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Homogeneous darboux polynomials and generalising integrable ODE systems
A self-consistent dynamical system with multiple absolutely continuous invariant measures
Laboratoire de Probabilités, Statistique et Modélisation (LPSM), Sorbonne Université, Université de Paris, 4 Place Jussieu, 75005 Paris, France |
In this paper we study a class of self-consistent dynamical systems, self-consistent in the sense that the discrete time dynamics is different in each step depending on current statistics. The general framework admits popular examples such as coupled map systems. Motivated by an example of [
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer Verlag, Berlin, 1998.
doi: 10.1007/978-3-662-12878-7. |
[2] |
V. Baladi and L-S. Young,
On the spectra of randomly perturbed expanding maps, Commun. Math. Phys., 2 (1993), 355-385.
doi: 10.1007/BF02098487. |
[3] |
V. Baladi, A. Kondah and B. Schmitt,
Random correlations for small perturbations of expanding maps, Random Comput. Dyn., 2-3 (1996), 179-204.
|
[4] |
P. Bálint, G. Keller, F. Sélley and I. P. Tóth, Synchronization versus stability of the invariant distribution for a class of globally coupled maps, Nonlinearity, 8 (2018), 3770.
doi: 10.1088/1361-6544/aac5b0. |
[5] |
J.-B. Bardet, G. Keller and R. Zweimüller,
Stochastically stable globally coupled maps with bistable thermodynamic limit, Commun. Math. Phys., 1 (2009), 237-270.
doi: 10.1007/s00220-009-0854-9. |
[6] |
M. Blank and G. Keller, Random perturbations of chaotic dynamical systems: Stability of the spectrum, Nonlinearity, 5 (1998), 1351.
doi: 10.1088/0951-7715/11/5/010. |
[7] |
M. Blank,
Collective phenomena in lattices of weakly interacting maps, Dokl. Akad. Nauk., 3 (2010), 300-304.
doi: 10.1134/S1064562410010126. |
[8] |
M. Blank,
Self-consistent mappings and systems of interacting particles, Dokl. Math., 1 (2011), 49-52.
doi: 10.1134/S1064562411010133. |
[9] |
M. Blank, Ergodic averaging with and without invariant measures, Nonlinearity, 8 (2017), 4649.
doi: 10.1088/1361-6544/aa8fe8. |
[10] |
T. Bogenschütz,
Stochastic stability of invariant subspaces, Ergod. Theor. Dyn. Syst., 3 (2000), 663-680.
doi: 10.1017/S0143385700000353. |
[11] |
J. Buzzi,
Absolutely continuous SRB measures for random Lasota–Yorke maps, T. Am. Math. Soc., 7 (2000), 3289-3303.
doi: 10.1090/S0002-9947-00-02607-6. |
[12] |
A. Boyarsky, P. Góra and C. Keefe,
Absolutely continuous invariant measures for non-autonomous dynamical systems, J. Math. Anal. Appl., 1 (2019), 159-168.
doi: 10.1016/j.jmaa.2018.09.060. |
[13] |
J-R. Chazottes and B. Fernandez, Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, Springer Science & Business Media, 2005. |
[14] |
J. Ding, Q. Du and T-Y. Li,
High order approximation of the Frobenius–Perron operator, Appl. Math. Comput., 2-3 (1993), 151-171.
doi: 10.1016/0096-3003(93)90099-Z. |
[15] |
J. Ding and A. Zhou,
Finite approximations of Frobenius–Perron operators. A solution of {U}lam's conjecture to multi-dimensional transformations, Physica D, 1-2 (1996), 61-68.
doi: 10.1016/0167-2789(95)00292-8. |
[16] |
G. Froyland,
Finite approximation of Sinai-Bowen-Ruelle measures for Anosov systems in two dimensions, Random Comput. Dyn., 4 (1995), 251-264.
|
[17] |
G. Froyland, Ulam's method for random interval maps, Nonlinearity, 4 (1999), 1029.
doi: 10.1088/0951-7715/12/4/318. |
[18] |
G. Froyland, C. González-Tokman and A. Quas, Stability and approximation of random invariant densities for Lasota–Yorke map cocycles, Nonlinearity, 4 (2014), 647.
