Robustness of nonuniform and random exponential dichotomies with applications to differential equations - Sousa, Alexandre do Nascimento Oliveira (2022)
In this thesis, we study hyperbolicity for deterministic and random nonautonomous dynamical systems and their applications to differential equations. More precisely, we present results for the following topics: nonuniform hyperbolicity for evolution processes and hyperbolicity for nonautonomous random dynamical systems. Concerning the first one, we study the robustness of the nonuniform exponential dichotomy for continuous and discrete evolution processes. We present an example of an infinite-dimensional differential equation that admits a nonuniform exponential dichotomy and apply the robustness result. Moreover, we study the persistence of nonuniform hyperbolic solutions in semilinear differential equations. Furthermore, we introduce a new concept of nonuniform exponential dichotomy, provide examples, and prove a stability result under perturbations for it. For the second topic, we introduce exponential dichotomies for random and nonautonomous dynamical systems. We prove a robustness result for this notion of hyperbolicity and study its applications to random and nonautonomous differential equations. Among these applications, we study the existence and continuity of random hyperbolic solutions and their associated unstable manifolds. As a consequence, we obtain continuity and topological structural stability for nonautonomous random attractors.