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Topological entropy by unit length for the Ginzburg-Landau equation on the line

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  • 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 ($k\geq 0$).
    Mathematics Subject Classification: Primary: 37B40, 35Q56, 35B41, 37L30, 35R15, 37C45, 28D20, 47A35.

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  • [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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