Numerical detection and continuation of saddle-node homoclinic bifurcations of codimension one and two

An extension of an existing truncated boundary-value method for the numerical continuation of connecting orbits is proposed to deal with homoclinic orbits to a saddle-node equilibrium. In contrast to previous numerical work by Schecter and Friedman and Doedel, the method is based on (linear) projection boundary conditions. These boundary conditions, with extra defining conditions for a saddle-node, allow the continuation of codimension-one curves of saddle-node homoclinic orbits. A new test function is motivated for detecting codimension-two points at which loci of saddle-nodes and homoclinic orbits become detached. Two methods for continuing such codim 2 points in three parameters are discussed. The numerical methods are applied to two example systems, modelling a DC Josephson junction and CO oxidation. For the former model, existing numerical results are recovered and extended; for the latter, new dynamical features are uncovered. All computations are performed using AUTO.     

Criteria on existence of horseshoes near homoclinic tangencies of arbitrary orders

Consider (m + 1)-dimensional, m ≥ 1, diffeomorphisms that have a saddle fixed point O with m-dimensional stable manifold W^s(O), one-dimensional unstable manifold W^u(O), and with the saddle value σ different from 1. We assume that W^s(O) and W^u(O) are tangent at the points of some homoclinic orbit and we let the order of tangency be arbitrary. In the case when σ < 1, we prove necessary and sufficient conditions of existence of topological horseshoes near homoclinic tangencies. In the case when σ > 1, we also obtain the criterion of existence of horseshoes under the additional assumption that the homoclinic tangency is simple.     

一维,二维和三维Si基纳米线的制备

在Ar气氛中，SiC粉末分别在Fe、Co、Fe：Co(1：1)的催化下，经一步反应制备了一维、二维和三维si基纳米线。SEM、HRTEM、EDX分析表明一维线状和二维网状Si基纳米线由C、Si、O组成，存在两类纳米线，一类是SiO，包裹的Si纳米线；另一类是SiO，包裹的SiC纳米线。三维Si基纳米线组成象花一样的结构，仅由SiOx组成。SiOx和Si是无定形结构，SiC是β-SiC单晶。     

Gaussian profile estimation in one dimension

We present several new results on the classic problem of estimating Gaussian profile parameters from a set of noisy data, showing that an exact solution of the maximum likelihood equations exists for additive Gaussian-distributed noise. Using the exact solution makes it possible to obtain analytic formulas for the variances of the estimated parameters. Finally, we show that the classic formulation of the problem is actually biased, but that the bias can be eliminated by a straightforward algorithm.     

Numerical solution of dispersion equation in one dimension

Abstract

An improved hybrid method for one‐dimensional advection‐diffusion problems, based on the Holly‐Preissmann two‐point fourth‐order and Crank‐Nicholson numerical schemes, has been proposed to handle the problem with Courant numbers (Cr) greater than 1. Extensive test runs and analyses have been performed for a schematic advection‐diffusion problem. Through a comparison of the analytical solution with the computed results, the accuracy and stability of this improved hybrid method are discussed. Satisfactory results are found for both weak and strong diffusion problems under large Courant number conditions. The sensitivity of the improved method to the temporal weighting factor has also been demonstrated. For strong diffusion problems, the use of a larger temporal weighting factor becomes necessary to eliminate the phenomenon of instability.     

Prism for in-line beam expansion in one dimension

A four-sided prism is described that expands or compresses a beam in one dimension without altering the direction of the beam or displacing its centerline.     

Lattice gas simulations of dynamical geometry in one dimension

We present numerical results obtained using a lattice gas model with dynamical geometry. The (irreversible) macroscopic behaviour of the geometry (size) of the lattice is discussed in terms of a simple scaling theory and obtained numerically. The emergence of irreversible behaviour from the reversible microscopic lattice gas rules is discussed in terms of the constraint that the macroscopic evolution be reproducible. The average size of the lattice exhibits power-law growth with exponent at late times. The deviation of the macroscopic behaviour from reproducibility for particular initial conditions ('rogue states') is investigated as a function of system size. The number of such 'rogue states' is observed to decrease with increasing system size. Two mean-field analyses of the macroscopic behaviour are also presented.     

一维杆状声子晶体的带隙特性

用有限元方法计算了一维杆状声子晶体的振动传输特性及其振动模态,从振形角度分析了一维杆状声子晶体相关参数(物理参数、尺寸参数)对带隙起始频率及带隙宽度的影响,发现带隙的形成与其模态振形密切相关,各参数对声子晶体带隙的影响主要是影响了个模态所在的频率,从而改变了其带隙的起始位置及宽度。     

Optimal finite element solutions to diffusion–convection problems in one dimension

Petrov–Galerkin methods have been proposed by several authors to eliminate the inaccuracies and oscillations obtained with Galerkin methods when applied to diffusion–convection problems at high Péclet numbers: the difficulty is to select the appropriate test space for a given trial space. We investigate here choices of test space which either exactly or approximately symmetrize the associated bilinear form and so retain the optimal character of the approximate solution. This is the key to high accuracy and superconvergence, and optimal recovery techniques are proposed to obtain the maximum information from the approximations. Examples are given to show how the position and thickness of boundary layers may be estimated with relatively coarse meshes.     

