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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), 1001-1048.
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, 70. Springer, New York, 2014.
doi: 10.1007/978-1-4939-0419-8. |
[3] |
W. Bahsoun, Rigorous numerical approximation of escape rates, Nonlinearity, 19 (2006), 2529-2542.
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), 4481-4495.
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), 525-538.
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), 443-463.
doi: 10.1007/s002200100427. |
[7] |
C. Bose, G. Froyland, C. Gonzales-Tokman and R. Murray, Ulam's Method for Lasota Yorke maps with holes, arXiv:1204.2329v2 |
[8] |
M. D. Boshernitzan, Quantitative recurrence results, Inv. Math., 113 (1993), 617-631.
doi: 10.1007/BF01244320. |
[9] |
M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems, Handbook of dynamical systems, Elsevier, 2 (2002), 221-264.
doi: 10.1016/S1874-575X(02)80026-1. |
[10] |
G. Froyland, Extracting dynamical behaviour via Markov models, in Alistair Mees, editor, Nonlinear Dynamics and Statistics: Proceedings, Newton Institute, (Cambridge 1998), 281-321, Birkhauser, 2001. |
[11] |
G. Froyland, Computer-assisted bounds for the rate of decay of correlations, Comm. Math. Phys., 189 (1997), 237-257.
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), 958-985.
doi: 10.1137/130911044. |
[13] |
S. Galatolo, Dimension and hitting time in rapidly mixing systems, Math. Res. Lett., 14 (2007), 797-805.
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, arXiv:1402.5918 (to appear on Erg. Th. Dyn. Sys.). |
[15] |
B. Hunt, Estimating invariant measures and Lyapunov exponents, Erg. Th. Dyn. Sys., 16 (1996), 735-749.
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), 141-152. |
[17] |
O. Ippei, Computer-assisted verification method for invariant densities and rates of decay of correlations, SIAM J. Applied Dynamical Systems, 10 (2011), 788-816.
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), 317-324.
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), 481-488.
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), 463-490.
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 "Ennio De Giorgi'': Proceedings. Published by the Scuola Normale Superiore in Pisa, 2004. |
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), 1001-1048.
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, 70. Springer, New York, 2014.
doi: 10.1007/978-1-4939-0419-8. |
[3] |
W. Bahsoun, Rigorous numerical approximation of escape rates, Nonlinearity, 19 (2006), 2529-2542.
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), 4481-4495.
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), 525-538.
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), 443-463.
doi: 10.1007/s002200100427. |
[7] |
C. Bose, G. Froyland, C. Gonzales-Tokman and R. Murray, Ulam's Method for Lasota Yorke maps with holes, arXiv:1204.2329v2 |
[8] |
M. D. Boshernitzan, Quantitative recurrence results, Inv. Math., 113 (1993), 617-631.
doi: 10.1007/BF01244320. |
[9] |
M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems, Handbook of dynamical systems, Elsevier, 2 (2002), 221-264.
doi: 10.1016/S1874-575X(02)80026-1. |
[10] |
G. Froyland, Extracting dynamical behaviour via Markov models, in Alistair Mees, editor, Nonlinear Dynamics and Statistics: Proceedings, Newton Institute, (Cambridge 1998), 281-321, Birkhauser, 2001. |
[11] |
G. Froyland, Computer-assisted bounds for the rate of decay of correlations, Comm. Math. Phys., 189 (1997), 237-257.
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), 958-985.
doi: 10.1137/130911044. |
[13] |
S. Galatolo, Dimension and hitting time in rapidly mixing systems, Math. Res. Lett., 14 (2007), 797-805.
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, arXiv:1402.5918 (to appear on Erg. Th. Dyn. Sys.). |
[15] |
B. Hunt, Estimating invariant measures and Lyapunov exponents, Erg. Th. Dyn. Sys., 16 (1996), 735-749.
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), 141-152. |
[17] |
O. Ippei, Computer-assisted verification method for invariant densities and rates of decay of correlations, SIAM J. Applied Dynamical Systems, 10 (2011), 788-816.
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), 317-324.
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), 481-488.
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), 463-490.
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 "Ennio De Giorgi'': Proceedings. Published by the Scuola Normale Superiore in Pisa, 2004. |
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