Numerical computation of ε-entropy for parabolic equations with analytic solutions

We report on numerical experiments computing the e-entropy for parabolic partial differential equations. The e-entropy is a measure of the spatial density of complexity for the dynamics on an invariant set in function space and has been studied analytically by a number of authors. The e-entropy only requires solutions of the equation to the accuracy of the parameter e and the resulting number is (asymptotically) independent of domain size. We consider the complex Ginzburg-Landau equation as an example where a number of analytic results exist and the Kuramoto-Sivashinsky equation where the accompanying theory has yet to be fully developed. Our numerical results for the Kuramoto-Sivashinsky equation do not contradict the conjectured linear scaling of the dimension with domain size. © 2004 Elsevier B.V. All rights reserved.