The scaling of Arnol'd tongues for differentiable homeomorphisms of the circle

We show both for diffeomorphisms of the circle and for differentiable homeomorphisms that are not diffeomorphisms, that the widths of the Arnol'd tongues in a one parameter family scale asq ^–3 whenq is the denominator of the rotation number.Research supported by NSERC     

Rational rotation numbers for maps of the circle

We consider families of maps of the circle of degree 1 which are homeomorphisms but not diffeomorphisms, that is maps like

Delay-coupled discrete maps: Synchronization, bistability, and quasiperiodicity

The synchronization transition is studied in delay-coupled logistic maps. For low coupling, in-phase and out-of-phase synchronous dynamics coexist, and with increasing coupling there is a regime of quasiperiodicity before eventual attraction to a fixed point at a critical value of coupling that depends on the nonlinearity. The presence of a region of asynchrony separating two synchronized regimes—termed anomalous behaviour—has been observed earlier in continuous systems and is shown here to occur in delay mappings as well. There are regions of in-phase, anti-phase, and out-of-phase dynamics of periodic as well as chaotic attractors.     

Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study

We consider a two-parameter family of maps of the plane to itself. Each map has a fixed point in the first quadrant and is a diffeomorphism in a neighborhood of this point. For certain parameter values there is a Hopf bifurcation to an invariant circle, which is smooth for parameter values in a neighborhood of the bifurcation point. However, computer simulations show that the corresponding invariant set fails to be even topologically a circle for parameter values far from the bifurcation point. This paper is an attempt to elucidate some of the mechanisms involved in this loss of smoothness and alteration of topological type.     

Bifurcations of coupled bistable maps

We show how to analytically determine the existence and stability properties of fixed points of piecewise-linear coupled map lattices, then use this technique to investigate the bifurcations undergone by systems of diffusively-coupled bistable maps. The behaviour of various piecewise-linear and smooth models is compared, and features peculiar to piecewise-linear models are highlighted. Some examples of counter-intuitive behaviour enforced by the bifurcation scenario are given.     

Deterministic ratchets, circle maps, and current reversals

In this work we transform the deterministic dynamics of an overdamped tilting ratchet into a discrete dynamical map by looking stroboscopically at the continuous motion originally ruled by differential equations. We show that, for the simple and widely used case of periodic dichotomous driving forces, the resulting discrete map belongs to the class of circle homeomorphisms. This approach allows us to apply the well-known properties of such maps to derive the necessary and sufficient conditions that the ratchet potential must satisfy in order to have a vanishing current. Furthermore, as a consequence of the above, we show (i) that there is a class of periodic potentials which do not exhibit the rectification phenomenon in spite of their asymmetry and (ii) that current reversals occur in the deterministic case for a large class of ratchet potentials.     

Josephson junctions and circle maps

The return map of a differential equation for the current driven Josephson junction, or the damped driven pendulum, is shown numerically to be a circle map. Phase locking, noise and hysteresis, can thus be understood in a simple and coherent way. The transition to chaos is related to the development of a cubic inflection point. Recent theoretical results on universal behavior at the transition to chaos can readily be checked experimentally by studying I–V characteristics.     

Rotation sets for networks of circle maps

We consider continuous maps of the torus, homotopic to the identity, that arise from systems of coupled circle maps and discuss the relationship between network architecture and rotation sets. Our main result is that when the map on the torus is invertible, network architecture can force the set of rotation vectors to lie in a low-dimensional subspace. In particular, the rotation set for an all-to-all coupled system of identical cells must be a subset of a line.     

Universality and self-similarity in the bifurcations of circle maps

The bifurcation structure in a two-parameter family of circle maps is considered. These maps have a (topological) degree that may be different from one. A generalization of the rotation number is given and symmetries of the bifurcations in parameter space are described. Continuity arguments are used to establish the existence of periodic orbits. By plotting the locus of parameter values associated with superstable cycles, self-similar bifurcations are found. These bifurcations are a generalization of the familiar period-doubling cascade in maps with one extrema, to two-parameter maps with two extrema. Finally, a scheme for the global organization of bifurcation in these maps is proposed.     

Near-critical behavior for one-parameter families of circle maps

Past studies of systems showing mixed-mode oscillations have revealed behavior along arbitrarily chosen parameter paths similar to that on the critical surface marking the break-up of invariant tori. Observations of this behavior in a model of the Belousov-Zhabotinskii reaction is presented. Using the theory of circle maps, it is shown that near-critical behavior can arise along one-parameter paths.     