doi: 10.1088/0951-7715/27/4/647. |
[19] |
G. Froyland, C. González-Tokman and R. D. A. Murray,
Quenched stochastic stability for eventually expanding-on-average random interval map cocycles, Ergod. Theor. Dyn. Syst., 10 (2019), 2769-2792.
doi: 10.1017/etds.2017.143. |
[20] |
M. Jiang and Y. B. Pesin,
Equilibrium measures for coupled map lattices: Existence, uniqueness and finite-dimensional approximations, Commun. Math. Phys., 3 (1998), 675-711.
doi: 10.1007/s002200050344. |
[21] |
K. Kaneko, Globally coupled chaos violates the law of large numbers but not the central limit theorem, Phys. Rev. Lett., 12 (1990), 1391.
doi: 10.1103/PhysRevLett.65.1391. |
[22] |
C. Liverani and G. Keller, A spectral gap for a one-dimensional lattice of coupled piecewise expanding interval maps, in Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, Springer, 2005,115–151.
doi: 10.1007/11360810_6. |
[23] |
G. Keller, R. Klages and P. J. Howard, Continuity properties of transport coefficients in simple maps, Nonlinearity, 8 (2008), 1719.
doi: 10.1088/0951-7715/21/8/003. |
[24] |
G. Keller, An ergodic theoretic approach to mean field coupled maps, Prog. Probab., (2000), 183–208. |
[25] |
S. Klus, P. Koltai and C. Schütte, On the numerical approximation of the Perron–Frobenius and Koopman operator, J. Comput. Dyn., 1 (2016), 51.
doi: 10.3934/jcd.2016003. |
[26] |
T-Y. Li,
Finite approximation for the Frobenius–Perron operator. A solution to Ulam's conjecture, J. Approx Theory, 2 (1976), 177-186.
doi: 10.1016/0021-9045(76)90037-X. |
[27] |
R. D. A. Murray, Discrete Approximation of Invariant Densities, University of Cambridge, 1997. Google Scholar |
[28] |
W. Parry,
On the $\beta$-expansions of real numbers, Acta Math. Hung., 3-4 (1960), 401-416.
doi: 10.1007/BF02020954. |
[29] |
W. Parry,
Representations for real numbers, Acta Math. Hung., 1-2 (1964), 95-105.
doi: 10.1007/BF01897025. |
[30] |
A. Rényi,
Representations for real numbers and their ergodic properties, Acta Math. Hung., 3-4 (1957), 477-493.
doi: 10.1007/BF02020331. |
[31] |
F. M. Sélley, Asymptotic Properties of Mean Field Coupled Maps, Ph.D thesis, Budapest University of Technology and Economics, 2019. Google Scholar |
[32] |
W. Ott, M. Stenlund and L-S. Young,
Memory loss for time-dependent dynamical systems, Math. Res. Lett., 2 (2009), 463-475.
doi: 10.4310/MRL.2009.v16.n3.a7. |
[33] |
S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, New York, 1960. |
show all references
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer Verlag, Berlin, 1998.
doi: 10.1007/978-3-662-12878-7. |
[2] |
V. Baladi and L-S. Young,
On the spectra of randomly perturbed expanding maps, Commun. Math. Phys., 2 (1993), 355-385.
doi: 10.1007/BF02098487. |
[3] |
V. Baladi, A. Kondah and B. Schmitt,
Random correlations for small perturbations of expanding maps, Random Comput. Dyn., 2-3 (1996), 179-204.
|
[4] |
P. Bálint, G. Keller, F. Sélley and I. P. Tóth, Synchronization versus stability of the invariant distribution for a class of globally coupled maps, Nonlinearity, 8 (2018), 3770.
doi: 10.1088/1361-6544/aac5b0. |
[5] |
J.-B. Bardet, G. Keller and R. Zweimüller,
Stochastically stable globally coupled maps with bistable thermodynamic limit, Commun. Math. Phys., 1 (2009), 237-270.
