We study parameter families of quasiperiodically forced (qpf) circle maps with Diophantine frequency. Under certain C 1 -open conditions concerning their geometry, we prove that these families exhibit nonuniformly hyperbolic behaviour, often referred to as the existence of strange nonchaotic attractors, on parameter sets of positive measure. This provides a nonlinear version of results by Young on quasiperiodic SL(2, R)-cocycles and complements previous results in this direction which hold for sets of frequencies of positive measure, but did not allow for an explicit characterisation of these frequencies.As an application, we study a qpf version of the Arnold circle map and show that the Arnold tongue corresponding to rotation number 1/2 collapses on an open set of parameters.The proof requires to perform a parameter exclusion with respect to some twist parameter and is based on the multiscale analysis of the dynamics on certain dynamically defined critical sets. A crucial ingredient is to obtain good control on the parameter-dependence of the critical sets. Apart from the presented results, we believe that this step will be important for obtaining further information on the behaviour of parameter families like the qpf Arnold circle map.
Poincaré's classification of the dynamics of homeomorphisms of the circle is one of the earliest, but still one of the most elegant, classification results in dynamical systems. Here we generalize this to quasiperiodically forced circle homeomorphisms homotopic to the identity, which have been the subject of considerable interest in recent years. Herman already showed two decades ago that a unique rotation number exists for all orbits in the quasiperiodically forced case. However, unlike the unforced case, no a priori bounds exist for the deviations from the average rotation. This plays an important role in the attempted classification, and in fact we define a system as ρ-bounded if such deviations are bounded and as ρ-unbounded otherwise. For the ρ-bounded case we prove a close analogue of Poincaré's result: if the rotation number is rationally related to the rotation rate on the base then there exists an invariant strip (the appropriate analogue for fixed or periodic points in this context), otherwise the system is semi-conjugate to an irrational translation of the torus. In the ρ-unbounded case, where neither of these two alternatives can occur, we show that the dynamics are always topologically transitive.
We give an equivalent condition for the existence of a semi-conjugacy to an irrational rotation for conservative homeomorphisms of the two-torus. This leads to an analogue of Poincaré's classification of circle homeomorphisms for conservative toral homeomorphisms with unique rotation vector and a certain bounded mean motion property. For minimal toral homeomorphisms, the result extends to arbitrary dimensions. Further, we provide a basic classification for the dynamics of toral homeomorphisms with all points non-wandering.
In the study of quasiperiodically forced systems invariant graphs have a special significance. In some cases, it was already possible to deduce statements about the invariant graphs of certain classes of systems from properties of the fibre maps. Here, we study quasiperiodically forced interval maps which are monotonically increasing and have negative Schwarzian derivative. First, we derive some basic results which only require monotonicity. Then we give a classification, with respect to the number and to the Lyapunov exponents of invariant graphs, for this class of systems. It turns out that the possibilities for the invariant graphs are exactly analogous to those for the fixed points of the unperturbed fibre maps.
We propose a general mechanism by which strange non-chaotic attractors (SNA) are created during the collision of invariant curves in quasiperiodically forced systems. This mechanism, and its implementation in different models, is first discussed on an heuristic level and by means of simulations. In the considered examples, a stable and an unstable invariant circle undergo a saddle-node bifurcation, but instead of a neutral invariant curve there exists a strange non-chaotic attractor-repeller pair at the bifurcation point. This process is accompanied by a very characteristic behaviour of the invariant curves prior to their collision, which we call 'exponential evolution of peaks'.This observation is then used to give a rigorous description of non-smooth saddle-node bifurcations and to prove the existence of SNA in certain parameter families of quasiperiodically forced interval maps. The non-smoothness of the bifurcations and the occurrence of SNA is established via the existence of 'sinksource-orbits', meaning orbits with positive Lyapunov exponent both forwards and backwards in time.The important fact is that the presented approach allows for a certain amount of flexibility, which makes it possible to treat different models at the same time -even if the results presented here are still subject to a number of technical constraints. This is unlike previous proofs for the existence of SNA, which are all restricted to very specific classes and depend on very particular properties of the considered systems. In order to demonstrate this flexibility, we also discuss the application of the results to the Harper map, an example which is well-known from the study of discrete Schrödinger operators with quasiperiodic potentials. Further, we prove the existence of strange non-chaotic attractors with a certain inherent symmetry, as they occur in non-smooth pitchfork bifurcations.
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