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Topological entropy by unit length for the Ginzburg-Landau equation on the line
1. | Université Internationale de Rabat, Technopolis 11 100 Sala el Jadida, Morocco |
References:
[1] |
R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.
doi: 10.1090/S0002-9947-1973-0338317-X. |
[2] |
M. Brin and G. Stuck, "Introduction to Dynamical Systems," Cambridge University Press, Cambridge, 2002.
doi: 10.1017/CBO9780511755316. |
[3] |
A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Apllications, Vol. 25, North Holland Publishing Co., Amsterdam, 1992. |
[4] |
A. V. Babin and M. I. Vishik, Attractors of partial differential evolution equations in an unbounded domain, Procceding of Royal Society of Edinburgh Sect. A, 116 (1990), 221-243.
doi: 10.1017/S0308210500031498. |
[5] |
P. Collet and J.-P. Eckmann, Extensive properties of the complex Ginzburg-Landau equation, Commun. Math. Phys., 200 (1999), 699-722.
doi: 10.1007/s002200050546. |
[6] |
P. Collet and J.-P. Eckmann, The definition and measurement of the topological entropy per unit volume in parabolic PDEs, Nonlinearity, 12 (1999), 451-473.
doi: 10.1088/0951-7715/12/3/002. |
[7] |
P. Collet and J.-P. Eckmann, Topological entropy and $\varepsilon$-entropy for damped hyperbolic equations, Ann. Henri Poincaré, 1 (2000), 715-752.
doi: 10.1007/PL00001013. |
[8] |
M. A. Efendiev and S. V. Zelik, The attractor of a nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure Appl. Math., 54 (2001), 625-688.
doi: 10.1002/cpa.1011. |
[9] |
M. A. Efendiev and S. V. Zelik, Upper and lower bounds for the Kolmogorov entropy of the attractor for the RDE in an unbounded domain, J. Dynam. Differential Equations, 14 (2002), 369-403.
doi: 10.1023/A:1015130904414. |
[10] |
O. Goubet and N. Maaroufi, Entropy by unit length for the Ginzburg-Landau equation on the line. A Hilbert space framework, Commun. Pure Appl. Anal., 11 (2012), 1253-1267.
doi: 10.3934/cpaa.2012.11.1253. |
[11] |
B. Hasselblatt and A. Katok, Principal structures, in "Handbook of Dynamical Systems, Vol. 1A," North-Holland, Amsterdam, (2002), 1-203.
doi: 10.1016/S1874-575X(02)80003-0. |
[12] |
A. N. Kolmogorov and V. M. Tihomirov, $\varepsilon$-entropy and $\varepsilon$-capacity of sets in functional spaces, Uspehi Mat. Nauk, 14 (1959), 3-86. |
[13] |
N. Maaroufi, "Quelques Proprietes Ergodiques de l'Attracteur Donne par le Systeme Dynamique Relatif a l'Equation de Ginzburg Landau Complexe Cubique sur un Domaine Non Borne," Ph.D thesis, 2010. |
[14] |
A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains-existence and comparaison, Nonlinearity, 8 (1995), 743-768.
doi: 10.1088/0951-7715/8/5/006. |
[15] |
A. Mielke and S. V. Zelik, Infinite-dimensional hyperbolic sets and spatio-temporal chaos in reaction-diffusion systems in $\mathbbR^n$, J. Dynam. Differential Equations, 19 (2007), 333-389.
doi: 10.1007/s10884-006-9058-6. |
[16] |
A. Miranville and S. V. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in "Handbook of Differential Equations: Evolutionary Equations. Vol. IV," Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2008), 103-200.
doi: 10.1016/S1874-5717(08)00003-0. |
[17] |
H. Queffelec and C. Zuily, "Element d'Analyse," Paris, Dunod, 2002. |
[18] |
P. Taráč, P. Bollerman, A. Doelman, A. van Harten and E. S. Titi, Analyticity of essentially bounded solutions to semlinear parabolic systems and validity of the Ginzburg-Landau equation, SIAM J. Math. Anal., 27 (1996), 424-448.
doi: 10.1137/S0036141094262518. |
[19] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[20] |
D. Turaev and S. V. Zelik, Analytical proof of space-time chaos in Ginzburg-Landau equations, Discrete Contin. Dyn. Syst., 28 (2010), 1713-1751.
doi: 10.3934/dcds.2010.28.1713. |
[21] |
M. I. Vishik and V. V. Chepyzhov, Kolmogorov $\varepsilon$-entropy of attractors of reaction-diffusion systems, Mat. Sb., 189 (1998), 81-110.
doi: 10.1070/SM1998v189n02ABEH000301. |
[22] |
P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982. |
[23] |
S. V. Zelik, An attractor of a nonlinear system of reaction-diffusion equations in $\mathbbR^n$ and estimates for its $\varepsilon$-entropy, Mat. Zametki, 65 (1999), 941-944.
doi: 10.1007/BF02675597. |
[24] |
S. V. Zelik, Multiparameter semigroups and attractors of reaction-diffusion equations in $\mathbbR^n$, Tr. Mosk. Mat. Obs., 65 (2004), 114-174; translation in Trans. Moscow Math. Soc., (2004), 105-160. |
[25] |
S. V. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity, Comm. Pure Appl. Math., 56 (2003), 584-637.
doi: 10.1002/cpa.10068. |
[26] |
S. V. Zelik, Spatial and dynamical chaos generated by reaction-diffusion systems in unbounded domains, J. Dynam. Differential Equations, 19 (2007), 1-74.
