Resumen de Topological attractors of quasi-periodically forced one-dimensional maps
The thesis is concerned with the topological attractors of some quasi-periodically forced one-dimensional maps. Our investigations display some very essential features of the dynamics of this kind of systems. The quasi-periodically forced systems consist of two generic types, the pinched ones and the non-pinched. The dynamical behaviours of these two types have an intrinsic distinction, presented on their attractors. Roughly speaking, the dynamical structures of the unforced one-dimensional maps, can be seen preserved by suitable forcing term in non-pinched systems, but this is not the case in pinched ones. Our Theorem A says that the attractors of a pinched system must be in one piece, with the unique $\omega$-limit set of pinched points inside all of them. Hence, if there are different attracting orbits in the unforced one-dimensional map, all of them can be represented by corresponding invariant subsets in the non-pinched systems; however in the case of a pinched system, the situation depends much on the pinched condition. These features of two types of systems are exhibited clearly by the typical families that we choose as examples.
More concretely, we study two types of typical families, whose unforced one-dimensional maps are already very well studied. The first type consists of two different quasi-periodically forced increasing real maps, both of them are classic examples of saddle-node bifurcation. We elaborate different states of their attractors by Theorem B and Theorem C respectively in the third chapter. They show evidently that the qualitative behaviours of non-pinched systems are exactly the same with the corresponding unforced real systems, which the pinched ones are affected in a great degree by the pinched conditions. The second type of families is the quasi-periodically forced S-unimodal maps. S-unimodal maps are prototypes of periodic behaviours. We propose the mechanism for the states of periodicity in forced systems according to the forced terms, which is based on rigorous analysis of the S-unimodal maps and is substantiated by numerical evidences.
Moreover, our analysis of S-unimodal maps also demonstrates the mechanism of bifurcations that happen on the attractors of cycles of chaotic intervals. This is a new result for us about these well studied systems, given by Theorem D. Bifurcations of this type have been reported for decades, but they are only described in physical context. We explain their mechanism mathematically and show each of them is the reverse of a corresponding bifurcation of periodic orbit. A special significance of these reverse bifurcations is that, each correspondence pairs of them forms a unit of similarity in the transition of an S-unimodal family.
