Approximation of box dimension of attractors using the subdivision algorithm

In this paper we use the subdivision algorithm to approximate the box dimension of attractors of dynamical systems. Although in theory the subdivision algorithm provides a covering of the attractor with boxes of arbitrarily small diameter, in practice we have to overcome two obstructions: (1) ensure that the covering is (almost) minimal and (2) enhance the speed of convergence to the box dimension. We solve both problems and apply our results to the Hénon, Lorenz, Rössler and Chua attractors. The method suggested in this paper uses information from several subdivision steps and converges to the box dimension much faster than the expression in the definition of the box dimension which uses only one covering of the attractor with boxes of a prescribed diameter.     

High-dimensional chaotic attractors in a gyrotron with nonfixed field structure

The chaotic dynamics of a gyrotron with nonfixed field structure is numerically simulated and the Lyapunov exponents of chaotic attractors are calculated. The dimensions of chaotic attractors estimated using the Kaplan-Yorke formula prove to be anomalously high. This fact is related to the presence of a large number of high-Q eigenmodes in the gyrotron resonator operating in the vicinity of a critical frequency of the electrodynamic system.     

Basins of attraction for chaotic attractors in coupled systems with period doubling

This article studies basins of attraction for chaotic attractors and their evolution when the attractors are restructured through variations in the control parameters. Numerical calculations show the evolution of the boundaries of the attractors, the changes within simply-connected regions, etc. Some of the regularities could be identified in a physical experiment. Pis’ma Zh. Tekh. Fiz. 23, 40–46 (February 26, 1997)     

Investigating the precision of linear dimension measurements

Conclusions It is necessary to oblige the Committee of Standards'laboratories and institutes to supervise systematically the level of errors in production measurements, to generalize the results thus obtained, and to provide industry with information on the accumulated data. It is necessary to improve the qualifications of the personnel engaged in measurements and, in raising the wage scales of workers, to consider without fail their metrological qualifications.Translated from Izmeritel'naya Tekhnika, No. 5, pp. 33–34, May, 1968.     

齿轮-轴承系统非线性振动混沌吸引子周期轨道控制

考虑轴承支撑齿轮传动系统建立了含多间隙的系统非线性动力学模型,模型中考虑了时变啮合刚度和综合传动误差等因素。基于OGY混沌控制改进策略对高维非双曲齿轮系统实施了多周期混沌控制,采用Newton-Raphson迭代法搜寻了P1、P2、P4和P8等多组不稳定周期轨道(Unstable Periodic Orbits,UPO)不动点,求解了各UPO解对应的Jacobi矩阵特征值和局部参数敏感度矢量,结合Poincaré截面等工具解析了混沌吸引子向P10周期轨道转换时的轨道间隔及迁移特性。在2 000周期步下对混沌吸引子实施了P1、P2、P4、P8和P10等多种周期组合式控制,结果表明在状态转换阶段,尤其30周期步内控制参数摄动量发生激增,此后恢复稳定且保持与目标控制轨道相同周期的状态演化;多周期轨道持续控制时周期状态越高,控制难度越大,所需参数摄动量相应增加。研究结果在理论上有助于齿轮系统混沌响应减振控制。     

Information transmission using a synchronous chaotic response with filtering in the communication channel

A principle is proposed for obtaining a chaotic synchronous response in a driven nonlinear oscillatory system under conditions where the signal of the driving chaotic system is filtered by the coupling channel. Music and speech signals are used to demonstrate the efficiency of applying this principle to transmit information through a channel having a limited frequency band. Pis’ma Zh. Tekh. Fiz. 25, 71–77 (August 26, 1999)     

Reconstruction of chaotic oscillations after passage through linear filters

An analysis is made of the diagnostics of chaotic oscillations generated by a dynamic system and distorted by some linear inertial circuit. A method is developed to reconstruct the initial signal and determine the characteristics of the distorting filter using a time series of the process under study. Pis’ma Zh. Tekh. Fiz. 25, 55–60 (February 26, 1999)     

一种单站纯方位目标跟踪中的线性近似化滤波算法

针对EKF算法中存在初始化困难的缺陷，利用近似线性化的方法，构建基于修正极坐标系下的近似线性最小二乘滤波算法ALF；以ALF作为EKF滤波器的初始化算法，实现联合ALF和EKF的两阶段滤波算法，并将它应用在水下单站纯方位目标跟踪中。仿真结果表明，由ALF和EKF组成的两阶段滤波算法具有很好的稳定性，算法精度较高，是一种有效的算法，对潜艇实施隐蔽探测与跟踪具有重要意义．     

Determination of the critical dimension of macrocracks originating at the continuation of a linear rigid inclusion

Physicomechanical Institute, Academy of Sciences of the Ukrainian SSR, Lvov. Translated from Problemy Prochnosti, No. 2, pp. 68–71, February, 1989.     