doi: 10.1007/s00220-009-0854-9. |
[6] |
M. Blank and G. Keller, Random perturbations of chaotic dynamical systems: Stability of the spectrum, Nonlinearity, 5 (1998), 1351.
doi: 10.1088/0951-7715/11/5/010. |
[7] |
M. Blank,
Collective phenomena in lattices of weakly interacting maps, Dokl. Akad. Nauk., 3 (2010), 300-304.
doi: 10.1134/S1064562410010126. |
[8] |
M. Blank,
Self-consistent mappings and systems of interacting particles, Dokl. Math., 1 (2011), 49-52.
doi: 10.1134/S1064562411010133. |
[9] |
M. Blank, Ergodic averaging with and without invariant measures, Nonlinearity, 8 (2017), 4649.
doi: 10.1088/1361-6544/aa8fe8. |
[10] |
T. Bogenschütz,
Stochastic stability of invariant subspaces, Ergod. Theor. Dyn. Syst., 3 (2000), 663-680.
doi: 10.1017/S0143385700000353. |
[11] |
J. Buzzi,
Absolutely continuous SRB measures for random Lasota–Yorke maps, T. Am. Math. Soc., 7 (2000), 3289-3303.
doi: 10.1090/S0002-9947-00-02607-6. |
[12] |
A. Boyarsky, P. Góra and C. Keefe,
Absolutely continuous invariant measures for non-autonomous dynamical systems, J. Math. Anal. Appl., 1 (2019), 159-168.
doi: 10.1016/j.jmaa.2018.09.060. |
[13] |
J-R. Chazottes and B. Fernandez, Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, Springer Science & Business Media, 2005. |
[14] |
J. Ding, Q. Du and T-Y. Li,
High order approximation of the Frobenius–Perron operator, Appl. Math. Comput., 2-3 (1993), 151-171.
doi: 10.1016/0096-3003(93)90099-Z. |
[15] |
J. Ding and A. Zhou,
Finite approximations of Frobenius–Perron operators. A solution of {U}lam's conjecture to multi-dimensional transformations, Physica D, 1-2 (1996), 61-68.
doi: 10.1016/0167-2789(95)00292-8. |
[16] |
G. Froyland,
Finite approximation of Sinai-Bowen-Ruelle measures for Anosov systems in two dimensions, Random Comput. Dyn., 4 (1995), 251-264.
|
[17] |
G. Froyland, Ulam's method for random interval maps, Nonlinearity, 4 (1999), 1029.
doi: 10.1088/0951-7715/12/4/318. |
[18] |
G. Froyland, C. González-Tokman and A. Quas, Stability and approximation of random invariant densities for Lasota–Yorke map cocycles, Nonlinearity, 4 (2014), 647.
doi: 10.1088/0951-7715/27/4/647. |
[19] |
G. Froyland, C. González-Tokman and R. D. A. Murray,
Quenched stochastic stability for eventually expanding-on-average random interval map cocycles, Ergod. Theor. Dyn. Syst., 10 (2019), 2769-2792.
doi: 10.1017/etds.2017.143. |
[20] |
M. Jiang and Y. B. Pesin,
Equilibrium measures for coupled map lattices: Existence, uniqueness and finite-dimensional approximations, Commun. Math. Phys., 3 (1998), 675-711.
doi: 10.1007/s002200050344. |
[21] |
K. Kaneko, Globally coupled chaos violates the law of large numbers but not the central limit theorem, Phys. Rev. Lett., 12 (1990), 1391.
doi: 10.1103/PhysRevLett.65.1391. |
[22] |
C. Liverani and G. Keller, A spectral gap for a one-dimensional lattice of coupled piecewise expanding interval maps, in Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, Springer, 2005,115–151.
doi: 10.1007/11360810_6. |
[23] |
G. Keller, R. Klages and P. J. Howard, Continuity properties of transport coefficients in simple maps, Nonlinearity, 8 (2008), 1719.
doi: 10.1088/0951-7715/21/8/003. |
[24] |
G. Keller, An ergodic theoretic approach to mean field coupled maps, Prog. Probab., (2000), 183–208. |
[25] |
S. Klus, P. Koltai and C. Schütte, On the numerical approximation of the Perron–Frobenius and Koopman operator, J. Comput. Dyn., 1 (2016), 51.
