圆映象的符号动力学

用符号动力学的方法对圆映象的Farey序列、Farey序列与M.S.S.序列的＊积、二元树及各Farey数的新生轨道进行了研究。 关键词：     

An embedding of the Farey web in the parameter space of simple families of circle maps

We describe an embedding of the Farey web, an extension of the better-known Farey tree, in the parameter space of simple families of circle maps. We also discuss some consequences of this embedding on the organization of frequency-locking and on topological properties of the boundary between simple and complicated dynamics.     

Mode-locking of self-generated oscillations in a semiconductor model for low-temperature impurity breakdown

The nonlinear dynamic behaviour of a model for self-generated, spatially coupled current oscillations in two separated parts of a semiconductor is analysed. The model involves impurity impact ionization, nonlinear energy relaxation of hot carriers, and energy exchange between the two subsystems. Quasiperiodicity and mode-locking are obtained, and characterized by a suitably defined rotation number and a spectral bifurcation diagram. The mode-locking structure is found to obey the Farey tree ordering, and can be understood on the basis of the circle map theory, assuming a particular path in the two-dimensional phase diagram of the circle map.     

Thermodynamics of the Farey Fraction Spin Chain

We consider the Farey fraction spin chain, a one-dimensional model defined on (the matrices generating) the Farey fractions. We extend previous work on the thermodynamics of this model by introducing an external field h. From rigorous and more heuristic arguments, we determine the phase diagram and phase transition behavior of the extended model. Our results are fully consistent with scaling theory (for the case when a “marginal” field is present) despite the unusual nature of the transition for h = 0, and the presence of long-range forces.     

Counting spanning trees in a small-world Farey graph

The problem of spanning trees is closely related to various interesting problems in the area of statistical physics, but determining the number of spanning trees in general networks is computationally intractable. In this paper, we perform a study on the enumeration of spanning trees in a specific small-world network with an exponential distribution of vertex degrees, which is called a Farey graph since it is associated with the famous Farey sequence. According to the particular network structure, we provide some recursive relations governing the Laplacian characteristic polynomials of a Farey graph and its subgraphs. Then, making use of these relations obtained here, we derive the exact number of spanning trees in the Farey graph, as well as an approximate numerical solution for the asymptotic growth constant characterizing the network. Finally, we compare our results with those of different types of networks previously investigated.     

Recurrence Times in Quasi-Periodic Motion: Statistical Properties,Role of Cell Size,Parameter Dependence

We investigate the probabilistic properties of recurrence times for the simplest form of aperiodic deterministic dynamics, quasi-periodic motion. Previous results using number theory techniques predict two fundamental recurrence times for uniform quasi-periodic motion on a two-dimensional torus, while no analogous analytic result seems to exist for higher dimensional tori. The two-dimensional uniform case is reanalyzed from a more geometric point of view and new, workable expressions are derived that enable us fully to understand and predict the recurrence phenomenon and to analyze its parameter dependence. Emphasis is placed on the statistical properties and, in particular, on the variability of recurrence times around their mean, in relation to local Farey tree structure. Higher-dimensional tori are considered, and seen to also display a high variability in their finite-time recurrence behavior. The results are finally extended to the non-uniform quasi-periodic case.     

A Fully Magnetizing Phase Transition

We analyze the Farey spin chain, a one-dimensional spin system with long-range multibody interactions. Using a polymer model technique, we show that when the temperature is decreased below the (single) critical temperature T _c=1/2, the magnetization jumps from zero to one.     

多种形式的加权广义Farey组织网络金字塔的复杂性

构建了三类确定性加权广义Farey组织的网络金字塔.理论推导并数值计算了网络金字塔的拓扑性质(度分布、平均最短路径、平均聚类系数和相称性系数等),进而将Farey序列作为网络节点的确定、随机和混合的三种权重值,以此为基础计算并拟合了三类网络金字塔的点的强度分布和边的权重分布.计算结果初步揭示了加权广义Farey组织的网络金字塔的复杂性特征,有助于了解一些实际网络的复杂性和多样性.     

