Title:

Low dimensional dynamics : bifurcations of Cantori and realisations of uniform hyperbolicity

This thesis is in two parts. The first part is about a bifurcation of a type of invariant set called a Cantorus. Cantori are most familiar as the Denjoy counterexample in the theory of circle homeomorphisms, but this example is not very smooth. More representative, smooth examples of Cantori occur in noninvertible circle mappings and in area preserving twist maps. In these systems, as the parameters are varied, Cantori can be created or destroyed or change their structure. In other words, they undergo bifurcation. In the first chapter I study a particular bifurcation which destroys a Cantorus in a family of bimodal circle maps. I show that this bifurcation takes place in a way that is reminiscent of a saddlenode bifurcation destroying a pair of fixed points. The second part of this thesis describes attempts to construct physically realistic examples of uniformly hyperbolic systems. The theory of uniform hyperbolicity is one of the great results from nonlinear dynamics. It provides many powerful tools for analysing dynamical systems, and even though people now realise that a lot of interesting systems are not uniformly hyperbolic, the study of these systems consists, to a large extent, of taking the ideas and methods developed for uniformly hyperbolic systems and then pushing them a little bit further. There are, of course, many standard examples of uniformly hyperbolic systems. The Smale Horseshoe, hyperbolic toral automorphisms and geodesic flows on surfaces of negative curvature to name but a few. All of these examples are, however, rather mathematical and abstract. It is all very well having a nice theory, but is it something that a Physicist could go and measure in a lab? I wanted to demonstrate the physical relevance of uniform hyperbolicity by finding a system that was as "real" as possible and which was provably uniformly hyperbolic. In fact I found two such systems. In Chapter 2 I give the construction of a time periodic flow in the plane which has a global attractor that is a uniformly hyperbolic Plykin attractor. The techniques used here could be applied more generally to construct other folding and stretching flows.
