Topological attractors of quasi-periodically forced one-dimensional maps

Abstract

This thesis is concerned with topological attractors of some quasi-periodically forced one-dimensional maps. The main aim of our study is to under- stand the states of the attractors by analyzing the mechanisms which rule the dynamics of the maps. Concretely we investigate two types of quasi- periodically forced one-dimensional families. The first type consists of two di erent quasi-periodically forced increasing systems. We present rigorous proofs for the states of their attractors. The second type of systems that we consider are those quasi-periodically forced S-unimodal systems. We pro- pose the mechanism for their changes of periodicity according to the forced terms, which is based on elaborate analysis of the S-unimodal maps and is substantiated by numerical evidence. The motivation of our research is the problem of Strange Nonchaotic Attractors. we rst explore shortly the general topological properties of the pinched closed invariant sets, which are of particular important for SNAs. We prove that, the !-limit set of pinched points is the unique minimal set in a pinched closed invariant set, and any continuous graph contained in a pinched closed invariant set must be invariant. Our first main result shows that, in a pinched system the !-limit set of pinched points is the only mini- mal set which must be contained in every invariant sets. We prove rigorously the states of attractors with respect to two parameters of two families which are forced monotone increasing maps, with some concave or convex struc- tures on bre maps. The last chapter is devoted to the quasi-periodically forced S-unimodal maps. In this chapter we propose the mechanism of the change of the pe- riodicity of their attractors, which works with respect to a parameter who controls the forcing term when the forced S-unimodal map is fixed. This mechanism of forced systems is based on the topological structures of the forced S-uniform maps. The similar merging and collision are also reported for S-unimodal maps in physical context for decades, we prove this is true later in this chapter. The facts that we present in the proof reveal the topo- logical structures that rule the change of periodicity of the forced S-unimodal systems.