doi: 10.3934/jcd.2016003. |
[26] |
T-Y. Li,
Finite approximation for the Frobenius–Perron operator. A solution to Ulam's conjecture, J. Approx Theory, 2 (1976), 177-186.
doi: 10.1016/0021-9045(76)90037-X. |
[27] |
R. D. A. Murray, Discrete Approximation of Invariant Densities, University of Cambridge, 1997. Google Scholar |
[28] |
W. Parry,
On the $\beta$-expansions of real numbers, Acta Math. Hung., 3-4 (1960), 401-416.
doi: 10.1007/BF02020954. |
[29] |
W. Parry,
Representations for real numbers, Acta Math. Hung., 1-2 (1964), 95-105.
doi: 10.1007/BF01897025. |
[30] |
A. Rényi,
Representations for real numbers and their ergodic properties, Acta Math. Hung., 3-4 (1957), 477-493.
doi: 10.1007/BF02020331. |
[31] |
F. M. Sélley, Asymptotic Properties of Mean Field Coupled Maps, Ph.D thesis, Budapest University of Technology and Economics, 2019. Google Scholar |
[32] |
W. Ott, M. Stenlund and L-S. Young,
Memory loss for time-dependent dynamical systems, Math. Res. Lett., 2 (2009), 463-475.
doi: 10.4310/MRL.2009.v16.n3.a7. |
[33] |
S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, New York, 1960. |









$t $ | $\overline{\text{var}}$ | $\overline{\beta}$ | $t $ | $\overline{\text{var}}$ | $\overline{\beta}$ |
0 | 7.1815 | 2.0065 | 0 | 5.1121 | 1.9993 |
5 | 1.1544 | 2.0039 | 5 | 0.9983 | 2.0130 |
10 | 0.9559 | 2.0020 | 10 | 0.9415 | 2.0117 |
15 | 0.9856 | 2.0015 | 15 | 0.9490 | 2.0134 |
20 | 0.9901 | 2.0012 | 20 | 0.9429 | 2.0149 |
25 | 0.9917 | 2.0011 | 25 | 0.9371 | 2.0160 |
30 | 0.9928 | 2.0010 | 30 | 0.9341 | 2.0165 |
35 | 0.9934 | 2.0009 | 35 | 0.9323 | 2.0168 |
40 | 0.9940 | 2.0008 | 40 | 0.9330 | 2.0171 |
45 | 0.9944 | 2.0008 | 45 | 0.9320 | 2.0174 |
50 | 0.9947 | 2.0007 | 50 | 0.9309 | 2.0176 |
55 | 0.9949 | 2.0007 | 55 | 0.9298 | 2.0178 |
60 | 0.9950 | 2.0007 | 60 | 0.9289 | 2.0179 |
65 | 0.9952 | 2.0007 | 65 | 0.9280 | 2.0180 |
70 | 0.9953 | 2.0007 | 70 | 0.9274 | 2.0181 |
75 | 0.9954 | 2.0007 | 75 | 0.9270 | 2.0181 |
80 | 0.9954 | 2.0007 | 80 | 0.9269 | 2.0181 |
85 | 0.9955 | 2.0007 | 85 | 0.9269 | 2.0181 |
90 | 0.9955 | 2.0006 | 90 | 0.9269 | 2.0181 |
95 | 0.9956 | 2.0006 | 95 | 0.9270 | 2.0181 |
100 | 0.9956 | 2.0006 | 100 | 0.9270 | 2.0181 |
