Geometry and topology of escape. II. Homotopic lobe dynamics

We continue our study of the fractal structure of escape-time plots for chaotic maps. In the preceding paper, we showed that the escape-time plot contains regular sequences of successive escape segments, called epistrophes, which converge geometrically upon each end point of every escape segment. In the present paper, we use topological techniques to: (1) show that there exists a minimal required set of escape segments within the escape-time plot; (2) develop an algorithm which computes this minimal set; (3) show that the minimal set eventually displays a recursive structure governed by an "Epistrophe Start Rule:" a new epistrophe is spawned Delta=D+1 iterates after the segment to which it converges, where D is the minimum delay time of the complex.     

平行电磁场中里德堡氢原子的自相似结构研究

利用半经典方法研究了平行电磁场中里德堡氢原子的分形自相似现象. 通过研究平行电磁场中里德堡氢原子的逃逸时间和初始出射角间的关系, 发现了逃逸时间图的自相似结构, 并通过研究与图中冰柱对应的逃逸轨道, 得到了自相似结构和逃逸轨道之间的关系, 发现了该类自相似逃逸轨道满足的规律. 进一步研究了标度能量和标度磁场对体系动力学的影响, 表明标度能量和标度磁场均控制体系的分形自相似结构. 当标度能量或标度磁场比较小时, 没有自相似现象, 随着标度能量或标度磁场的增大, 自相似出现, 体系变复杂.     

Fractal dynamics in the ionization of helium Rydberg atoms

We study the ionization of helium Rydberg atoms in an electric field above the classical ionization threshold within the semiclassical theory.By introducing a fractal approach to describe the chaotic dynamical behavior of the ionization,we identify the fractal self-similarity structure of the escape time versus the distribution of the initial launch angles of electrons,and find that the self-similarity region shifts toward larger initial launch angles with a decrease in the scaled energy.We connect the fractal structure of the escape time plot to the escape dynamics of ionized electrons.Of particular note is that the fractal dimensions are sensitively controlled by the scaled energy and magnetic field,and exhibit excellent agreement with the chaotic extent of the ionization systems for both helium and hydrogen Rydberg atoms.It is shown that,besides the electric and magnetic fields,core scattering is a primary factor in the fractal dynamics.     

Gravitational collapse with standard and dark energy in the teleparallel equivalent of general relativity

A perfect fluid with self-similarity of the second kind is studied within the framework of the teleparallel equivalent of general relativity(TEGR).A spacetime which is not asymptotically flat is derived.The energy conditions of this spacetime are studied.It is shown that after some time the strong energy condition is not enough to satisfy showing a transition from standard matter to dark energy.The singularities of this solution are discussed.     

First integral of the Euler-Lagrange equation and boundary conditions for stochastic processes: Bistable potential driven by colored noise

Starting from the variational problem of finding the maximum probability path, which is obtained by applying a condition of a mobile end point of the extremal path, we set a first integral of the Euler-Lagrange (EL) equation; this constant of motion, together with appropriate boundary conditions, leads to an initial value problem. We consider the one-dimensional stochastic process of a particle on a bistable potential, driven by colored noise, and obtain the escape rate across the barrier. By solving numerically the initial value problem, without any approximation, we plot the τ-dependence of the extremal action for the particle to go from the bottom to the top of the potential. The behavior of S(τ) for this problem has not been reported in the literature, and it is qualitatively similar to a result we obtained recently for the oscillator potential.     

偏心环形开放台球中粒子逃逸动力学的分形性质

二维台球体系因为能够体现混沌现象的基本特征且数值运算相对简单,从而成为研究微观体系混沌动力学的理想模型,近年来一直广受关注.本文研究非同心的环形开放台球中粒子逃逸的混沌动力学性质,它体现了与初条件密切相关的奇异性.采用简化的盒计数 (box-counting)算法,计算了分形维数,结果定量地反映了粒子逃逸前与环壁碰撞次数随粒子入射角变化的函数关系.其中,特别关注环形台球的偏心率对体系混沌性质的影响.     

