Robustness of nonuniform and random exponential dichotomies with applications to differential equations

In this thesis, we study hyperbolicity for deterministic and random nonautonomous dynamical systems and their applications to differential equations. More precisely, we present results in the following topics: nonuniform hyperbolicity for evolution processes and hyperbolicity for nonautonomous random dynamical systems. In the first topic, we study the robustness of the nonuniform exponential dichotomy for continuous and discrete evolution processes. We present an example of an infinitedimensional 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. In 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

Chaotic Diffusion on Periodic Orbits: The Perturbed Arnol'd Cat Map

Chaotic diffusion on periodic orbits (POs) is studied for the perturbed Arnol'd cat map on a cylinder, in a range of perturbation parameters corresponding to an extended structural-stability regime of the system on the torus. The diffusion coefficient is calculated using the following PO formulas: (a) The curvature expansion of the Ruelle zeta function. (b) The average of the PO winding-number squared, $w^{2}$, weighted by a stability factor. (c) The uniform (nonweighted) average of $w^{2}$. The results from formulas (a) and (b) agree very well with those obtained by standard methods, for all the perturbation parameters considered. Formula (c) gives reasonably accurate results for sufficiently small parameters corresponding also to cases of a considerably nonuniform hyperbolicity. This is due to {\em uniformity sum rules} satisfied by the PO Lyapunov eigenvalues at {\em fixed} $w$. These sum rules follow from general arguments and are supported by much numerical evidence.Comment: 6 Tables, 2 Figures (postscript); To appear in Physical Review