$t $ | $\overline{\text{var}}$ | $\overline{\beta}$ | $t $ | $\overline{\text{var}}$ | $\overline{\beta}$ |
0 | 7.1815 | 2.0065 | 0 | 5.1121 | 1.9993 |
5 | 1.1544 | 2.0039 | 5 | 0.9983 | 2.0130 |
10 | 0.9559 | 2.0020 | 10 | 0.9415 | 2.0117 |
15 | 0.9856 | 2.0015 | 15 | 0.9490 | 2.0134 |
20 | 0.9901 | 2.0012 | 20 | 0.9429 | 2.0149 |
25 | 0.9917 | 2.0011 | 25 | 0.9371 | 2.0160 |
30 | 0.9928 | 2.0010 | 30 | 0.9341 | 2.0165 |
35 | 0.9934 | 2.0009 | 35 | 0.9323 | 2.0168 |
40 | 0.9940 | 2.0008 | 40 | 0.9330 | 2.0171 |
45 | 0.9944 | 2.0008 | 45 | 0.9320 | 2.0174 |
50 | 0.9947 | 2.0007 | 50 | 0.9309 | 2.0176 |
55 | 0.9949 | 2.0007 | 55 | 0.9298 | 2.0178 |
60 | 0.9950 | 2.0007 | 60 | 0.9289 | 2.0179 |
65 | 0.9952 | 2.0007 | 65 | 0.9280 | 2.0180 |
70 | 0.9953 | 2.0007 | 70 | 0.9274 | 2.0181 |
75 | 0.9954 | 2.0007 | 75 | 0.9270 | 2.0181 |
80 | 0.9954 | 2.0007 | 80 | 0.9269 | 2.0181 |
85 | 0.9955 | 2.0007 | 85 | 0.9269 | 2.0181 |
90 | 0.9955 | 2.0006 | 90 | 0.9269 | 2.0181 |
95 | 0.9956 | 2.0006 | 95 | 0.9270 | 2.0181 |
100 | 0.9956 | 2.0006 | 100 | 0.9270 | 2.0181 |
$t $ | $\overline{\text{var}}$ | $\overline{\beta}$ | $t $ | $\overline{\text{var}}$ | $\overline{\beta}$ |
0 | 0.0000 | 2.0000 | 0 | 0.0000 | 2.0000 |
25 | 1.0000 | 2.0000 | 25 | 1.0000 | 2.0000 |
50 | 1.0000 | 2.0000 | 50 | 0.9993 | 2.0003 |
75 | 1.0000 | 2.0000 | 75 | 0.9783 | 2.0072 |
100 | 0.9998 | 2.0000 | 100 | 0.9362 | 2.0160 |
125 | 0.9995 | 2.0001 | 125 | 0.9307 | 2.0177 |
150 | 0.9987 | 2.0002 | 150 | 0.9268 | 2.0181 |
175 | 0.9978 | 2.0003 | 175 | 0.9269 | 2.0181 |
200 | 0.9969 | 2.0005 | 200 | 0.9269 | 2.0181 |
225 | 0.9966 | 2.0005 | 225 | 0.9269 | 2.0181 |
250 | 0.9966 | 2.0005 | 250 | 0.9269 | 2.0181 |
275 | 0.9964 | 2.0005 | 275 | 0.9269 | 2.0181 |
300 | 0.9962 | 2.0006 | 300 | 0.9269 | 2.0181 |
$t $ | $\overline{\text{var}}$ | $\overline{\beta}$ | $t $ | $\overline{\text{var}}$ | $\overline{\beta}$ |
0 | 0.0000 | 2.0000 | 0 | 0.0000 | 2.0000 |
25 | 1.0000 | 2.0000 | 25 | 1.0000 | 2.0000 |
50 | 1.0000 | 2.0000 | 50 | 0.9993 | 2.0003 |
75 | 1.0000 | 2.0000 | 75 | 0.9783 | 2.0072 |
100 | 0.9998 | 2.0000 | 100 | 0.9362 | 2.0160 |
125 | 0.9995 | 2.0001 | 125 | 0.9307 | 2.0177 |
150 | 0.9987 | 2.0002 | 150 | 0.9268 | 2.0181 |
