Forced Circle Maps Research Articles

$$\rho $$-bounded orbits and minimal sets for generalized quasiperiodically forced circle maps

$$\rho $$-bounded orbits and minimal sets for generalized quasiperiodically forced circle maps

Density of mode-locking property for quasi-periodically forced Arnold circle maps

Abstract We show that the mode-locking region of the family of quasi-periodically forced Arnold circle maps with a topologically generic forcing function is dense. This gives a rigorous verification of certain numerical observations in [M. Ding, C. Grebogi and E. Ott. Evolution of attractors in quasiperiodically forced systems: from quasiperiodic to strange nonchaotic to chaotic. Phys. Rev. A 39(5) (1989), 2593–2598] for such forcing functions. More generally, under some general conditions on the base map, we show the density of the mode-locking property among dynamically forced maps (defined in [Z. Zhang. On topological genericity of the mode-locking phenomenon. Math. Ann. 376 (2020), 707–72]) equipped with a topology that is much stronger than the $C^0$ topology, compatible with smooth fiber maps. For quasi-periodic base maps, our result generalizes the main results in [A. Avila, J. Bochi and D. Damanik. Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts. Duke Math. J.146 (2009), 253–280], [J. Wang, Q. Zhou and T. Jäger. Genericity of mode-locking for quasiperiodically forced circle maps. Adv. Math.348 (2019), 353–377] and Zhang (2020).

Genericity of mode-locking for quasiperiodically forced circle maps

We show that a generic quasiperiodically forced circle homeomorphism is mode-locked: the rotation number in the fibres is rationally related to the rotation number in the base and it is stable under small perturbations of the system. As a consequence, this implies that for a generic parameter family of quasiperiodically forced circle homeomorphisms satisfying a twist condition, the graph of the rotation number as a function of the parameter is a devil's staircase.

Abundance of Mode-Locking for Quasiperiodically Forced Circle Maps

We show that a generic quasiperiodically forced circle homeomorphism is mode-locked: the rotation number in the fibres is rationally related to the rotation number in the base and it is stable under small perturbations of the system. As a consequence, this implies that for a generic parameter family of quasiperiodically forced circle homeomorphisms satisfying a twist condition, the graph of the rotation number as a function of the parameter is a devil's staircase.

Universal scaling of rotation intervals for quasi-periodically forced circle maps

We introduce a simplifying assumption which makes it possible to approximate the rotation number of an invertible quasi-periodically forced circle map by an integral in the limit of large forcing. We use this to describe universal scaling laws for the width of the non-trivial rotation interval of non-invertible quasi-periodically forced circle maps in this limit, and compare the results with numerical simulations.

Subdiffusion due to strange nonchaotic dynamics: A numerical study

We numerically investigate diffusion phenomena in quasiperiodically forced systems with spatially periodic potentials using a lift of the quasiperiodically forced circle map and a quasiperiodically forced damped pendulum. These systems exhibit several types of dynamics: quasiperiodic, strange nonchaotic, and chaotic. The strange nonchaotic and chaotic dynamics induce deterministic diffusion of orbits. The diffusion type gradually changes from logarithmic to subdiffusive within a strange nonchaotic regime and finally becomes normal in a chaotic regime. Fractal time-series analysis shows that the subdiffusion is caused by the antipersistence property of strange nonchaotic motion.

Strange Non-Chaotic Attractors in Quasiperiodically Forced Circle Maps

The occurrence of strange non-chaotic attractors (SNA) in quasiperiodically forced systems has attracted considerable interest over the last two decades, in particular since it provides a rich class of examples for the possibility of complicated dynamics in the absence of chaos. Their existence was first described by Millions̆c̆ikov (and later by Vinograd and also Herman) for quasiperiodic $${\rm SL(2, {\mathbb R})}$$ -cocycles and by Grebogi et al (and later Keller) for so-called pinched skew products. However, except for these two particular classes there are still hardly any rigorous results on the topic, despite a large number of numerical studies confirming the widespread existence of SNA in quasiperiodically forced systems. Here, we prove the existence of SNA in quasiperiodically forced circle maps under rather general conditions, which can be stated in terms of $${{\mathcal C}^1}$$ -estimates. As a consequence, we obtain the existence of SNA for parameter sets of positive measure in suitable parameter families. These SNA carry the unique physical measure of the system, which determines the behaviour of Lebesgue-almost all initial conditions. Finally, we show that the dynamics are minimal in the considered situations. The results apply in particular to a forced version of the Arnold circle map. For this example, we also describe how the first Arnold tongue collapses and looses its regularity due to the presence of strange non-chaotic attractors and a related unbounded mean motion property.

ON PERSISTENCE AND STABILITY OF COUPLED MAP LATTICES

Linear stability of homogeneous and inhomogeneous steady state solutions for coupled map lattices (CML) are studied. Some persistence problems in CML are discussed and applied to immune and an economic models. The stability results are applied to CML motivated by forced circle map, Winfree model, Josephson junction arrays and an ecological CML model. Global stability results for some CML systems are given.

The structure of mode-locked regions in quasi-periodically forced circle maps

Using a mixture of analytic and numerical techniques we show that the mode-locked regions of quasi-periodically forced Arnold circle maps form complicated sets in parameter space. These sets are characterized by ‘pinched-off’ regions, where the width of the mode-locked region becomes very small. By considering general quasi-periodically forced circle maps we show that this pinching occurs in a broad class of such maps having a simple symmetry.

Fine structure of mode-locked regions of the quasi-periodically forced circle map

Perturbation theory is used to describe the structure of mode-locked tongues for the quasi-periodically forced circle map near parameter values at which the tongues width is very small. The analysis reveals that the tongues consist of interlocking `bubbles' bounded by saddlenode bifurcations.

Intermittency and strange nonchaotic attractors in quasi-periodically forced circle maps

A possible mechanism for the creation of strange nonchaotic attractors close to the boundary of mode-locked tongues in a family of maps of the torus is described. This mechanism is based on the numerical observation that there are parameter values on the boundary of the mode-locked tongues at which the saddlenode bifurcation of invariant curves is not smooth, and assumptions about the nature of intermittency just outside the mode-locked tongues.

Evolution of complex oscillations in a quasiperiodically forced chain

We investigate complex dynamics along the chain of quasiperiodically forced circle maps. We present numerical evidence for the development of a strange nonchaotic behavior within a wide range of spatial parameters. The bifurcations and properties of nontrivial attracting sets are studied.

Strange non-chaotic attractor in a quasiperiodically forced circle map

We show that in the quasiperiodically forced circle map strange non-chaotic attractors can appear for non-linearities far from the border of chaos. The destruction of a two-frequency quasiperiodic torus connected with the appearance of a strange non-chaotic attractor is described in detail. This strange non-chaotic attractor is characterized by logarithmically slow diffusion of the phase. It is shown that in this regime the high-order phase-locking states disappear and the rotation number varies rather smoothly with the parameters.