Universality of k·3n and k·4n bifurcations in area-preserving maps

We study period-trebling and period-quadrupling bifurcations in two-dimensional reversible area-preserving maps. Our numerical results show that there are unique universal limiting behaviors in each of the period-trebling and period-quadrupling sequences.     

Universality and self-similarity of an energy-constrained sandpile model with random neighbors

We study an energy-constrained sandpile model with random neighbors. The critical behavior of the model is in the same universality class as the mean-field self-organized criticality sandpile. The critical energy E(c) depends on the number of neighbors n for each site, but the various exponents are independent of n. A self-similar structure with n-1 major peaks is developed for the energy distribution p(E) when the system approaches its stationary state. The avalanche dynamics contributes to the major peaks appearing at E(p(k))=2k/(2n-1) with k=1,2,...,n-1, while the fine self-similar structure is a natural result of the way the system is disturbed.     

Scalings in circle maps II

In this paper we consider one parameter families of circle maps with nonlinear flat spot singularities. Such circle maps were studied in [Circles I] where in particular we studied the geometry of closest returns to the critical interval for irrational rotation numbers of constant type. In this paper we apply those results to obtain exact relations between scalings in the parameter space to dynamical scalings near parameter values where the rotation number is the golden mean. Then results on [Circles I] can be used to compute the scalings in the parameter space. As far as we are aware, this constitutes the first case in which parameter scalings can be rigorously computed in the presence of highly nonlinear (and nonhyperbolic) dynamics.     

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.     

Universality from resonance renormalization of Hamiltonian maps

The Hamiltonian for a single island chain (pth-resonance) of the standard mapping is obtained using secular perturbation theory and the method of averaging. A local standard mapping is reconstituted, approximately, for a single island of that chain. The relation between the stochasticity parameter K? of the local mapping, and the parameter K of the original mapping is obtained, which constitutes a renormalization of p mapping iterations. Setting K?=K then determines a value of K for which each p-island chain is self-similar in all orders.     

Critical circle maps near bifurcation

We estimate harmonic scalings in the parameter space of a one-parameter family of critical circle maps. These estimates lead to the conclusion that the Hausdorff dimension of the complement of the frequency-locking set is less than 1 but not less than 1/3. Moreover, the rotation number is a Hölder continuous function of the parameter.Partially supported by KBN grant Iteracje i Fraktale #210909101.Partially supported by NSF Grant #DMS-9206793 and the Sloan Research Fellowship.     

On the fixed points for circle maps

Renormalization group transformations have been developed to study the critical behavior of circle maps. When the winding number equals the golden mean, the fixed point functions must satisfy two functional equations. However, it turns out that one of these equations already determines the fixed point solutions. It is shown that under certain conditions the second functional equation is automatically satisfied.     

Scalings in circle maps (I)

Letf be a flat spot circle map with irrational rotation number. Located at the edges of the flat spot are non-flat critical points (S: xAx ^v ,v1). First, we define scalings associated with the closest returns of the orbit of the critical point. Under the assumption that these scalings go to zero, we prove that the derivative of long iterates of the critical value can be expressed in the scalings. The asymptotic behavior of the derivatives and the scalings can then be calculated. We concentrate on the cases for which one can prove the above assumption. In particular, let one of the singularities be linear. These maps arise for example as the lower bound of the non-decreasing truncations of non-invertible bimodal circle maps. It follows that the derivatives grow at a sub-exponential rate.     

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

Trajectory scaling function for bifurcations in area-preserving maps on the plane

The trajectory scaling function for area-preserving maps on the plane is found using a calculation of the unstable manifold for the renormalization group operator R·T=Λ·T²·Λ^-1 with $\text{Λ=}\text{α 0}\text{0 β}$. Internal self-similarities of high order cycles and of power spectra are deduced.     

Further scaling properties for circle maps

It is shown that certain iterations of (k–1)tuples of commuting invertible circle maps whose rotation numbers are algebraic of degree k, show very similar scaling properties to those found by Feigenbaum et al. in the case k=2.     

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.     

Differentiable circle maps with a flat interval

We study weakly order preserving circle maps with a flat interval, which are differentiable even on the boundary of the flat interval. We obtain estimates on the Lebesgue measure and the Hausdorff dimension of the non-wandering set. Also, a sharp transition is found from degenerate geometry to bounded geometry, depending on the degree of the singularities at the boundary of the flat interval.Partially supported by KBN grant Iteracje i Fraktale #210909101.     

Quasi-genericity of bifurcations to high dimensional invariant tori for maps

We consider a family of maps in a Banach spaceE near the situation when the derivative at the fixed point has two pairs of complex eigenvalues lying on the unit circle, the other part of the spectrum being strictly inside the unit disc. We focus our attention on the region of the parameter space where the truncated normal form of the maps shows a bifurcation of a family of invariantT ¹-circles into a family of invariantT ²-tori. We show that this problem needs a 3 dimensional parameter unfolding and that, for the complete maps, bifurcation occurs at points _,, where is the rotation number on the non-normally hyperbolicT ¹-circle, ande ^±2i are the eigenvalues of the constant matrix conjugated to the non-contracting part of the linearization on the normal fiber bundle overT ¹. Making some non-resonance and diophantine assumptions on (, ) leading to a positive measure Cantor set inT ², we show that in paraboloïdal regions of the 3 dim. parameter space we have clean bifurcations as for the truncated normal form. The complement of these regions forms a set of bubbles such as the ones obtained by Chenciner in [Chen] for a codimension 2 problem for maps in ². The main tool here is a generalization for a matrix function onT ¹, close to a constant, of the quasi-conjugacy to a constant, modulo a minimum of additional parameters (moved quasi-conjugacy). For the infinite dimensional case we use aC decoupling result on the angular dependent linear parts into a contraction, still angular dependent, and another part quasi-conjugated to a constant matrix. This type of analysis applies for a wide range of problems, where truncated normal forms of the maps give bifurcations fromT ⁿ toT ⁿ⁺¹ tori, and this needs a (n+1)-dimensional parameter unfolding.We gratefully acknowledge the DRET (contrat 86/1445) who supported one of the authors (J.L.) during this work. This research has been also supported by the E.E.C. contract No. ST 2J-0316-C (EDB) on Mathematical problems in nonlinear Mechanics     

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.     

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.