175 | 0.9978 | 2.0003 | 175 | 0.9269 | 2.0181 |
200 | 0.9969 | 2.0005 | 200 | 0.9269 | 2.0181 |
225 | 0.9966 | 2.0005 | 225 | 0.9269 | 2.0181 |
250 | 0.9966 | 2.0005 | 250 | 0.9269 | 2.0181 |
275 | 0.9964 | 2.0005 | 275 | 0.9269 | 2.0181 |
300 | 0.9962 | 2.0006 | 300 | 0.9269 | 2.0181 |
$t$ | $\overline{\text{int}}_{f_*(0.1)}$ | $\overline{\text{int}}_{f_*(0.2)}$ |
0 | 0.4096 | 0.5018 |
10 | 0.0101 | 0.0321 |
20 | 0.0049 | 0.0174 |
30 | 0.0028 | 0.0097 |
40 | 0.0017 | 0.0064 |
50 | 0.0011 | 0.0040 |
60 | 0.0007 | 0.0021 |
70 | 0.0005 | 0.0007 |
80 | 0.0004 | 0.0003 |
90 | 0.0003 | 0.0002 |
100 | 0.0002 | 0.0001 |
110 | 0.0002 | 0.0001 |
120 | 0.0002 | 0.0001 |
130 | 0.0001 | 0.0001 |
140 | 0.0001 | 0.0001 |
150 | 0.0001 | 0.0001 |
160 | 0.0001 | 0.0001 |
170 | 0.0001 | 0.0001 |
180 | 0.0001 | 0.0001 |
190 | 0.0001 | 0.0001 |
200 | 0.0001 | 0.0001 |
$t$ | $\overline{\text{int}}_{f_*(0.1)}$ | $\overline{\text{int}}_{f_*(0.2)}$ |
0 | 0.4096 | 0.5018 |
10 | 0.0101 | 0.0321 |
20 | 0.0049 | 0.0174 |
30 | 0.0028 | 0.0097 |
40 | 0.0017 | 0.0064 |
50 | 0.0011 | 0.0040 |
60 | 0.0007 | 0.0021 |
70 | 0.0005 | 0.0007 |
80 | 0.0004 | 0.0003 |
90 | 0.0003 | 0.0002 |
100 | 0.0002 | 0.0001 |
110 | 0.0002 | 0.0001 |
120 | 0.0002 | 0.0001 |
130 | 0.0001 | 0.0001 |
140 | 0.0001 | 0.0001 |
150 | 0.0001 | 0.0001 |
160 | 0.0001 | 0.0001 |
170 | 0.0001 | 0.0001 |
180 | 0.0001 | 0.0001 |
190 | 0.0001 | 0.0001 |
200 | 0.0001 | 0.0001 |
$t $ | $ \varepsilon=1 $ | $ \varepsilon=2.5$ | $t $ | $\varepsilon=1 $ | $ \varepsilon=35$ | $t$ | $\varepsilon=1$ | $ \varepsilon=400 $ |
0 | 6.4694 | 5.0341 | 0 | 5.3766 | 6.4806 | 0 | 7.7582 | 5.1444 |
5 | 1.2483 | 1.1951 | 5 | 1.2370 | 1.1785 | 5 | 1.3852 | 1.0419 |
10 | 0.9808 | 0.8849 | 10 | 0.9882 | 0.9788 | 10 | 0.5224 | 0.5990 |
15 | 0.9914 | 0.8471 | 15 | 0.6884 | 0.9873 | 15 | 0.0210 | 0.1435 |
20 | 0.9976 | 0.8447 | 20 | 0.0474 | 0.5994 | 20 | 0.0007 | 0.0026 |
25 | 0.9994 | 0.8275 | 25 | 0.0015 | 0.1093 | 25 | 0.0000 | 0.0002 |
30 | 0.9999 | 0.8418 | 30 | 0.0000 | 0.0347 | 30 | 0.0000 | 0.0000 |
35 | 1.0000 | 0.8305 | 35 | 0.0000 | 0.0023 | 35 | 0.0000 | 0.0000 |
40 | 1.0000 | 0.8468 | 40 | 0.0000 | 0.0001 | 40 | 0.0000 | 0.0000 |