On the existence of global attractors in principal bundles

This article discusses necessary and sufficient conditions for the existence of global attractors for transformation semigroups on principal bundles. Since the global attractor is a compact set, the discussion involves the compactness of the fibres. A compact structure group is a necessary condition for the existence of the global attractor. In specific situations, the global attractor exists if the structure group is compact.     

An explanation of disproportionate growth using linear 3-dimensional informetrics and its relation with the fractal dimension

Summary We study new and existing data sets which show that growth rates of sources usually are different from growth rates of items. Examples: references in publications grow with a rate that is different (usually higher) from the growth rate of the publications themselves; article growth rates are different from journal growth rates and so on. In this paper we interpret this phenomenon of “disproportionate growth' in terms of Naranan's growth model and in terms of the self-similar fractal dimension of such an information system, which follows from Naranan's growth model. The main part of the paper is devoted to explain disproportionate growth. We show that the “simple' 2-dimensional informetrics models of source-item relations are not able to explain this but we also show that linear 3-dimensional informetrics (i.e. adding a new source set) is capable to model disproportionate growth. Formulae of such different growth rates are presented using Lotkaian informetrics and new and existing data sets are presented and interpreted in terms of the used linear 3-dimensional model.     

The general properties including accuracy and resolution of linear filtering methods for strain estimation

The vast majority of strain imaging systems applies linear filtering to estimate strain from displacement data. Methods such as piecewise-linear least squares regression and staggered strain estimation have come to be widely known and applied, but the properties of these estimators have rarely (or never) been compared quantitatively. Given their tractable properties, careful analysis of linear filters allows us to make numerous observations that are simple, yet valuable. We consider accuracy and resolving power, which raises the question of whether any particular filter offers the best possible accuracy at a given resolution. Our surprising results provide insight at two levels: They highlight general considerations affecting the type of filter that is appropriate for practical applications, and indicate promising avenues for further research.     

Lorenz attractors in unfoldings of homoclinic-flip bifurcations

Lorenz-like attractors are known to appear in unfoldings from certain codimension two homoclinic bifurcations for differential equations in ?³ that possess a reflectional symmetry. This includes homoclinic loops under a resonance condition and the inclination-flip homoclinic loops. We show that Lorenz-like attractors also appear in the third possible codimension two homoclinic bifurcation (for homoclinic loops to equilibria with real different eigenvalues); the orbit-flip homoclinic bifurcation. We moreover provide a bifurcation analysis computing the bifurcation curves of bifurcations from periodic orbits and discussing the creation and destruction of the Lorenz-like attractors. Known results for the inclination flip are extended to include a bifurcation analysis.     

Dynamics near the product of planar heteroclinic attractors

Motivated by problems in equivariant dynamics and connection selection in heteroclinic networks, Ashwin and Field investigated the product of planar dynamics where at least one of the factors was a planar homoclinic attractor. However, they were only able to obtain partial results in the case of a product of two planar homoclinic attractors. We give general results for the product of planar homoclinic and heteroclinic attractors. We show that the likely limit set of the basin of attraction of the product of two planar heteroclinic attractors is always the unique one-dimensional heteroclinic network which covers the heteroclinic attractors in the factors. The method we use is general and likely to apply to products of higher dimensional heteroclinic attractors as well as to situations where the product structure is broken but the cycles are preserved.     

无界域上椭圆型方程的正解

讨论了无界域上的一类拟线性椭圆型方程正解的存在性,利用山路引理得到了一个存在性结果。     

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.     

Distribution and enumeration of attractors in probabilistic boolean networks

Many mathematical models for gene regulatory networks have been proposed. In this study, the authors study attractors in probabilistic Boolean networks (PBNs). They study the expected number of singleton attractors in a PBN and show that it is (2 - (1=2)/sup L-1)/sup n/, where n is the number of nodes in a PBN and L is the number of Boolean functions assigned to each node. In the case of L = 2, this number is simplified into 1.5/sup n/. It is an interesting result because it is known that the expected number of singleton attractors in a Boolean network (BN) is 1. Then, we present algorithms for identifying singleton and small attractors and perform both theoretical and computational analyses on their average case time complexities. For example, the average case time complexities for identifying singleton attractors of a PBN with L = 2 and L = 3 are O(1.601/sup n/) and O(1.763/sup n/), respectively. The results of computational experiments suggest that these algorithms are much more efficient than the naive algorithm that examines all possible 2/sup n/ states.