Singularities of the susceptibility of a Sinai-Ruelle-Bowen measure in the presence of stable-unstable tangencies

Let ρ be a Sinai-Ruelle-Bowen (SRB or 'physical') measure for the discrete time evolution given by a map f, and let ρ(A) denote the expectation value of a smooth function A. If f depends on a parameter, the derivative δρ(A) of ρ(A) with respect to the parameter is formally given by the value of the so-called susceptibility function Ψ(z) at z=1. When f is a uniformly hyperbolic diffeomorphism, it has been proved that the power series Ψ(z) has a radius of convergence r(Ψ)>1, and that δρ(A)=Ψ(1), but it is known that r(Ψ)<1 in some other cases. One reason why f may fail to be uniformly hyperbolic is if there are tangencies between the stable and unstable manifolds for (f,ρ). The present paper gives a crude, non-rigorous, analysis of this situation in terms of the Hausdorff dimension d of ρ in the stable direction. We find that the tangencies produce singularities of Ψ(z) for |z|<1 if d<1/2, but only for |z|>1 if d>1/2. In particular, if d>1/2, we may hope that Ψ(1) makes sense, and the derivative δρ(A)=Ψ(1) thus has a chance to be defined.     

Calibrating a scanning electron microscope in two coordinates by the use of one certified dimension

A method is proposed for calibrating a scanning electron microscope that corresponds completely to national standards providing unification of measurements in nanotechnology. __________ Translated from Izmeritel’naya Tekhnika, No. 6, pp. 18–20, June, 2008.     

一维Bi2Te3基纳米材料的制备、结构控制与热电性能

热电材料的低维化可以改善材料电输运与热传输的矛盾,特别是一维纳米热电材料明显的晶体各向异性和强烈的量子禁闭效应,可大幅度提高材料的热电优值和热电转换效率。Bi2Te3是制造低温热电材料的最常用材料,在温差发电和半导体制冷方面具有广阔的商业应用前景。以一维Bi2Te3基纳米热电材料的制备技术为评述线索,重点论述一维Bi2Te3基纳米热电材料形貌参数(包括直径、长径比)、晶面取向等微观结构的调控方法、生长机理以及显微结构对热电性能的影响规律。指出发展新的一维Bi2Te3基纳米热电材料结构控制方法,研究一维纳米热电材料的定向排布及组装技术,从更深层次揭示一维结构与热电性能的关系,以及开发一维Bi2Te3基纳米热电材料在各领域的实际应用是未来研究的发展方向。     

Calibrating a scanning electron microscope by means of a linear measure having one certified dimension

A method is proposed for calibrating a scanning electron microscope that enables one to use a linear measure with a single certified dimension to determine all the basic parameters needed for making linear measurements: image magnification, electron-probe diameter, and correction parameter for deriving the true dimensions of relief elements from the distance between video-signal maxima.Translated from Izmeritel'naya Tekhnika, No. 7, pp. 68–70, July, 1994.     

A strange attractor in the unfolding of an orbit-flip homoclinic orbit

An orbit-flip homoclinic orbit o of a vector field defined on R 3 is a homoclinic orbit to an equilibrium point for which the one-dimensional unstable manifold of the equilibrium point is connected to the one-dimensional strong stable manifold. In this paper, we show that in a generic unfolding of such a homoclinic orbit, there exists a positive Lebesgue measure set in the parameter space for which the corresponding vector field possesses a suspended strange attractor. To prove the result, we propose a rescaling in the phase space and a blowing up in the parameter space, and in the new system, we show that the Poincaré return map is close to the map (x,y) M (1 - āx2,bx) when b is close to 0. With a similar rescaling/blowing up, we also obtain a similar result in the case where o is an inclination-flip homoclinic orbit.     

New aspects in the unfolding of the nilpotent singularity of codimension three

This paper is concerned with three-dimensional vector fields and more specifically with the study of dynamics in unfoldings of the nilpotent singularity of codimension three. The ultimate goal is to understand the dynamics and bifurcations in the unfolding of the singularity. However, it is clear from the literature that the bifurcation diagram is very complicated and a complete study is far beyond the current possibilities, not only from a theoretical point of view but also from a numerical point of view, despite recent developments of computational methods for dynamical systems. Since all complicated dynamical behaviour is known to be of small amplitude, shrinking to the singularity for parameter values tending to the bifurcation parameter, the aim in this paper is especially to focus on a different aspect that might be interesting in the study of global bifurcation problems in the presence of such a nilpotent singularity of codimension three. The notion is introduced of 'traffic regulator' and the specific sets called the 'inset' and 'outset', which give a new framework for studying a transition map in a cylindrical neighbourhood of the singularity that contains all the non-trivial dynamics that can bifurcate from the singularity, focusing on the domains on which the transition map is defined. A list is also given of open problems which are believed to be helpful for future investigation of the bifurcations from the nilpotent triple zero singularity in 3.     

The interaction of thermocapillary perturbations with 3D waves in a heated falling liquid film

The evolution of three-dimensional (3D) waves into thermocapillary-wave structures in a locally heated water film flowing down a vertical plate has been experimentally studied. The interaction of hydrodynamic perturbations with thermocapillary instabilities was monitored using a high-speed high-resolution IR imager. The existence of thermocapillary structures has been observed for the first time at high Reynolds numbers (Re = 150) in regime A, which is characterized by high temperature gradients on the film surface and a definite value of the ??most dangerous?? wavelength of instability that arises when the heat flux density exceeds a certain threshold. It is established that structures periodically appear in which the distance between temperature maxima is much shorter than that observed before. As the heat flux density is increased, the thermal entry length exhibits a sharp drop when thermocapillary instabilities arise in the vicinity of the upper edge of the heater.     

细晶粒BaTiO3陶瓷电阻率的尺寸效应

在不同的测试电压条件下,对不同晶粒尺寸的BaTiO3陶瓷室温下的电阻率进行了测量;结合室温下的复阻抗谱,采用简化的等效电路模型和净极化电流的理论,分析了高、低频率下晶粒和晶界对瓷体电阻的影响.