Dynamics of Finger Formation in Laplacian Growth Without Surface Tension

We study the dynamics of “finger” formation in Laplacian growth without surface tension in a channel geometry (the Saffman–Taylor problem). We present a pedagogical derivation of the dynamics of the conformal map from a strip in the complex plane to the physical channel. In doing so we pay attention to the boundary conditions (no flux rather than periodic) and derive a field equation of motion for the conformal map. We first consider an explicit analytic class of conformal maps that form a basis for solutions in infinitely long channels, characterized by meromorphic derivatives. The great bulk of these solutions can lose conformality due to finite time singularities. By considerations of the nature of the analyticity of these solutions, we show that those solutions which are free of such singularities inevitably result in a single asymptotic “finger” whose width is determined by initial conditions. This is in contradiction with the experimental results that indicate selection of a finger of width 1/2. In the last part of this paper we show that such a solution might be determined by the boundary conditions of a finite body of fluid, e.g. finiteness can lead to pattern selection.     

Time-periodic perturbation of a Liénard equation with an unbounded homoclinic loop

We consider a quadratic Liénard equation with an unbounded homoclinic loop, which is a solution tending in forward and backward time to a non-hyperbolic equilibrium point located at infinity. Under small time-periodic perturbation, this equilibrium becomes a normally hyperbolic line of singularities at infinity. We show that the perturbed system may present homoclinic bifurcations, leading to the existence of transverse intersections between the stable and unstable manifolds of such a normally hyperbolic line of singularities. The global study concerning the infinity is performed using the Poincaré compactification in polar coordinates, from which we obtain a system defined on a set equivalent to a solid torus in R³, whose boundary plays the role of the infinity. The transversality of the manifolds is proved using the Melnikov method and implies, via the Birkhoff-Smale Theorem, a complex dynamical behaviour of the perturbed system solutions in the finite part of the phase space. Numerical simulations are performed in order to illustrate this behaviour, which could be called “the chaos arising from infinity”, since it depends on the global structure of the Liénard equation, including the points at infinity. Although applied to a particular case, the analysis presented provides a geometrical approach to study periodic perturbations of homoclinic (or heteroclinic) loops to infinity of any planar polynomial vector field.     

Resonance theory in atom-surface scattering

We study the problem of analytic extension of the resolvent for Hamiltonians arising in scattering of atoms by a quantum surface. We prove that the resolvent extends holomorphically to some regions of the lower half plane with isolated singularities called Landau resonances which are branch points of the resolvent. We study also the effect of impurities on the singularities of the resolvent and show that the presence of impurities adds poles to the Landau resonances.     

The early universe and cosmogenesis

We extrapolate the Cosmological Standard Model to the past, determine initial geometrical conditions in the early universe, and consider a new cosmogenesis paradigm based on the concept of black-and-white holes with integrable singularities.     

On Borel singularities in quantum field theory

We consider the effective one-loop lagrangian in a constant electric field. It is shown that perturbation theory behaves as n!, giving rise to singularities in the Borel plane. Comparing with the exact result we show how to integrate these singularities. We suggest that renormalons in QED and QCD should be integrated in a similar way. We make a speculation on a possible interpretation of this integration.     

Multiparameter family of collapsing solutions to the critical nonlinear Schrödinger equation in dimension <Emphasis Type="Italic">D</Emphasis>=2

We consider the critical nonlinear Schrödinger equation in dimension D=2 and obtain a system consisting of three equations describing the collapse of solutions. The system admits a five-parameter family of solutions. Almost everywhere, except for an exponentially narrow region near the collapse point, the tunneling processes are negligible. The relation between initial data and the condition of occurrence of the collapse is investigated. The separatrix, which divides the collapse domain and expansion regions having no singularities in a finite time interval, is found.     

Vortex structures with complex points singularities in two-dimensional Euler equations. New exact solutions

In this work we present a new class of exact stationary solutions for two-dimensional (2D) Euler equations. Unlike already known solutions, the new ones contain complex singularities. We consider point singularities which have a vector field index greater than 1 as complex. For example, the dipole singularity is complex because its index is equal to 2. We present in explicit form a large class of exact localized stationary solutions for 2D Euler equations with a singularity whose index is equal to 3. The solutions obtained are expressed in terms of elementary functions. These solutions represent a complex singularity point surrounded by a vortex satellite structure. We also discuss the motion equation of singularities and conditions for singularity point stationarity which provide the stationarity of the complex vortex configuration.     

The effect of astigmatism on the phase singularities of focused beams

The phase singularities of plane beams focused by an aperture lens with astigmatism are studied. Numerical calculation results are given to illustrate the dependence of phase singularities of focused plane beams on the astigmatic coefficient and the Fresnel number. It is shown that as the Fresnel number gradually increases, the phase singularities shift not only towards the z-axis but also towards the geometrical focal plane in the presence of astigmatism, whereas the phase singularities move along the focal plane and always line in the focal plane for the astigmatism-free case. With increasing astigmatic coefficient, the phase singularities shift not only towards the z-axis but also far away from the geometrical focal plane.     

