

G. Keller and C. Liverani, Stability of the spectrum for transfer operators,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141. Google Scholar

[17]

O. Ippei, Computer-assisted verification method for invariant densities and rates of decay of correlations,, SIAM J. Applied Dynamical Systems, 10 (2011), 788. doi: 10.1137/09077864X. Google Scholar

[18]

O. E. Lanford III, Informal remarks on the orbit structure of discrete approximations to chaotic maps,, Exp. Math., 7 (1998), 317. doi: 10.1080/10586458.1998.10504377. Google Scholar

[19]

A. Lasota and J. Yorke, On the existence of invariant measures for piecewise monotonic transformations,, Trans. Amer. Math. Soc., 186 (1973), 481. doi: 10.1090/S0002-9947-1973-0335758-1. Google Scholar

[20]

C. Liverani, Rigorous numerical investigations of the statistical properties of piecewise expanding maps-A feasibility study,, Nonlinearity, 14 (2001), 463. doi: 10.1088/0951-7715/14/3/303. Google Scholar

[21]

C. Liverani, Invariant Measures and Their Properties. A Functional Analytic Point of View,, Dynamical Systems. Part II: Topological Geometrical and Ergodic Properties of Dynamics. Centro di Ricerca Matematica, (2004). Google Scholar

[1]

Anna Belova. Rigorous enclosures of rotation numbers by interval methods. Journal of Computational Dynamics, 2016, 3 (1) : 81-91. doi: 10.3934/jcd.2016004
[2]

Hjörtur Björnsson, Sigurdur Hafstein, Peter Giesl, Enrico Scalas, Skuli Gudmundsson. Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4247-4269. doi: 10.3934/dcdsb.2019080
[3]

Christopher Bose, Rua Murray. The exact rate of approximation in Ulam's method. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 219-235. doi: 10.3934/dcds.2001.7.219
[4]

Xin-Guo Liu, Kun Wang. A multigrid method for the maximal correlation problem. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 785-796. doi: 10.3934/naco.2012.2.785
[5]

H. Merdan, G. Caginalp. Decay of solutions to nonlinear parabolic equations: renormalization and rigorous results. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 565-588. doi: 10.3934/dcdsb.2003.3.565
[6]

Rua Murray. Ulam's method for some non-uniformly expanding maps. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 1007-1018. doi: 10.3934/dcds.2010.26.1007
[7]

Paweł Góra, Abraham Boyarsky. Stochastic perturbations and Ulam's method for W-shaped maps. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1937-1944. doi: 10.3934/dcds.2013.33.1937
[8]

Eric Cancès, Claude Le Bris. Convergence to equilibrium of a multiscale model for suspensions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 449-470. doi: 10.3934/dcdsb.2006.6.449
[9]

M. Bauer, A. Lopes. A billiard in the hyperbolic plane with decay of correlation of type $n^{-2}$. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 107-116. doi: 10.3934/dcds.1997.3.107
[10]

Lukas Neumann, Christian Schmeiser. A kinetic reaction model: Decay to equilibrium and macroscopic limit. Kinetic & Related Models, 2016, 9 (3) : 571-585. doi: 10.3934/krm.2016007
[11]

Frederike Kissling, Christian Rohde. The computation of nonclassical shock waves with a heterogeneous multiscale method. Networks & Heterogeneous Media, 2010, 5 (3) : 661-674. doi: 10.3934/nhm.2010.5.661
[12]

Eric A. Carlen, Süleyman Ulusoy. Localization, smoothness, and convergence to equilibrium for a thin film equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4537-4553. doi: 10.3934/dcds.2014.34.4537
[13]

Benoît Merlet, Morgan Pierre. Convergence to equilibrium for the backward Euler scheme and applications. Communications on Pure & Applied Analysis, 2010, 9 (3) : 685-702. doi: 10.3934/cpaa.2010.9.685
[14]

Rafael Granero-Belinchón, Martina Magliocca. Global existence and decay to equilibrium for some crystal surface models. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2101-2131. doi: 10.3934/dcds.2019088
[15]

Jian Hou, Liwei Zhang. A barrier function method for generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1091-1108. doi: 10.3934/jimo.2014.10.1091
[16]

Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A penalty method for generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2012, 8 (1) : 51-65. doi: 10.3934/jimo.2012.8.51
[17]

R. Baier, M. Dellnitz, M. Hessel-von Molo, S. Sertl, I. G. Kevrekidis. The computation of convex invariant sets via Newton's method. Journal of Computational Dynamics, 2014, 1 (1) : 39-69. doi: 10.3934/jcd.2014.1.39
[18]

Leonardi Filippo. A projection method for the computation of admissible measure valued solutions of the incompressible Euler equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 941-961. doi: 10.3934/dcdss.2018056
[19]

Thomas Leroy. Relativistic transfer equations: Comparison principle and convergence to the non-equilibrium regime. Kinetic & Related Models, 2015, 8 (4) : 725-763. doi: 10.3934/krm.2015.8.725
[20]

Miguel Escobedo, Minh-Binh Tran. Convergence to equilibrium of a linearized quantum Boltzmann equation for bosons at very low temperature. Kinetic & Related Models, 2015, 8 (3) : 493-531. doi: 10.3934/krm.2015.8.493

 Impact Factor: 

Tools

Metrics

  • PDF downloads (19)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences