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

We report on numerical experiments computing the ε-entropy for parabolic partial differential equations. The ε-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 ε-entropy only requires solutions of the equation to the accuracy of the parameter ε 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.