Chaotic electronic scattering with He(+)

We investigate classical electronic collisions with a He(+) ion. Scattering functions, such as the scattering angle, collisional time, or energy of the outgoing electron, all exhibit an interesting hierarchial self-similar structure, which can be interpreted in terms of the indefinite number of electronic returns to the vicinity of the nucleus, encounters between electrons, and Keplerian excursions of electrons during the collisional processes. Based on this mechanism a binary coding is introduced to organize the dynamics of this three-body system and to provide an understanding of the self-similarity among generations of scale magnification, which yields escape rates that vary with the sectional cut into the parameter space. The self-similarity displayed within a single generation, on the other hand, can be simply tied to the periods of the two independent electronic excursions. The physical interpretation and the symbolic dynamics introduced here are generally useful for three-body collisional systems, including atomic, molecular, or stellar collisions.     

Large Deviations for Non-Uniformly Expanding Maps

We obtain large deviation bounds for non-uniformly expanding maps with non-flat singularities or criticalities and for partially hyperbolic non-uniformly expanding attracting sets. That is, given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the map, showing that the Lebesgue measure of the set of points whose time averages stay away from the space average tends to zero exponentially fast with the number of iterates involved. As easy by-products we deduce escape rates from subsets of the basins of physical measures for these types of maps. The rates of decay are naturally related to the metric entropy and pressure function of the system with respect to a family of equilibrium states. 2000 Mathematics Subject Classification: 37D25, 37A50, 37B40, 37C40     

Light propagation in anisotropic metamaterials. II. Reflection from the half-space

We consider reflection of light from half-space of anisotropic metamaterial at arbitrary direction of optical axis in the plane of light incidence. We obtain conditions at which total reflection and refraction do not depend on the polarization of incident light and the reflection does not depend on also the angle of incidence. We also study the possibilities and conditions for using the considered system as beam splitter, omnidirectional reflector, phase retarder, and so on.     

Self-similar spacetimes: Geometry and dynamics

The nature and uses of self-similarity in general relativity are discussed. A spacetime may be self-similar (homothetic) along surfaces of any dimensionality, from 1 to 4. A geometric construction is given for all self-similar spacetimes. As an important special case, the spatially self-similar cosmological models are introduced, and their dynamical properties are studied in some detail: The initial-value problem is posed, the ADM formulation is established (when applicable), and it is shown that the evolution equations preserve a self-similarity of initial data. The existence of a conserved quantity is deduced from self-similarity. Possible applications to cosmology and singularities are mentioned. Supported in part by the National Science Foundation [GP-36687X].     

Photoionization of the Bound Systems at High Energies

We consider photoionization of a system bound by the central potential V(r). We demonstrate that the high energy nonrelativistic asymptotics of the photoionization cross section can be obtained without solving the wave equation. The asymptotics can be expressed in terms of the Fourier transform of the potential by employing the Lippmann–Schwinger equation. We find the asymptotics for the screened Coulomb field. We demonstrate that the leading corrections to this asymptotics are described by the universal factor. The high energy nonrelativistic asymptotics is found to be determined by the analytic properties of the potential V(r). We show that the energy dependence of the asymptotics of photoionization cross sections of fullerenes is to large extent model-dependent. We demonstrate that if the fullerene field V(r) is approximated by the function with singularities in the complex plane, the power drop of the asymptotics is reached at the energies which are so high that the cross section becomes unobservably small. The preasymptotic behavior with a faster decrease of the cross sections becomes important in these cases.     

Reflection principles for biased random walks and application to escape time distributions

We present a reflection principle for an arbitrarybiased continuous time random walk (comprising both Markovian and non-Markovian processes) in the presence of areflecting barrier on semi-infinite and finite chains. For biased walks in the presence of a reflecting barrier this principle (which cannot be derived from combinatorics) is completely different from its familiar form in the presence of an absorbing barrier. The result enables us to obtain closed-form solutions for the Laplace transform of the conditional probability for biased walks on finite chains for all three combinations of absorbing and reflecting barriers at the two ends. An important application of these solutions is the calculation of various first-passage-time and escape-time distributions. We obtain exact results for the characteristic functions of various kinds of escape time distributions for biased random walks on finite chains. For processes governed by a long-tailed event-time distribution we show that the mean time of escape from bounded regions diverges even in the presence of a bias—suggesting, in a sense, the absence of true long-range diffusion in such frozen processes.