دانلود کتاب Gromov Hausdorff پایداری سیستم‌های دینامیکی و کاربردها در PDE

A common definition of a dynamical system is any phenomenon of nature evolving in “time”. One of the most important classes of dynamical systems is described by differential equations or difference equations.We can classify dynamical systems according to discrete or continuous time. Specific questions in the qualitative theory of dynamical systems are the long-term behavior of solutions and the stability problem. A main goal is to study the preservation of geometric structure of solutions under perturbations. For this problem, in early 1960s, Smale used the theory of differential topology to understand the structure of solutions on compact smooth manifolds. One of the most successful notions is hyperbolicity, which has been well developed due to Anosov, Smale, Palis, Mañé, etc. In the light of the rich consequence of differentiable dynamical systems, there are many works that have studied the dynamics properties from a topological viewpoint. For example, instead of hyperbolic systems, it was also shown that any expansive system with shadowing property is topologically stable and admits the spectral decomposition. These results are known as Walters’ stability theorem and spectral decomposition theorem, respectively. One of the important concepts in geometry is the Gromov-Haudorff distance. Roughly, the Gromov-Haudorff distance measures how far two compact metric spaces are from being isometric. Motivated from this, Arbieto and Morales [5] introduced the Gromov- Hausdorff distance between twomaps on two compact metric spaces and applied it to study the stability of dynamical systems under perturbations of both maps and phase spaces. We observe that the Gromov-Hausdorff distance is a strong tool to study the stability of dynamical systems. In the first part of the book, we introduce the notion of Gromov-Hausdorff distance DGH between two dynamical systems and study the stability of dynamical systems under Gromov-Hausdorff perturbations. Note that distance DGH induces a topology on the collection DS of all dynamical systems up to isometries. In the second part of the book, we study the stability of dynamical systems induced by dissipative partial differential equations under perturbations of the domain and equation. One of the difficulties in this direction is that the phase space of the induced dynamical system can be changed as we perturb the domain. To overcome this difficulty, we use the Gromov-Hausdorff distance between two dynamical systems.