# American Institute of Mathematical Sciences

February  2014, 34(2): 647-662. doi: 10.3934/dcds.2014.34.647

## 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

Received  March 2011 Revised  May 2013 Published  August 2013

In this paper we study the notion of topological entropy by unit length for the dynamical system given by the complex Ginzburg-Landau equation on the line (CGL). This equation has a global attractor $\mathcal{A}$ that attracts all the trajectories. We first prove the existence of the topological entropy by unit length for the topological dynamical system $(\mathcal{A},S)$ in a Hilbert space framework, where $S(t)$ is the semi-flow defined by CGL. Next we show that this topological entropy by unit length is bounded by the product of the upper fractal dimension per unit length (see [10]) with the expansion rate. Finally, we prove that this quantity is invariant for all $H^k$ metrics ($k\geq 0$).
Citation: N. Maaroufi. Topological entropy by unit length for the Ginzburg-Landau equation on the line. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 647-662. doi: 10.3934/dcds.2014.34.647
##### References:
 [1] R. Bowen, Topological entropy for noncompact sets,, Trans. Amer. Math. Soc., 184 (1973), 125. doi: 10.1090/S0002-9947-1973-0338317-X. [2] M. Brin and G. Stuck, "Introduction to Dynamical Systems,", Cambridge University Press, (2002). doi: 10.1017/CBO9780511755316. [3] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", Studies in Mathematics and its Apllications, (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. 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. 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. 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. 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. 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. 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. doi: 10.3934/cpaa.2012.11.1253. [11] B. Hasselblatt and A. Katok, Principal structures,, in, (2002), 1. 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. [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. 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. 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, (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. [17] H. Queffelec and C. Zuily, "Element d'Analyse,", Paris, (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. doi: 10.1137/S0036141094262518. [19] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Applied Mathematical Sciences, 68 (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. 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. doi: 10.1070/SM1998v189n02ABEH000301. [22] P. Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, 79 (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. 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. [25] S. V. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity,, Comm. Pure Appl. Math., 56 (2003), 584. 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. doi: 10.1007/s10884-006-9007-4.

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##### References:
 [1] R. Bowen, Topological entropy for noncompact sets,, Trans. Amer. Math. Soc., 184 (1973), 125. doi: 10.1090/S0002-9947-1973-0338317-X. [2] M. Brin and G. Stuck, "Introduction to Dynamical Systems,", Cambridge University Press, (2002). doi: 10.1017/CBO9780511755316. [3] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", Studies in Mathematics and its Apllications, (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. 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. 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. 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. 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. 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. 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. doi: 10.3934/cpaa.2012.11.1253. [11] B. Hasselblatt and A. Katok, Principal structures,, in, (2002), 1. 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. [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. 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. 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, (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. [17] H. Queffelec and C. Zuily, "Element d'Analyse,", Paris, (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. doi: 10.1137/S0036141094262518. [19] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Applied Mathematical Sciences, 68 (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. 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. doi: 10.1070/SM1998v189n02ABEH000301. [22] P. Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, 79 (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. 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. [25] S. V. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity,, Comm. Pure Appl. Math., 56 (2003), 584. 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. doi: 10.1007/s10884-006-9007-4.
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