doi: 10.1007/s10884-006-9007-4. |
show all references
References:
[1] |
R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.
doi: 10.1090/S0002-9947-1973-0338317-X. |
[2] |
M. Brin and G. Stuck, "Introduction to Dynamical Systems," Cambridge University Press, Cambridge, 2002.
doi: 10.1017/CBO9780511755316. |
[3] |
A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Apllications, Vol. 25, North Holland Publishing Co., Amsterdam, 1992. |
[4] |
A. V. Babin and M. I. Vishik, Attractors of partial differential evolution equations in an unbounded domain, Procceding of Royal Society of Edinburgh Sect. A, 116 (1990), 221-243.
doi: 10.1017/S0308210500031498. |
[5] |
P. Collet and J.-P. Eckmann, Extensive properties of the complex Ginzburg-Landau equation, Commun. Math. Phys., 200 (1999), 699-722.
doi: 10.1007/s002200050546. |
[6] |
P. Collet and J.-P. Eckmann, The definition and measurement of the topological entropy per unit volume in parabolic PDEs, Nonlinearity, 12 (1999), 451-473.
doi: 10.1088/0951-7715/12/3/002. |
[7] |
P. Collet and J.-P. Eckmann, Topological entropy and $\varepsilon$-entropy for damped hyperbolic equations, Ann. Henri Poincaré, 1 (2000), 715-752.
doi: 10.1007/PL00001013. |
[8] |
M. A. Efendiev and S. V. Zelik, The attractor of a nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure Appl. Math., 54 (2001), 625-688.
doi: 10.1002/cpa.1011. |
[9] |
M. A. Efendiev and S. V. Zelik, Upper and lower bounds for the Kolmogorov entropy of the attractor for the RDE in an unbounded domain, J. Dynam. Differential Equations, 14 (2002), 369-403.
doi: 10.1023/A:1015130904414. |
[10] |
O. Goubet and N. Maaroufi, Entropy by unit length for the Ginzburg-Landau equation on the line. A Hilbert space framework, Commun. Pure Appl. Anal., 11 (2012), 1253-1267.
doi: 10.3934/cpaa.2012.11.1253. |
[11] |
B. Hasselblatt and A. Katok, Principal structures, in "Handbook of Dynamical Systems, Vol. 1A," North-Holland, Amsterdam, (2002), 1-203.
doi: 10.1016/S1874-575X(02)80003-0. |
[12] |
A. N. Kolmogorov and V. M. Tihomirov, $\varepsilon$-entropy and $\varepsilon$-capacity of sets in functional spaces, Uspehi Mat. Nauk, 14 (1959), 3-86. |
[13] |
N. Maaroufi, "Quelques Proprietes Ergodiques de l'Attracteur Donne par le Systeme Dynamique Relatif a l'Equation de Ginzburg Landau Complexe Cubique sur un Domaine Non Borne," Ph.D thesis, 2010. |
[14] |
A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains-existence and comparaison, Nonlinearity, 8 (1995), 743-768.
doi: 10.1088/0951-7715/8/5/006. |
[15] |
A. Mielke and S. V. Zelik, Infinite-dimensional hyperbolic sets and spatio-temporal chaos in reaction-diffusion systems in $\mathbbR^n$, J. Dynam. Differential Equations, 19 (2007), 333-389.
doi: 10.1007/s10884-006-9058-6. |
[16] |
A. Miranville and S. V. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in "Handbook of Differential Equations: Evolutionary Equations. Vol. IV," Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2008), 103-200.
doi: 10.1016/S1874-5717(08)00003-0. |
[17] |
H. Queffelec and C. Zuily, "Element d'Analyse," Paris, Dunod, 2002. |
[18] |
P. Taráč, P. Bollerman, A. Doelman, A. van Harten and E. S. Titi, Analyticity of essentially bounded solutions to semlinear parabolic systems and validity of the Ginzburg-Landau equation, SIAM J. Math. Anal., 27 (1996), 424-448.
doi: 10.1137/S0036141094262518. |
[19] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[20] |
D. Turaev and S. V. Zelik, Analytical proof of space-time chaos in Ginzburg-Landau equations, Discrete Contin. Dyn. Syst., 28 (2010), 1713-1751.
doi: 10.3934/dcds.2010.28.1713. |
[21] |
M. I. Vishik and V. V. Chepyzhov, Kolmogorov $\varepsilon$-entropy of attractors of reaction-diffusion systems, Mat. Sb., 189 (1998), 81-110.
doi: 10.1070/SM1998v189n02ABEH000301. |
[22] |
P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982. |
[23] |
S. V. Zelik, An attractor of a nonlinear system of reaction-diffusion equations in $\mathbbR^n$ and estimates for its $\varepsilon$-entropy, Mat. Zametki, 65 (1999), 941-944.
doi: 10.1007/BF02675597. |
[24] |
S. V. Zelik, Multiparameter semigroups and attractors of reaction-diffusion equations in $\mathbbR^n$, Tr. Mosk. Mat. Obs., 65 (2004), 114-174; translation in Trans. Moscow Math. Soc., (2004), 105-160. |
[25] |
S. V. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity, Comm. Pure Appl. Math., 56 (2003), 584-637.
doi: 10.1002/cpa.10068. |
[26] |
S. V. Zelik, Spatial and dynamical chaos generated by reaction-diffusion systems in unbounded domains, J. Dynam. Differential Equations, 19 (2007), 1-74.
doi: 10.1007/s10884-006-9007-4. |
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