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.     

Comb structure in hairy boundaries: Some transition problems for circle maps

For some good two-parameter families of circle maps such as the sine family F_a,b (θ) = θ + a − (b/2π) sin 2πθ, we show that both the boundary of chaos and the boundary of the parameter region corresponding to a uniquely defined rotation number are not locally connected. We discuss the effects of this irregularity on the observation of the onset of chaos.     

Invariant circles and rotation bands in monotone twist maps

In this paper we show that the existence of certain orbits or minimal sets in an area-preserving monotone twist map is necessary and sufficient for the non-existence of invariant circles with specified rotation numbers. The necessity of these conditions follows from classic results of Birkhoff and recent results of Mather. The sufficiency of these conditions depends on the notion of a rotation band which associates a set of rotation numbers with a given orbit or invariant set. We also make some remarks on Mather's paper [M4]. In particular, we use his main theorem to give a lower bound on the width of the interval of rotation numbers associated with the zone of instability that contains the irrational whenf has no invariant circle with rotation number .     

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.     

Map dependence of the fractal dimension deduced from iterations of circle maps

Every orientation preserving circle mapg with inflection points, including the maps proposed to describe the transition to chaos in phase-locking systems, gives occasion for a canonical fractal dimensionD, namely that of the associated set of for whichf =+g has irrational rotation number. We discuss how this dimension depends on the orderr of the inflection points. In particular, in the smooth case we find numerically thatD(r)=D(r ^–1)=r ^–1/8.     

An integration formula for the square of moment maps of circle actions

The integration of the exponential of the square of the moment map of the circle action is studied by a direct stationary phase computation and by applying the Duistermaat-Heckman formula. Both methods yield two distinct formulas expressing the integral in terms of contributions from the critical set of the square of the moment map. Certain cohomological pairings on the symplectic quotient are computed explicitly using the asymptotic behavior of the two formulas.     

On the existence of fixed points of the composition operator for circle maps

In the theory of circle maps with golden ratio rotation number formulated by Feigenbaum, Kadanoff, and Shenker [FKS], and by Ostlund, Rand, Sethna, and Siggia [ORSS], a central role is played by fixed points of a certain composition operator in map space. We define a common setting for the problem of proving the existence of these fixed points and of those occurring in the theory of maps of the interval. We give a proof of the existence of the fixed points for a wide range of the parameters on which they depend.     

Computation of derivatives of the rotation number for parametric families of circle diffeomorphisms

In this paper we present a numerical method to compute derivatives of the rotation number for parametric families of circle diffeomorphisms with high accuracy. Our methodology is an extension of a recently developed approach to compute rotation numbers based on suitable averages of iterates of the map and Richardson extrapolation. We focus on analytic circle diffeomorphisms, but the method also works if the maps are differentiable enough. In order to justify the method, we also require the family of maps to be differentiable with respect to the parameters and the rotation number to be Diophantine. In particular, the method turns out to be very efficient for computing Taylor expansions of Arnold Tongues of families of circle maps. Finally, we adapt these ideas to study invariant curves for parametric families of planar twist maps.     

End-pumped optical bistability Tm,Ho:YLF laser

We experimentally demonstrate strong optical bistability in a 2 μm continuous wave Tm,Ho:YLF laser with a bulk crystal. The laser is end-pumped by a 792 nm fiber-coupled laser diode. The bistable region is as wide as 1.6 W and the jump power at the turning point is as high as 82.5 mW when the temperature of the laser crystal is kept at 253 K. The influences of the cavity length and the transmission of the output coupler on the characteristics of optical bistability are obtained. The theoretical analysis shows that the bistable output in the Tm,Ho:YLF lasers comes from the combined effects of the nonlinear saturation of ground state reabsorption, the energy transfer upconversion, and the excited state absorption.     

Theory of circle maps and the problem of one-dimensional optical resonator with a periodically moving wall

We consider the electromagnetic field in a cavity with a periodically oscillating perfectly reflecting boundary and show that the mathematical theory of circle maps leads to several physical predictions. Notably, well-known results in the theory of circle maps (which we review briefly) imply that there are intervals of parameters where the waves in the cavity get concentrated in wave packets whose energy grows exponentially. Even if these intervals are dense for typical motions of the reflecting boundary, in the complement there is a positive measure set of parameters where the energy remains bounded.