Stern-Brocot trees in cascades of mixed-mode oscillations and canards in the extended Bonhoeffer-van der Pol and the FitzHugh-Nagumo models of excitable systems

We report a systematic two-parameter study of the organization of mixed-mode oscillations and period-adding sequences observed in an extended Bonhoeffer-van der Pol and in a FitzHugh-Nagumo oscillator. For both systems, we construct isospike diagrams and show that the number of spikes of their periodic oscillations are organized in a remarkable hierarchical way, forming a Stern-Brocot tree. The Stern-Brocot tree is more general than the Farey tree. We conjecture the Stern-Brocot tree to also underlie the hierarchical structure of periodic oscillations of other systems supporting mixed-mode oscillations.     

Scalings of mixed-mode regimes in a simple polynomial three-variable model of nonlinear dynamical systems

We describe scaling laws for a control parameter for various sequences of bifurcations of the LSn mixed-mode regimes consisting of single large amplitude maximum followed by n small amplitude peaks. These regimes are obtained in a normalized version of a simple three-variable polynomial model that contains only one nonlinear cubic term. The period adding bifurcations for LSn patterns scales as 1/n at low n and as 1/n2 at sufficiently large values of n. Similar scaling laws 1/k at low k and 1/k2 at sufficiently high values of k describe the period adding bifurcations for complex k(LSn)(LS(n + 1)) patterns. A finite number of basic LSn patterns and infinite sequences of complex k(LSn)(LS(n + 1)) patterns exist in the model. Each periodic pattern loses its stability by the period doubling bifurcations scaled by the Feigenbaum law. Also an infinite number of the broken Farey trees exists between complex periodic orbits. A family of 1D return maps constructed from appropriate Poincaré sections is a very fruitful tool in studies of the dynamical system. Analysis of this family of maps supports the scaling laws found using the numerical integration of the model.     

Quantum accelerator modes from the Farey tree

We show that mode locking finds a purely quantum nondissipative counterpart in atom-optical quantum accelerator modes. These modes are formed by exposing cold atoms to periodic kicks in the direction of the gravitational field. They are anchored to generalized Arnol'd tongues, parameter regions where driven nonlinear classical systems exhibit mode locking. A hierarchy for the rational numbers known as the Farey tree provides an ordering of the Arnol'd tongues and hence of experimentally observed accelerator modes.     

Molecule formation and the Farey tree in the one-dimensional Falicov-Kimball model

The ground-state configurations of the one-dimensional Falicov-Kimball model are studied exactly with numerical calculations revealing unexpected effects for small interaction strength. In neutral systems we observe molecular formation, phase separation, and changes in the conducting properties; while in nonneutral systems the phase diagram exhibits Farey tree order (Aubry sequence) and a devil's staircase structure. Conjectures are presented for the boundary of the segregated domain and the general structure of the ground states.Dedicated to Prof. Philippe Choquard, on the occasion of his 65th birthday, and, by anticipation, in honor of his retirement.     

一个一维CI相变模型的进一步研究

对一个一维CI相变模型进行进一步研究。外势周期D具有一个临界值Dc。D < c时的相图与D > c时的相图非常地不同。当D < c时,相图随D而变化,且在D的大部分区域显示出Farey树结构。当D > c时,相图不再变化,且不满足Farey树结构。通过有效势F(u)研究了相变过程。     

Trace maps as 3D reversible dynamical systems with an invariant

One link between the theory of quasicrystals and the theory of nonlinear dynamics is provided by the study of so-called trace maps. A subclass of them are mappings on a one-parameter family of 2D surfaces that foliate ³ (and also ³). They are derived from transfer matrix approaches to properties of 1D quasicrystals. In this article, we consider various dynamical properties of trace maps. We first discuss the Fibonacci trace map and give new results concerning boundedness of orbits on certain subfamilies of its invariant 2D surfaces. We highlight a particular surface where the motion is integrable and semiconjugate to an Anosov system (i.e., the mapping acts as a pseudo-Anosov map). We identify properties of symmetry and reversibility (time-reversal symmetry) in the Fibonacci trace map dynamics and discuss the consequences for the structure of periodic orbits. We show that a conservative period-boubling sequence can be identified when moving through the one-parameter family of 2D surfaces. By using generator trace maps, in terms of which all trace maps obtained from invertible two-letter substitution rules can be expressed, we show that many features of the Fibonacci trace map hold in general. The role of the Fricke character , its symmetry group, and reversibility for the Nielsen trace maps are described algebraically. Finally, we outline possible higher-dimensional generalizations.     

