| This dissertation has two parts. In both parts we use geometric methods to study dynamical systems with two scales.; In the first part we study the Maxwell-Bloch equations for a two-level laser in a ring cavity. For Class A lasers, these equations have two distinct time scales, and form a singularly perturbed, semilinear hyperbolic system with two distinct characteristics. We extend Fenichel's geometric singular perturbation theory [40] to infinite dimensions by proving the persistence of a smooth, slow manifold under an unbounded perturbation. The proof relies on the energy preserving nature of the nonlinearity, and the existence of two characteristics. The slow manifold is a globally attracting, positively invariant manifold, with infinite dimension and codimension, that contains the attractor of the system. We rigorously decouple the slow and fast time scales) and obtain a reduced (but still infinite dimensional) dynamical system described by a functional differential equation. We also present the results of numerical computations. These demonstrate the applicability of our analysis and reveal a new type of spatiotemporal chaos in the Maxwell-Bloch equations. Independent of these scaling assumptions, we prove that the attractor of these equations has Gevrey regularity. Finally, we prove similar invariant manifold theorems for a class of infinite dimensional dynamical systems with relaxation.; In the second part we study the weak limits of gradient dynamical systems with two spatial scales. These are higher dimensional generalisations of a model for the kinetics of martensitic phase transitions proposed by Abeyaratne, Chu and James [2]. We derive averaged equations in certain regions of phase space, but these equations typically do not have unique solutions. For the two dimensional problem we find that generically the phase space breaks into a countable number of domains, in the interior of which the homogenized dynamics are rectilinear. These domains have a Cantor set structure caused by the bifurcations of circle maps. Consequently, the homogenized equations vary on all scales, and we consider the implications for such models in materials science. V. P. Smyshlyaev has studied this problem independently, and some of our results are similar. |
