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Symmetry exploiting control of hybrid mechanical systems
An elementary way to rigorously estimate convergence to equilibrium and escape rates
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 |
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. |
[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. |
[3] |
W. Bahsoun, Rigorous numerical approximation of escape rates,, Nonlinearity, 19 (2006), 2529.
doi: 10.1088/0951-7715/19/11/002. |
[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. |
[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. |
[6] |
L. Barreira and B. Saussol, Hausdorff dimension of measures via Poincaré recurrence,, Comm. Math. Phys., 219 (2001), 443.
doi: 10.1007/s002200100427. |
[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. |
[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. |
[10] |
G. Froyland, Extracting dynamical behaviour via Markov models,, in Alistair Mees, (1998), 281.
|
[11] |
G. Froyland, Computer-assisted bounds for the rate of decay of correlations,, Comm. Math. Phys., 189 (1997), 237.
doi: 10.1007/s002200050198. |
[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. |
[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. |
[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. |
[16] |
G. Keller and C. Liverani, Stability of the spectrum for transfer operators,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[3] |
W. Bahsoun, Rigorous numerical approximation of escape rates,, Nonlinearity, 19 (2006), 2529.
doi: 10.1088/0951-7715/19/11/002. |
[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. |
[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. |
[6] |
L. Barreira and B. Saussol, Hausdorff dimension of measures via Poincaré recurrence,, Comm. Math. Phys., 219 (2001), 443.
doi: 10.1007/s002200100427. |
[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. |
[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. |
[10] |
G. Froyland, Extracting dynamical behaviour via Markov models,, in Alistair Mees, (1998), 281.
|
[11] |
G. Froyland, Computer-assisted bounds for the rate of decay of correlations,, Comm. Math. Phys., 189 (1997), 237.
doi: 10.1007/s002200050198. |
[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. |
[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. |
[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. |
[16] |
G. Keller and C. Liverani, Stability of the spectrum for transfer operators,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141.
|
[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. |
[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. |
[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. |
[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. |
[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 |
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