切换电路系统的振荡行为及其非光滑分岔机理

探讨了周期时间开关及控制阈值下在两个Rayleigh型子系统之间切换的电路系统随参数变化的复杂动力学演化过程, 通过对子系统平衡点的分析, 给出了参数空间中Fold分岔和Hopf分岔的条件, 考察了切换面处广义Jacobian矩阵特征值随辅助参数变化的分布情况, 得到了切换面处系统可能存在的各种分岔行为, 进而讨论了系统不同行为的产生机理, 指出系统的相轨迹存在分别由周期开关和控制阈值决定的两类不同的分界点, 而系统轨迹与非光滑分界面的多次碰撞将导致系统由周期倍化分岔导致混沌振荡.     

Renormalization group limit cycles and first-order phase transitions in superconductors and abelian Higgs models

Renormalization group methods are used to investigate the thermodynamic behaviour of a multicomponent abelian Higgs model. Interpreted as a Ginzburg-Landau-type model for superconductivity, the model interpolates, via the replica trick, between superconductors with and without quenched random impurities. For a range of values of the numbers of complex order parameter components and of replicas, the renormalization group trajectories exhibit a stable focus, surrounded by an unstable limit cycle. By evaluating the free energy and the equation of state, we find that flows within the limit cycle correspond to a type of critical behaviour with oscillatory modulations. Outside the limit cycle, there is a region of the parameter space in which runaway of the trajectories may be interpreted in terms of a fluctuation-induced first-order phase transition, but in a third region, no clear interpretation is possible.     

Geometro-Differential Conception of Extended Particles and the Semigroup of Trajectories in Minkowski Space-Time

The semigroup of trajectories in Minkowski space-time and its induced representations are constructed as a generalization of the Galilei case. They describe relativistic pointlike particles and yield the free propagator as a path integral in the space of trajectories parametrized by a fifth parameter. This non physical propagator in a five-dimensional space is integrated over the fifth parameter to yield the physical propagator in Minkowski space. Thereafter, this notion is applied to a model of extended particles with internal Poincaré symmetry and moving in an external Minkowski space. The geometrical structure is of Hilbert bundles and the interaction is introduced as a connection. The propagator is a path integral with respect to either the internal and external trajectories and reduces to a product of an internal and an external propagator when the interaction is ignored, just as has been found in a previous work with representations of the group rather than those of the semigroup.     

Basin entropy in Boolean network ensembles

The information processing capacity of a complex dynamical system is reflected in the partitioning of its state space into disjoint basins of attraction, with state trajectories in each basin flowing towards their corresponding attractor. We introduce a novel network parameter, the basin entropy, as a measure of the complexity of information that such a system is capable of storing. By studying ensembles of random Boolean networks, we find that the basin entropy scales with system size only in critical regimes, suggesting that the informationally optimal partition of the state space is achieved when the system is operating at the critical boundary between the ordered and disordered phases.     

Fractional standard map

Properties of the phase space of the standard map with memory are investigated. This map was obtained from a kicked fractional differential equation. Depending on the value of the map parameter and the fractional order of the derivative in the original differential equation, this nonlinear dynamical system demonstrates attractors (fixed points, stable periodic trajectories, slow converging and slow diverging trajectories, ballistic trajectories, and fractal-like structures) and/or chaotic trajectories. At least one type of fractal-like sticky attractors in the chaotic sea was observed.     

Compressibility of random walker trajectories on growing networks

We find that the simple coupling of network growth to the position of a random walker on the network generates a traveling wave in the probability distribution of nodes visited by the walker. We argue that the entropy of this probability distribution is bounded as the network size tends to infinity. This means that the growth of a space coupled to a random walker situated in it constrains its dynamics to a set of typical random walker trajectories, and walker trajectories inside the growing space are compressible.     

On a Class of World Surfaces for the Linear String Model of a Baryon

For the linear string model of a baryon where three material points (three quarks) are connected in series by relativistic strings, a class of physically realizable world surfaces has been found which admit parametrization such that the equations of motion and the boundary conditions are linear due to the proportionality of the parameter associated with the quark trajectories to the natural parameter. The surfaces of this class are represented as a Fourier series of the eigenfunctions of some boundary-value problem. It is shown that the series generating such surfaces contains a finite number of terms. In particular, in the 3+1-dimensional Minkowski space, one and only one surface of the class under consideration is realizable, namely, the helicoid.     

