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.     

Universality and scaling of period-doubling bifurcations in a dissipative distributed medium

A homogeneous medium, consisting of nonlinear elements, demonstrating transition to chaos via period-doubling bifurcations, is considered. The coupling between the elements is supposed to be of a dissipative type, i.e. it tends to equalize their instantaneous states. Using the renormalization group approach, the following scaling law for weakly inhomogeneous states near the critical point is obtained: at each period doubling the spatial scale increases by β=√2. On the basis of this law the scaling hypotheses for the transition to chaos in the semi-infinite and finite systems are proposed. The scaling properties are verified by the numerical calculations with a simple model.     

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.     

Control of degenerate Hopf bifurcations in three-dimensional maps

A feedback control method is proposed to create a degenerate Hopf bifurcation in three-dimensional maps at a desired parameter point. The particularity of this bifurcation is that the system admits a stable fixed point inside a stable Hopf circle, between which an unstable Hopf circle resides. The interest of this solution structure is that the asymptotic behavior of the system can be switched between stationary and quasi-periodic motions by only tuning the initial state conditions. A set of critical and stability conditions for the degenerate Hopf bifurcation are discussed. The washout-filter-based controller with a polynomial control law is utilized. The control gains are derived from the theory of Chenciner's degenerate Hopf bifurcation with the aid of the center manifold reduction and the normal form evolution.     

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.     

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

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.     

Transitivity and blowout bifurcations in a class of globally coupled maps

A class of globally coupled one dimensional maps is studied. For the uncoupled one dimensional map it is possible to compute the spectrum of Liapunov exponents exactly, and there is a natural equilibrium measure (Sinai–Ruelle–Bowen measure), so the corresponding `typical' Liapunov exponent may also be computed. The globally coupled systems thus provide examples of blowout bifurcations in arbitrary dimension. In the two dimensional case these maps have parameter values at which there is a transitive (topological) attractor which is a filled-in quadrilateral and, simultaneously, the synchronized state is a Milnor attractor.     

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.     

Transition to topological chaos for circle maps

Many biperiodic flows can be modelled by maps of a circle to itself. For such maps the transition from zero to positive topological entropy can be achieved in several ways. We describe all the possible routes for smooth circle maps, and discuss the relevance of our results to the transition to chaos for two-frequency systems.     

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.     

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.