Multiple time scale analysis of two models for the peroxidase-oxidase reaction

A multiple time scale analysis of two four-variable models of the peroxidase-oxidase reaction, the DOP, and the Olsen model, is carried out. It is shown that autonomous limit cycle oscillations are exhibited by the fast subsets of these two models, but only in certain regions of parameter space, confirming the prior suggestion that the slow variable (NADH) is not essential for oscillatory behavior. However, it is found that the slow variable is essential for oscillatory behavior over other ranges of parameter values, and is always essential for complex oscillatory and chaotic behavior. This latter conclusion is based on a study involving driving the fast subset with a sinusoidally varying (NADH). This study suggests the level of coupling between fast and slow variables of an autonomous system necessary to cause the chaos observed in the DOP model. Further study of the driven system allows for the identification of a natural period of the nonoscillatory but bistable fast subsystem and a set of rules for applying a parametric driving in such a way as to generate a more complete Farey sequence from a truncated Farey sequence. These conclusions are used to compare the very similar DOP and Olsen models, which, nevertheless, exhibit quite different Farey sequences and routes to chaos. (c) 1995 American Institute of Physics.     

多种确定性广义Farey组织的网络金字塔

提出复杂网络的另一类家族的确定性金字塔,它是由确定性的不同连接方式构造的广义Farey组织的网络金字塔(GFONP),理论推导了该类金字塔网络的拓扑性质及其转变特点,发现三种GFONP网络的度分布都服从指数分布形式,随着时间t的不断演化网络金字塔群聚系数不断减小并趋于一个常数,同时网络金字塔从异配向同配转变,度-度相关系数也趋于一个正的常数.结果显示这类型网络的新特性具有应用潜力.     

Clusters of bound particles in the derivative δ-function Bose gas

In this paper we discuss a novel procedure for constructing clusters of bound particles in the case of a quantum integrable derivative δ-function Bose gas in one dimension. It is shown that clusters of bound particles can be constructed for this Bose gas for some special values of the coupling constant, by taking the quasi-momenta associated with the corresponding Bethe state to be equidistant points on a single circle in the complex momentum plane. We also establish a connection between these special values of the coupling constant and some fractions belonging to the Farey sequences in number theory. This connection leads to a classification of the clusters of bound particles associated with the derivative δ-function Bose gas and allows us to study various properties of these clusters like their size and their stability under the variation of the coupling constant.     

Self-segregation in chemical reactions,diffusion in a catalytic environment and an ideal polymer near a defect

We study a family of equivalent continuum models in one dimension. All these models map onto a single equation and include simple chemical reactions, diffusion in presence of a trap or a source and an ideal polymer chain near an attractive or repulsive site. We have obtained analytical results for the survival probability, total growth rate, statistical properties of nearest-neighbour distribution between a trap and unreacted particle and mean-squared displacement of the polymer chain. Our results are compared with the known asymptotic results in the theory of discrete random walks on a lattice in presence of a defect.     

Labyrinthine pathways towards supercycle attractors in unimodal maps

As an important preceding step for the demonstration of an uncharacteristic (q-deformed) statisticalmechanical structure in the dynamics of the Feigenbaum attractor we uncover previously unknown properties of the family of periodic superstable cycles in unimodal maps. Amongst the main novel properties are the following: i) The basins of attraction for the phases of the cycles develop fractal boundaries of increasing complexity as the period-doubling structure advances towards the transition to chaos. ii) The fractal boundaries, formed by the pre-images of the repellor, display hierarchical structures organized according to exponential clusterings that manifest in the dynamics as sensitivity to the final state and transient chaos. iii) There is a functional composition renormalization group (RG) fixed-point map associated with the family of supercycles. iv) This map is given in closed form by the same kind of q-exponential function found for both the pitchfork and tangent bifurcation attractors. v) There is final-stage ultra-fast dynamics towards the attractor, with a sensitivity to initial conditions which decreases as an exponential of an exponential of time. We discuss the relevance of these properties to the comprehension of the discrete scale-invariance features, and to the identification of the statistical-mechanical framework present at the period-doubling transition to chaos. This is the first of three studies (the other two are quoted in the text) which together lead to a definite conclusion about the applicability of q-statistics to the dynamics associated to the Feigenbaum attractor.