Editorial

Symbolic dynamics is a powerful tool in the study of dynamical systems. The purpose of symbolic dynamics is to provide a simplified picture of complicated dynamics, that gives some insight into its complexity. To this end, the state space of the system is partitioned in a finite number of pieces, and the exact trajectories of individual points are traded off by the trajectory relative to that partition. These so-called coarse-grained trajectories turn out to be realisations of a stationary random process with a finite alphabet. In particular, the entropy of a dynamical system can be approximated by the Shannon entropy of any of its symbolic dynamics (the finer the partition, the better the approximation). Today, symbolic dynamics is an independent field of theoretical physics and applied mathematics with applications to such important disciplines as cryptology, time series analysis, and data-compression.     

An improved random walk approach for yield optimization

In this paper, the traditional random walk approach for yield optimization is modified to further improve its efficiency. The orthogonal array of experiment design is employed as an alternative to generating sample points in the circuit parameter space. In addition, the step of the random walk is modified to be adapt. Finally, an example is given to show its high efficiency.     

‘Percolative’ dynamics in the hexagonal lattice

We study a percolative dynamic model for the hexagonal lattice. Random trajectories are generated and their critical behaviour is studied. The critical behaviour corresponds to that of simple percolatio in some of the parameter space, but elsewhere the exponents reveal new universality classes. We calculate the fractal dimension of extended trajectories for different critical points.     

Spatial Statistics and Information Geometry for Parametric Statistical Models of Galaxy Clustering

Poisson spatial processes of points and ofextended objects representing smoothed clusters ofgalaxies are considered; some results are obtained forplanar representations of random filaments, which may help interpret the findings of the Las CampanasRedshift Survey. Based on a model for the voidprobability function, a family of gamma-relateddistributions is investigated as a three-dimensionalmodel for the clustering of galaxies. The unclusteredmodels in this family correspond to the random case andto maximum information-theoretic entropy. The Riemannianinformation metric and Gaussian curvature are derived for the parameter space of thefamily of models, which provides a background on whichto write dynamics for cluster evolution.     

周期切换下Chen系统的振荡行为与非光滑分岔分析

研究了不同参数Chen系统之间进行周期切换时的分岔和混沌行为.基于平衡态分析,考虑Chen系统在不同稳态解时通过周期切换连接生成的复合系统的分岔特性,得到系统的不同周期振荡行为.在演化过程中,由于切换导致的非光滑性,复合系统不仅仅表现为两子系统动力特性的简单连接,而且会产生各种分岔,导致诸如混沌等复杂振荡行为.通过Poincaré映射方法,讨论了如何求周期切换系统的不动点和Floquet特征乘子.基于Floquet理论,判定系统的周期解是渐近稳定的.同时得到,随着参数变化,系统既可以由倍周期分岔序列进入混沌,也可以由周期解经过鞍结分岔直接到达混沌.研究结果揭示了周期切换系统的非光滑分岔机理.     

Transport of a passive scalar and Lagrangian chaos in a Hamiltonian hydrodynamic system

An experimental and numerical study is made of the chaotic behavior of Lagrangian trajectories and transport of a passive tracer in a quasi-two-dimensional four-vortex flow with a periodic time dependence of the Euler velocity field. Quantitative measurements are made of tracer transport between isolated vortices in physical space and in “action” variable space. The theory of adiabatic chaos is used to interpret the measurements. The simplest phenomenological models of liquid particle random walks are proposed to describe the anomalous transport in terms of the action.     

On the absence of stable periodic orbits in domains of separatrix crossings in nonsymmetric slow-fast Hamiltonian systems

We consider a two degree of freedom Hamiltonian system with one degree of freedom corresponding to fast motion and the other corresponding to slow motion. We assume that at frozen values of the slow variables there is a separatrix on the phase plane of the fast variables and there is a region in the phase space (the domain of separatrix crossings) where projections of phase points onto the plane of the fast variables repeatedly cross the separatrix in the process of evolution of the slow variables. Under rather general conditions, we prove that there are no stable periodic trajectories of any prescribed period inside the domain of separatrix crossings, except maybe for periodic trajectories passing anomalously close to the saddle point.     

