On the breakdown of regularity of invariant curves in quasi-periodically forced systems

Abstract: In this thesis we study the process of torus collisions in one-parameter families of quasi-periodically forced dynamical systems. Specifically, we study the process whereby two invariant curves (homeomorphic to circles), one attracting and one repelling, bifurcate into a strange non-chaotic attractor. In Paper A, the system is a quasi-periodically forced logistic family, but avoids period-doubling. We give an asymptotic analysis of some geometric properties of the attractor, as it approaches the repeller at the bifurcation point. In Paper B, we study the same type of questions as in Paper A, but instead for a class of quasi-periodic C^2 Schrödinger cocycles. In both papers the results confirm a conjecture that the distance between the curves is asymptotically linear in the parameter, for those classes of systems. In addition, we obtain results about the asymptotic growth of C^1-norms. In Paper C, we study the same class of systems as in Paper B, but instead look at the asymptotics of the (maximal) Lyapunov exponent at the bifurcation point. The results show that it has Hölder exponent exactly 1/2, as the energy parameter approaches the lowest energy of the spectrum. This confirms, for this class and setting, similar conjectures about this asymptotic behaviour.