The Classification Of Quasiperiodically Forced Circle Systems

The quasiperiodically forced circle system occur in various situations in physics and have been extensively studied as a source of examples of interesting dynamics. For instance, the qpf Arnold circle map can be taken as a model of oscillators forced with two or more incommensurate frequencies, and one specific form is the projection flow on the angle of quasiperiodic linear system. One of the classical example is Harper map, which serves as a model of quasiperiodic crystals([21]). In this thesis, we are de-voted to study the classification of quasiperiodically forced circle systems. Especially, we focus our attention on the local situations, i.e., about the perturbative systems.In the first chapter, we will introduce some notations and preliminaries that will be used in this thesis. We first introduce our research object:quasiperiodically forced circle systems, including the discrete version and the continuous version; and then we will present some other concepts:the fibred rotation number, q-invariant torus, lin-earization and some concepts and properties in the number theory.In the second chapter, we will consider the properties of quasiperiodically forced circle diffeomorphisms, for which the fibred rotation numbers are rational with respect to the forcing frequencies. We will discuss the existence of lower dimensional invariant tori for these systems. Firstly, we will prove for any analytic quasiperiodically forced circle diffeomorphisms(ω,εf), where f is fixed and ε is small, if ω is Diophantine and the fibred rotation number of the diffeomorphism remains constant in a unilateral neighborhood of ε=0, then the diffeomorphism has at least one analytic q-invariant torus, provided ε is small enough. Applying this result, we will obtain the same results for any analytic quasiperiodically forced circle diffeomorphisms (ω,<p/q,ω>+εf), where f is fixed and ε is small.In the third chapter, we will consider the properties of quasiperiodically forced cir-cle flows, provided the fibred rotation numbers are rational independent on the forcing frequencies. We shall discuss the linearization of these kind of systems beyond Brjuno condition. As for analytic nonlinear quasi-periodically forced circle flows (ω,ρ+f),with ω=(1, a),α∈R\Q, we will prove that for positive measure set of p, the system is C∞rotations linearizable and Cω almost linearizable provided that f is sufficiently small and a is not super-Liouvillean. Moreover, if ω=1/λ(1,a), we prove that the similar result holds and the smallness of the perturbation is independent of A. As a corollary, we obtain that the linearizable systems and systems with mode-locking is locally dense. These results can be viewed as the nonlinear counterpart of the reducibility results for quasi-periodic linear flow [25] and quasi-periodic cocycles [6].