Topological entropyby unit length for the Ginzburg-Landau equation on the line

N. Maaroufi

doi:10.3934/dcds.2014.34.647

Article Contents

2014, Volume 34, Issue 2: 647-662. Doi: 10.3934/dcds.2014.34.647

This issue Previous Article Generic property of irregular sets in systems satisfying the specification property Next Article Non-normal numbers with respect to Markov partitions

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

N. Maaroufi^1,

1.
Université Internationale de Rabat, Technopolis 11 100 Sala el Jadida

Received: February 28, 2011

Revised: April 30, 2013

Published: January 2014

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Abstract

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 ($k\geq 0$).

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Mathematics Subject Classification: Primary: 37B40, 35Q56, 35B41, 37L30, 35R15, 37C45, 28D20, 47A35.

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