Examples of dynamical systems in the interface between order and chaos

Abstract

This thesis deals with the complexity inherent in the long-term behavior of both chaotic and non-chaotic dynamical systems. Thereby, two particular examples form the starting point of our work. The first example is a simple model for the occurrence of a so-called strange non-chaotic attractor. We study fractal aspects of this attractor by determining several associated dimensional quantities. Interestingly, the considered model system shows a complex long-term behavior despite having zero topological entropy. It is hence natural to ask whether there exists another topological invariant which is able to detect this inherent complexity. This question is the origin for the investigation launched in the second part of the thesis where we introduce the notion of amorphic complexity. After examining basic properties of this new quantity, we study its applicability to almost sure 1-1 extensions of equicontinuous systems with the particular focus on Sturmian subshifts, Denjoy homeomorphisms on the circle and regular Toeplitz subshifts. The second motivating example of this thesis is closely related to a parameter family of sets of bounded orbits associated with the classical Farey map. This family of sets was recently studied as a generalization of the sets of bounded continued fraction expansions where several topological and dimensional properties were considered. In particular, it was shown that a natural associated bifurcation set plays a central role in the understanding of this family of sets. In the last part of the present dissertation, we extend these results to parameter families of sets of bounded orbits associated with more general continuous interval maps and thereby focus on topological aspects.