Conformal geometry and invariants of 3-strand Brownian braids

We propose a simple geometrical construction of topological invariants of 3-strand Brownian braids viewed as world lines of 3 particles performing independent Brownian motions in the complex plane z. Our construction is based on the properties of conformal maps of doubly-punctured plane z to the universal covering surface. The special attention is paid to the case of indistinguishable particles. Our method of conformal maps allows us to investigate the statistical properties of the topological complexity of a bunch of 3-strand Brownian braids and to compute the expectation value of the irreducible braid length in the non-Abelian case.     

Renormalization group operators for maps and universal scaling of universal scaling exponents

Renormalization group (RG) methods provide a unifying framework for understanding critical behaviour, such as transition to chaos in area-preserving maps and other dynamical systems, which have associated with them universal scaling exponents. Recently, de la Llave et al. (2007) [10] have formulated the Principle of Approximate Combination of Scaling Exponents (PACSE for short), which relates exponents for different criticalities via their combinatorial properties. The main objective of this paper is to show that certain integrable fixed points of RG operators for area-preserving maps do indeed follow the PACSE.     

Universal properties of maps on an interval

We consider itcrates of maps of an interval to itself and their stable periodic orbits. When these maps depend on a parameter, one can observe period doubling bifurcations as the parameter is varied. We investigate rigorously those aspects of these bifurcations which are universal, i.e. independent of the choice of a particular one-parameter family. We point out that this universality extends to many other situations such as certain chaotic regimes. We describe the ergodic properties of the maps for which the parameter value equals the limit of the bifurcation points.     

Universal properties of the transition from quasi-periodicity to chaos in dissipative systems

An exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi-periodic regime is held at a fixed irrational value. Our approach is a natural extension of K.A.M. theory to strong coupling. Most of our analysis is for analytic circle maps. We have found a strong coupling fixed point where invertibility is lost, which describes the universal features of the transition to chaos. We find numerically that any two such critical maps with the same winding number are C¹ conjugate. It follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency.When the winding number has a periodic continued fraction, our renormalization transform has a fixed point and spectra are self similar in addition. For a set of non-periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub-space of all critical maps. Physically one finds singular and universal spectra that do not scale.     

Universal scaling behaviour for iterated maps in the complex plane

According to the theory of Schröder and Siegel, certain complex analytic maps possess a family of closed invariant curves in the complex plane. We have made a numerical study of these curves by iterating the map, and have found that the largest curve is a fractal. When the winding number of the map is the golden mean, the fractal curve has universal scaling properties, and the scaling parameter differs from those found for other types of maps. Also, for this winding number, there are universal scaling functions which describe the behaviour asn→∞ of theQ _n ^th iterates of the map, whereQ _n is then ^th Fibonacci number.     

Cyclic star products and universalities in symbolic dynamics of trimodal maps

Cyclic star products for the triple superstable kneading (TSSK) sequences are presented for symbolic dynamics of trimodal maps of endomorphisms on the interval. Feigenbaum’s metric universalities in unimodal maps are generalized to trimodal maps. Four equal-value universal convergent rates {δ_a,δ_c,δ_η,δ_a,c,η} and three universal scaling factors {_C,_D,_E} are first obtained.     

Classical solutions of higher-dimensional nonlinear sigma models on spheres

We construct several classical solutions of higher-dimensional nonlinear sigma models on spheres. These solutions are characterized by typical topological maps, in particular, four famous Hopf maps and the universal maps of theK-theory.Dedicated to the late Professor Shichiro Oka.     

