98-50 Collet P., Eckmann J.-P.

Extensive Properties of the Complex Ginzburg-Landau Equation (272K, postcript) Feb 6, 98

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Abstract. We study the set of solutions of the complex Ginzburg-Landau equation in $\real^d$, d<3. We consider the global attracting set i.e., the forward map of the set of bounded initial data, and restrict it to a cube $Q_L$ of side $L$. We cover this set by a (minimal) number $N_{Q_L}(\epsilon )$ of balls of radius $\epsilon $ in $\Linfty(Q_L)$. We show that the Kolmogorov \epsilon-entropy per unit length, $H_\epsilon =\lim_{L\to\infty} L^{-d} \log N_{Q_L}(\epsilon)$ exists. In particular, we bound $H_\epsilon $ by $\OO(\log(1/\epsilon ))$, which shows that the attracting set is {\em smaller} than the set of bounded analytic functions in a strip. We finally give a positive lower bound: $H_\epsilon>\OO(\log(1/\epsilon )) $.

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