Polygonal Approaches to the Circular Billiard: the Lyapunov Exponent

The Lyapunov exponents for three polygonal approaches to the circular billiard quasiperiodic approach, random approach and equilateral approach are calculated, the chaotic behavior in polygonal billiards is discussed. The role of singularity presented by vertex angles is quantitatively discussed. It is found that for the equilateral approach, the Lyapunov exponents vary with the side number N according to a Poisson law, i.e., λ(N) = aN exp(-βN).     

Relationship between Continuum of Hurst Exponents of Noise-like Time Series and the Cantor Set

In this paper, we have modified the Detrended Fluctuation Analysis (DFA) using the ternary Cantor set. We propose a modification of the DFA algorithm, Cantor DFA (CDFA), which uses the Cantor set theory of base 3 as a scale for segment sizes in the DFA algorithm. An investigation of the phenomena generated from the proof using real-world time series based on the theory of the Cantor set is also conducted. This new approach helps reduce the overestimation problem of the Hurst exponent of DFA by comparing it with its inverse relationship with α of the Truncated Lévy Flight (TLF). CDFA is also able to correctly predict the memory behavior of time series.     

An Image Encryption Algorithm Using Logistic Map with Plaintext-Related Parameter Values

This paper deals with a plaintext-related image encryption algorithm that modifies the parameter values used by the logistic map according to plain image pixel intensities. The parameter values are altered in a row-wise manner, which enables the usage of the same procedure also during the decryption. Furthermore, the parameter modification technique takes into account knowledge about the logistic map, its fixed points and possible periodic cycles. Since the resulting interval of parameter values achieves high positive values of Lyapunov exponents, the chaotic behavior of the logistic map should be most pronounced. These assumptions are verified by a set of experiments and the obtained numerical values are compared with those reported in relevant papers. It is found that the proposed design that uses a simpler, but well-studied, chaotic map with mitigated issues obtains results comparable with algorithms that use more complex chaotic systems. Moreover, the proposed solution is much faster than other approaches with a similar purpose.     

一种基于重标极差方法的动力学结构突变检测新方法

提出了一种针对相关性时间序列动力学结构突变检测的新方法——滑动移除重标极差分析.数值试验表明,新方法能够准确的检测出理想序列中的动力学结构突变,其检测结果对于子序列的长度依赖性较小,而且具有很强的抗噪能力,弥补了滑动去趋势波动分析的不足. 关键词： 标度指数 滑动移除重标极差分析 滑动去趋势波动分析 动力学结构突变     

激光大气闪烁的高频谱特性

 采用三波长闪烁仪进行了大气边界层的光传输实验,结果表明：在相同的传播条件下,非平面波激光大气闪烁功率谱的高频部分与波长存在着一定的关系,功率谱无标度区间的斜率即标度指数是一个与湍流强度、波长等因素有关的函数,在满足弱起伏的条件下,标度指数一般与波长呈负相关,波长愈长则标度指数愈小,频谱下降的趋势愈快;波长愈短则标度指数愈大,频谱下降的趋势愈慢。     

Multiple Intersection Exponents for Planar Brownian Motion

Let p≥2, n ₁≤⋅⋅⋅≤n _p be positive integers and be independent planar Brownian motions started uniformly on the boundary of the unit circle. We define a p-fold intersection exponent ς _p (n ₁,…,n _p ), as the exponential rate of decay of the probability that the packets , i=1,…,p, have no joint intersection. The case p=2 is well-known and, following two decades of numerical and mathematical activity, Lawler et al. (Acta Math. 187:275–308, 2001) rigorously identified precise values for these exponents. The exponents have not been investigated so far for p>2. We present an extensive mathematical and numerical study, leading to an exact formula in the case n ₁=1, n ₂=2, and several interesting conjectures for other cases.     

Cellular automata simulations on nanocrystallization processes: From instantaneous growth approximation to limited growth

Cellular automata simulations have been performed to simulate the crystallization process under a limited growth approximation. This approximation resembles several characteristics exhibited by nanocrystalline microstructures and nanocrystallization kinetics. Avrami exponent decreases from a value n = 4 indicating interface controlled growth and constant nucleation rate to a value n ~ 1 indicating absence of growth. A continuous change of the growth contribution to the Avrami exponent from zero to 3 is observed as the composition of the amorphous phase becomes richer in the element present in the crystalline phase.     

Flux vector splitting solutions for coupling hydraulic transient of gas-liquid-solid three-phase flow in pipelines

The gas-liquid-solid three-phase mixed flow is the most general in multiphase mixed transportation. It is significant to exactly solve the coupling hydraulic transient problems of this type of multiphase mixed flow in pipelines. Presently, the method of characteristics is widely used to solve classical hydraulic transient problems. However, when it is used to solve coupling hydraulic transient problems, excessive interpolation errors may be introduced into the results due to unavoidable multiwave interpolated calculations. To deal with the problem, a finite difference scheme based on the Steger-Warming flux vector splitting is proposed. A flux vector splitting scheme is established for the coupling hydraulic transient model of gas-liquid-solid three-phase mixed flow in the pipelines. The flux subvectors are then discretized by the Lax-Wendroff central difference scheme and the Warming-Beam upwind difference scheme with second-order precision in both time and space. Under the Rankine-Hugoniot conditions and the corresponding boundary conditions, an effective solution to those points located at the boundaries is developed, which can avoid the problem beyond the calculation region directly induced by the second-order discrete technique. Numerical and experimental verifications indicate that the proposed scheme has several desirable advantages including high calculation precision, excellent shock wave capture capability without false numerical oscillation, low sensitivity to the Courant number, and good stability.     

Continuity of Lyapunov exponent for analytic quasi-periodic cocycles on higher-dimensional torus

It is known that the Lyapunov exponent is not continuous at certain points in the space of continuous quasi-periodic cocycles. We show that the Lyapunov exponent is continuous for a higher-dimensional analytic category in this paper. It has a modulus of continuity of the form exp(−∣logt∣ ^σ ) for some 0 < σ < 1.