Renormalization Group for Strongly Coupled Maps

Systems of strongly coupled chaotic maps generically exhibit collective behavior emerging out of extensive chaos. We show how the well-known renormalization group (RG) of unimodal maps can be extended to the coupled systems, and in particular to coupled map lattices (CMLs) with local diffusive coupling. The RG relation derived for CMLs is nonperturbative, i.e., not restricted to a particular class of configurations nor to some vanishingly small region of parameter space. After defining the strong-coupling limit in which the RG applies to almost all asymptotic solutions, we first present the simple case of coupled tent maps. We then turn to the general case of unimodal maps coupled by diffusive coupling operators satisfying basic properties, extending the formal approach developed by Collet and Eckmann for single maps. We finally discuss and illustrate the general consequences of the RG: CMLs are shown to share universal properties in the space-continuous limit which emerges naturally as the group is iterated. We prove that the scaling properly ties of the local map carry to the coupled systems, with an additional scaling factor of length scales implied by the synchronous updating of these dynamical systems. This explains various scaling laws and self-similar features previously observed numerically.     

Renormalization Approach to Optimal Limiter Control in 1-D Chaotic Systems

Optimal limiter control of chaos in 1-d systems is described by flat-topped maps. When we study the properties of this control by a bifurcation analysis of the latter, we find partial universal behavior. The optimality of the control method is expressed by an exponentially fast control onto selected periodic orbits, making targeting algorithms idle.     

Dynamics of simple one-dimensional maps under perturbation

It is known that the one-dimensional discrete maps having single-humped nonlinear functions with the same order of maximum belong to a single class that shows the universal behaviour of a cascade of period-doubling bifurcations from stability to chaos with the change of parameters. This paper concerns studies of the dynamics exhibited by some of these simple one-dimensional maps under constant perturbations. We show that the “universality” in their dynamics breaks down under constant perturbations with the logistic map showing different dynamics compared to the other maps. Thus these maps can be classified into two types with respect to their response to constant perturbations. Unidimensional discrete maps are interchangeably used as models for specific processes in many disciplines due to the similarity in their dynamics. These results prove that the differences in their behaviour under perturbations need to be taken into consideration before using them for modelling any real process.     

Fully developed chaotic 1−d maps

Single-hump 1–d maps are investigated which generate ergodic process on an interval mapped everywhere two-to-one onto itself. Introducing a new transformation transverse to conjugation it is shown that such maps are related by smooth transformations to each other. It is found that each of the families consisting of conjugate maps contains a map everywhere expanding and producing ergodic iterations according to the uniform probability density. The general framework is used to construct maps together with their probability density functions. Quantities characterizing the dynamics are calculated and their parameter dependence while maintaining the fully developed chaotic state is studied. Furthermore, universal maps exhibiting fully developed chaos are considered.     

Influence of dynamical noise on time series generated by nonlinear maps

We consider periodic and chaotic dynamics of discrete nonlinear maps in the presence of dynamical noise. We show that dynamical noise corrupting dynamics of a nonlinear map may be considered as a measurement “pseudonoise” with the distribution determined by the Jacobian of the map. The formula for the distribution of the measurement “pseudonoise” for one-dimensional quadratic maps has also been obtained in an explicit form. We expect that our results apply to an arbitrary distribution of low-level dynamical noise and hope that these results could help to find a universal method of discriminating dynamical from measurement noise.     

Preparation and characterization of polyethylene/TiO2 nanocomposites

The rheological behaviour, dispersion, crystallization behavior, mechanical properties, fracture surface morphology of polyethylene (PE)/TiO₂ nanocomposites prepared by melt compounding were investigated using rheometer, energy dispersive X-ray spectrometer (EDX), polarized microscopy, impact tester, universal testing machine and field-emission scanning electron microscopy (FE-SEM). The rheological analysis indicated a fine dispersion of TiO₂ during the melt compounding. The large scaled surface dispersion of TiO₂ nanoparticles was revealed by the EDX composition distribution maps. The introduction of 2.0 wt% TiO₂ in composites improved the mechanical properties significantly compared to neat PE, and resulted in 45% increase in notched impact strength. Moreover, the further analysis and discussion showed the mechanical properties of the composites were controlled by the dispersion conditions of TiO₂ and its nucleating effect on PE crystallization.     

