Rotation numbers for quasi-periodically forced monotone circle maps (original) (raw)

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

Dynamical Systems, 2012

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. Dedicated to the memory of Jaroslav Stark 1. Introduction Noninvertible circle maps have been used to model the breakdown of invariant tori in differential equations [5, 9], and there are many interesting results about the bifurcation structure for these maps [9, 10]. Whilst invertible maps of the circle have a unique rotation number measuring the average speed of orbits around the circle, noninvertible circle maps have a rotation interval: different orbits can lead to different rotation rates, but the set of all rotation rates is a closed interval, which may be a point [1, 9, 11]. If the differential equation being modelled is quasi-periodically forced then it is

Rotation sets for orbits of degree one circle maps

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 2002

Let F be the lifting of a circle map of degree one. In [Bamón et al., 1984] a notion of F-rotation interval of a point was given. In this paper we define and study a new notion of a rotation set of point which preserves more of the dynamical information contained in the sequences than the one preserved from [Bamón et al., 1984]. In particular, we characterize dynamically the endpoints of these sets and we obtain an analogous version of the Main Theorem of [Bamón et al., 1984] in our settings.

Observed Rotation Numbers in Families of Circle Maps

International Journal of Bifurcation and Chaos, 2001

Noninvertible circle maps may have a rotation interval instead of a unique rotation number. One may ask which of the numbers or sets of numbers within this rotation interval may be observed with positive probability in term of Lebesgue measure on the circle. We study this question numerically for families of circle maps. Both the interval and "observed" rotation numbers are computed for large numbers of initial conditions. The numerical evidence suggests that within the rotation interval only a very narrow band or even a unique rotation number is observed. These observed rotation numbers appear to be either locally constant or vary wildly as the parameter is changed. Closer examination reveals that intervals with wild variation contain many subintervals where the observed rotation numbers are locally constant. We discuss the formation of these intervals. We prove that such intervals occur whenever one of the endpoints of the rotation interval changes. We also examine the e...

Numerical computation of rotation numbers of quasi-periodic planar curves

Physica D: Nonlinear Phenomena, 2009

Recently, a new numerical method has been proposed to compute rotation numbers of analytic circle diffeomorphisms, as well as derivatives with respect to parameters, that takes advantage of the existence of an analytic conjugation to a rigid rotation. This method can be directly applied to the study of invariant curves of planar twist maps by simply projecting the iterates of the curve onto a circle. In this work we extend the methodology to deal with general planar maps. Our approach consists in computing suitable averages of the iterates of the map that allow us to obtain a new curve for which the direct projection onto a circle is well posed. Furthermore, since our construction does not use the invariance of the quasiperiodic curve under the map, it can be applied to more general contexts. We illustrate the method with several examples.

Optimal estimates on rotation number of almost periodic systems

Zeitschrift für angewandte Mathematik und Physik, 2006

In this paper, we will give some optimal estimates on the rotation number of the linear equationẍ + p(t)x = 0, and that of the asymmetric equation:ẍ + p(t)x + + q(t)x − = 0, where p(t) and q(t) are almost periodic functions and x + = max{x, 0}, x − = min{x, 0}. These estimates are obtained by introducing some kind of new norms in the spaces of almost periodic functions.

Rotation intervals of endomorphisms of the circle

Ergodic Theory and Dynamical Systems, 1984

The rotation number of a diffeomorphism f: S1 → S1 with lift is defined as . We investigate the case where f is an endomorphism. Then this limit may not exist and may depend on x. We investigate the set of limit points of , as a function of x.

Rotation numbers of discontinuous orientation-preserving circle maps revisited

The theory of circle homeomorphisms has a great number of deep results. However, sometimes continuity or single-valuedness of a circle map may be restrictive in theoretical constructions or applications. In this paper it is shown that some principal properties of circle homeomorphisms are inherited by the class of orientation-preserving circle maps. The latter class is rather broad and contains not only circle homeomorphisms but also a variety of non continuous maps arising in applications. Of course, even in cases when a property remains to be valid for orientation-preserving circle maps, absence of continuity sometimes results in noticeable changes of related proofs.

Computation and stability of periodic orbits of nonlinear mappings

2002

In this paper, we first present numerical methods that allow us to compute accurately periodic orbits in high dimensional mappings and demonstrate the effectiveness of our methods by computing orbits of various stability types. We then use a terminology for the different stability types, which is perfectly suited for systems with many degrees of freedom, since it clearly reflects the configuration of the eigenvalues of the corresponding monodromy matrix on the complex plane. Studying the distribution of these eigenvalues over the points of an unstable periodic orbit, we attempt to find connections between local dynamics and the global morphology of the orbit.

Self-rotation number using the turning angle

Physica D: Nonlinear Phenomena, 2000

The self-rotation number, as defined by Peckham, is the rotation rate of the image of a point about itself. Here we use the notion of "turning angle" to give a simplified algorithm to compute the selfrotation number for maps that "avoid an angle." We show that the orientation preserving Hénon map does avoid an angle. Moreover, the self-rotation number for orbits of the Hénon map can be computed once and for all at the anti-integrable limit by a simple algorithm depending upon the symbol sequence for the orbit.