Distance of attractors of evolutionary equations.

Resumen

This thesis is devoted to obtain good estimates of the distance of attractors once we know they behavecontinuously. Our aim is to study what we have to ask to the system to improve the known rate ofconvergence of attractors existing in the literature.We have obtained that, if we have a finite dynamical system generated by a Morse Smale gradient mapwhich has an attractor and we perturb this system in such way so that the perturbed system has also anattractor, then the rate of convergence of these attractors is of order the distance of the corresponding time one maps. Moreover, we generalize this result with the following one: If we have a finite dynamical systemgenerated by a Morse Smale gradient map which has an attractor, and we consider a non­autonomousperturbation of it, such that this non autonomous perturbed dynamical system has a pullback attractor, thenthe rate of convergence of the attractor and pullback attractor is of order the rate of convergence of thecorresponding time one maps. These results are obtained in a finite dimensional framework and the method of proof is using the Shadowing properties that Morse Smale maps have.Since our framework for these results is finite dimensional, and we eventually want to apply thistechnique to infinite dimensional systems, we need a tool that reduces the system to a finite dimensionalone, Inertial Manifolds. These smooth finite dimensional manifolds, positive invariant under the flow andexponentially attractive, contain the attractor if the system has one. Since we want to study the behavior of the systems under perturbations and obtain good rates for the convergence of the attractors, we study thebehavior of these inertial manifolds under perturbations of the system, providing good rates of itsconvergence in the C 0 topology and also in the C 1 topology.Finally, we apply the techniques mentioned above to address the problem of the distance ofattractors corresponding to a particular reaction diffusion equation and a perturbation of itdescribed in a thin domain. This thesis improves the rate of convergence of attractors for thisparticular problem existing in the literature.