45 | 1.0000 | 0.8321 | 45 | 0.0000 | 0.0000 | 45 | 0.0000 | 0.0000 |
50 | 0.9531 | 0.8358 | 50 | 0.0000 | 0.0000 | 50 | 0.0000 | 0.0000 |
55 | 0.5357 | 0.8382 | 55 | 0.0000 | 0.0000 | 55 | 0.0000 | 0.0000 |
60 | 0.1802 | 0.7651 | 60 | 0.0000 | 0.0000 | 60 | 0.0000 | 0.0000 |
65 | 0.0128 | 0.6366 | 65 | 0.0000 | 0.0000 | 65 | 0 | 0.0000 |
70 | 0.0004 | 0.5117 | 70 | 0.0000 | 0.0000 | 70 | 0 | 0 |
75 | 0.0000 | 0.4535 | 75 | 0 | 0.0000 | 75 | 0 | 0 |
80 | 0.0000 | 0.4269 | 80 | 0 | 0.0000 | 80 | 0 | 0 |
85 | 0.0000 | 0.4255 | 85 | 0 | 0 | 85 | 0 | 0 |
90 | 0.0000 | 0.4015 | 90 | 0 | 0 | 90 | 0 | 0 |
95 | 0.0000 | 0.3966 | 95 | 0 | 0 | 95 | 0 | 0 |
100 | 0.0000 | 0.3813 | 100 | 0 | 0 | 100 | 0 | 0 |
$t $ | $ \varepsilon=1 $ | $ \varepsilon=2.5$ | $t $ | $\varepsilon=1 $ | $ \varepsilon=35$ | $t$ | $\varepsilon=1$ | $ \varepsilon=400 $ |
0 | 6.4694 | 5.0341 | 0 | 5.3766 | 6.4806 | 0 | 7.7582 | 5.1444 |
5 | 1.2483 | 1.1951 | 5 | 1.2370 | 1.1785 | 5 | 1.3852 | 1.0419 |
10 | 0.9808 | 0.8849 | 10 | 0.9882 | 0.9788 | 10 | 0.5224 | 0.5990 |
15 | 0.9914 | 0.8471 | 15 | 0.6884 | 0.9873 | 15 | 0.0210 | 0.1435 |
20 | 0.9976 | 0.8447 | 20 | 0.0474 | 0.5994 | 20 | 0.0007 | 0.0026 |
25 | 0.9994 | 0.8275 | 25 | 0.0015 | 0.1093 | 25 | 0.0000 | 0.0002 |
30 | 0.9999 | 0.8418 | 30 | 0.0000 | 0.0347 | 30 | 0.0000 | 0.0000 |
35 | 1.0000 | 0.8305 | 35 | 0.0000 | 0.0023 | 35 | 0.0000 | 0.0000 |
40 | 1.0000 | 0.8468 | 40 | 0.0000 | 0.0001 | 40 | 0.0000 | 0.0000 |
45 | 1.0000 | 0.8321 | 45 | 0.0000 | 0.0000 | 45 | 0.0000 | 0.0000 |
50 | 0.9531 | 0.8358 | 50 | 0.0000 | 0.0000 | 50 | 0.0000 | 0.0000 |
55 | 0.5357 | 0.8382 | 55 | 0.0000 | 0.0000 | 55 | 0.0000 | 0.0000 |
60 | 0.1802 | 0.7651 | 60 | 0.0000 | 0.0000 | 60 | 0.0000 | 0.0000 |
65 | 0.0128 | 0.6366 | 65 | 0.0000 | 0.0000 | 65 | 0 | 0.0000 |
70 | 0.0004 | 0.5117 | 70 | 0.0000 | 0.0000 | 70 | 0 | 0 |
75 | 0.0000 | 0.4535 | 75 | 0 | 0.0000 | 75 | 0 | 0 |
80 | 0.0000 | 0.4269 | 80 | 0 | 0.0000 | 80 | 0 | 0 |
85 | 0.0000 | 0.4255 | 85 | 0 | 0 | 85 | 0 | 0 |
90 | 0.0000 | 0.4015 | 90 | 0 | 0 | 90 | 0 | 0 |
95 | 0.0000 | 0.3966 | 95 | 0 | 0 | 95 | 0 | 0 |
100 | 0.0000 | 0.3813 | 100 | 0 | 0 | 100 | 0 | 0 |
[1] |
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