The transition to chaos for a special solution of the area-preserving quadratic map

Following previous work on chaotic boundaries of half-plane Hamiltonian maps a special solution of the area-preserving quadratic map is introduced and investigated. The breakdown of regular bounded motion on invariant curves is found from the radius of convergence of a power series whose successive terms oscillate wildly due to the presence of small divisors. Previous techniques for taming such series are found to be insufficient and new ones are introduced.It is found that half-plane Hamiltonian maps appear to have certain universal features and that the chaotic boundary has similarities to the boundaries of Siegel domains of complex conformal maps.The chaotic boundary function α_c(ν) has some interesting new features which are not fully understood.     

Non-asymptotic corrections to orbital scaling in period doubling maps

An extension of Feigenbaum's scaling laws for orbital sequences is presented for one-dimensional maps with quadratic maxima. We find that two universal constants, and combinations of these with the Feigenbaum scaling factors, characterize this generalized approach to chaos. Suggestions for future investigations are presented.     

Quantitative Universality for a Class of Weakly Chaotic Systems

We consider a general class of intermittent maps designed to be weakly chaotic, i.e., for which the separation of trajectories of nearby initial conditions is weaker than exponential. We show that all its spatio and temporal properties, hitherto regarded independently in the literature, can be represented by a single characteristic function ?. A universal criterion for the choice of ? is obtained within the Feigenbaum’s renormalization-group approach. We find a general expression for the dispersion rate ζ(t) of initially nearby trajectories and we show that the instability scenario for weakly chaotic systems is more general than that originally proposed by Gaspard and Wang (Proc. Natl. Acad. Sci. USA 85:4591, 1988). We also consider a spatially extended version of such class of maps, which leads to anomalous diffusion, and we show that the mean squared displacement satisfies σ ²(t)～ζ(t). To illustrate our results, some examples are discussed in detail.     

Universal multifractal properties of circle maps from the point of view of critical phenomena I. Phenomenology

The strange attractor for maps of the circle at criticality has been shown to be characterized by a remarkable infinite set of exponents. This characterization by an infinite set of exponents has become known as the multifractal approach. The present paper reformulates the multifractal properties of the strange attractor in a way more akin to critical phenomena. This new approach allows one to study the universal properties of both the critical point and of its vicinity within the same framework, and it allows universal properties to be extracted from experimental data in a straightforward manner. Obtaining Feigenbaum's scaling function from the experimental data is, by contrast, much more difficult. In addition to the infinite set of exponents, universal amplitude ratios here appear naturally. To study the crossover region near criticality, a correlation time, which plays a role analogous to the correlation length in critical phenomena, is introduced. This new approach is based on the introduction of a joint probability distribution for the positive integer moments of the closest-return distances. This joint probability distribution is physically motivated by the large fluctuations of the multifractal moments with respect to the choice of origin. The joint probability distribution has scaling properties analogous to those of the free energy close to a critical point.     

Parameter renormalization of maps based on potential function

A systematic way for deriving the parameter renormalization group equation for one-dimensional maps is presented and the critical behavior of periodic doubling is investigated. Introducing a formal potential function in one-parameter cases, it is shown that accumulation points correspond to local potential maxima and universal constants are easily determined. The estimates of accumulation points and universal constants match the known values asymptotically when the order of potential grows large. The potential function shows scaling in the parameter space with the universal convergent rate at the accumulation point similar to the Feigenbaum universal function. For two-parameter cases, a parameter reduction transformation is found to be useful to determine some important fixed points. A locally defined potential function is introduced and its scaling property is discussed. (c) 1997 American Institute of Physics.     

Coherent state map quantization in a Hermitian-like setting

For vector bundles having an involution on the base space, Hermitian-like structures are defined in terms of such an involution. We prove a universality theorem for suitable self-involutive reproducing kernels on Hermitian-like vector bundles. This result relies on pullback operations involving the tautological bundle on the Grassmann manifold of a Hilbert space and exhibits the aforementioned reproducing kernels as pullbacks of universal reproducing kernels that live on the Hermitian-like tautological bundle. To this end we use a certain type of classifying morphisms, which are geometric versions of the coherent state maps from quantum theory. As a consequence of that theorem, we obtain some differential geometric properties of these reproducing kernels in this setting.