Observation of quantum particles on a large space-time scale

A quantum particle observed on a sufficiently large space-time scale can be described by means of classical particle trajectories. The joint distribution for large-scale multiple-time position and momentum measurements on a nonrelativistic quantum particle moving freely inR ^v is given by straight-line trajectories with probabilities determined by the initial momentum-space wavefunction. For large-scale toroidal and rectangular regions the trajectories are geodesics. In a uniform gravitational field the trajectories are parabolas. A quantum counting process on free particles is also considered and shown to converge in the large-space-time limit to a classical counting process for particles with straight-line trajectories. If the quantum particle interacts weakly with its environment, the classical particle trajectories may undergo random jumps. In the random potential model considered here, the quantum particle evolves according to a reversible unitary one-parameter group describing elastic scattering off static randomly distributed impurities (a quantum Lorentz gas). In the large-space-time weak-coupling limit a classical stochastic process is obtained with probability one and describes a classical particle moving with constant speed in straight lines between random jumps in direction. The process depends only on the ensemble value of the covariance of the random field and not on the sample field. The probability density in phase space associated with the classical stochastic process satisfies the linear Boltzmann equation for the classical Lorentz gas, which, in the limith0, goes over to the linear Landau equation. Our study of the quantum Lorentz gas is based on a perturbative expansion and, as in other studies of this system, the series can be controlled only for small values of the rescaled time and for Gaussian random fields. The discussion of classical particle trajectories for nonrelativistic particles on a macroscopic spacetime scale applies also to relativistic particles. The problem of the spatial localization of a relativistic particle is avoided by observing the particle on a sufficiently large space-time scale.     

Generic fractal structure of finite parts of trajectories of piecewise smooth Hamiltonian systems

Piecewise smooth Hamiltonian systems with tangent discontinuity are studied. A new phenomenon is discovered, namely, the generic chaotic behavior of finite parts of trajectories. The approach is to consider the evolution of Poisson brackets for smooth parts of the initial Hamiltonian system. It turns out that, near second-order singular points lying on a discontinuity stratum of codimension two, the system of Poisson brackets is reduced to the Hamiltonian system of the Pontryagin Maximum Principle. The corresponding optimization problem is studied and the topological structure of its optimal trajectories is constructed (optimal synthesis). The synthesis contains countably many periodic solutions on the quotient space by the scale group and a Cantor-like set of nonwandering points (NW) having fractal Hausdorff dimension. The dynamics of the system is described by a topological Markov chain. The entropy is evaluated, together with bounds for the Hausdorff and box dimension of (NW).     

Trajectory divergence for coupled relaxation oscillators: Measurements and models

The exponential divergence of nearby phase space trajectories is a hallmark of nonperiodic (chaotic) behavior in dynamical systems. We present the first laboratory of measurements of divergence rates (or characteristic exponents), using a system of coupled tunnel diode relaxation oscillators. This property of sensitive dependence on initial conditions is reliably associated with broadband spectra, and both methods are used to characterize the motion as a function of the coupling strength and natural frequency ratio of the two oscillators. A simple piecewise linear model correctly predicts the major periodic and non-periodic regions of the parameter space, thus confirming that the chaotic behavior involves only a few degrees of freedom.Work supported by the National Science Foundation and the Research Corporation.     

Directional Analysis of the Chaotic Superlattice around the Equilibrium Point in the Phase Space

The recently proposed method of our research group named as directional Lyapunov exponents(DLEs) is presented. Then, DLEs are used to analyze the eigenstructure of the output phase space around the equilibrium points. Finally, the impacts of the superlattice parameter changes on the characteristics of the output chaotic signal are analyzed. The experimental results show that parameter changes of the superlattice will affect the eigenstructure around the equilibrium points in the output phase space, and DLEs are sensitive to these changes.     

Renormalization group analysis of a simple hierarchical fermion model

A simple hierarchical fermion model is constructed which gives rise to an exact renormalization transformation in a 2-dimensional, parameter space. The behaviour of this transformation is studied. It has two hyperbolic fixed points for which the existence of aglobal critical line is proven. The asymptotic behaviour of the transformation is used to prove the existence of the thermodynamic limit in a certain domain in parameter space. Also the existence of a continuum limit for these theories is investigated using informatioin about the asymptotic renomralization behaviour. It turns out that the trivial fixed point gives rise to a twoparameter family of continuum limits corresponding to that part of parameter space where the renormalization trajectories originate at this fixed point. Although the model is not very realistic it serves as a simple example of the application of the renormalization group to proving the existence of the thermodynamic limit and the continuum limit of lattice models. Moreover, it illustrates possible complications that can arise in global renormalization group behaviour, and that might also be present in other models where no global analysis of the renormalization transformation has yet been achieved.A part of the material here presented was used in the author's thesis