# 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.  Google Scholar [2] M. Brin and G. Stuck, "Introduction to Dynamical Systems,", Cambridge University Press, (2002).  doi: 10.1017/CBO9780511755316.  Google Scholar [3] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", Studies in Mathematics and its Apllications, (1992).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] B. Hasselblatt and A. Katok, Principal structures,, in, (2002), 1.  doi: 10.1016/S1874-575X(02)80003-0.  Google Scholar [12] A. N. Kolmogorov and V. M. Tihomirov, $\varepsilon$-entropy and $\varepsilon$-capacity of sets in functional spaces,, Uspehi Mat. Nauk, 14 (1959), 3.   Google Scholar [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).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] H. Queffelec and C. Zuily, "Element d'Analyse,", Paris, (2002).   Google Scholar [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.  Google Scholar [19] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Applied Mathematical Sciences, 68 (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] P. Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, 79 (1982).   Google Scholar [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.  Google Scholar [24] S. V. Zelik, Multiparameter semigroups and attractors of reaction-diffusion equations in $\mathbbR^n$,, Tr. Mosk. Mat. Obs., 65 (2004), 114.   Google Scholar [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.  Google Scholar [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.  Google Scholar

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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.  Google Scholar [2] M. Brin and G. Stuck, "Introduction to Dynamical Systems,", Cambridge University Press, (2002).  doi: 10.1017/CBO9780511755316.  Google Scholar [3] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", Studies in Mathematics and its Apllications, (1992).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] B. Hasselblatt and A. Katok, Principal structures,, in, (2002), 1.  doi: 10.1016/S1874-575X(02)80003-0.  Google Scholar [12] A. N. Kolmogorov and V. M. Tihomirov, $\varepsilon$-entropy and $\varepsilon$-capacity of sets in functional spaces,, Uspehi Mat. Nauk, 14 (1959), 3.   Google Scholar [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).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] H. Queffelec and C. Zuily, "Element d'Analyse,", Paris, (2002).   Google Scholar [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.  Google Scholar [19] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Applied Mathematical Sciences, 68 (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] P. Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, 79 (1982).   Google Scholar [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.  Google Scholar [24] S. V. Zelik, Multiparameter semigroups and attractors of reaction-diffusion equations in $\mathbbR^n$,, Tr. Mosk. Mat. Obs., 65 (2004), 114.   Google Scholar [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.  Google Scholar [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.  Google Scholar
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