动力系统简介

本文对动力系统这一学科的历史、理论、应用以及与其他学科的联系作一简介.     

Stochastic modeling of a billiard in a gravitational field: Power law behavior of Lyapunov exponents

We consider the motion of a point particle (billiard) in a uniform gravitational field constrained to move in a symmetric wedge-shaped region. The billiard is reflected at the wedge boundary. The phase space of the system naturally divides itself into two regions in which the tangent maps are respectively parabolic and hyperbolic. It is known that the system is integrable for two values of the wedge half-angle ₁ and ₂ and chaotic for ₁<< ₂. We study the system at three levels of approximation: first, where the deterministic dynamics is replaced by a random evolution; second, where, in addition, the tangent map in each region is, replaced by its average; and third, where the tangent map is replaced by a single global average. We show that at all three levels the Lyapunov exponent exhibits power law behavior near ₁ and ₂ with exponents 1/2 and 1, respectively. We indicate the origin of the exponent 1, which has not been observed in unaccelerated billiards.     

Correlation properties of combiners with memory in stream ciphers

For pseudo-random generators where one or several LFSRs are combined by a memoryless function, it is known that the output sequences are correlated to certain LFSR-sequences whose correlation coefficients c _t satisfy the equation _i c ₂ ⁱ = 1. In this paper it is proved that a corresponding result also holds for generators whose LFSRs are connected to a combiner with memory.If correlation probabilities are conditioned on side information, e.g., on known output digits, it is shown that new or stronger correlations may occur. This is exemplified for the summation cipher with only two LFSRs where such correlations can be exploited in a known plaintext attack. A cryptanalytic algorithm is given which is shown to be successful for LFSRs of considerable length and with arbitrary feedback connection.A preliminary version of this paper was presented at Eurocrypt '90, May 21–24, Århus, Denmark, and has appeared in the proceedings, pp. 204–213.     

Forms of vibrations and structural changes in liquid state

Summary The forms of vibrations and displacements of particles in amorphous structures have been investigated. The particles, moving on highly non-linear amplitude, are responsible for the creation of disordered structures of amorphous bodies. The non-linear oscillators, even if &apos;few&apos; in concentration, are characterized by unpredictable trajectories in phase space. The non-linear oscillators are fully developed in the liquid state above the crossover temperature T_cr and between T_cr and T_g their number decreases. Under T_g they completely disappear. The interconnection between the linear oscillators in blocks plays the most important role in the characteristic time spectra in liquid state. Using the additive properties of elements polarizibilities, the number of acoustical units in individual blocks at T_cr is estimated to be about 600 units. The diameter of blocks at T_cr was estimated to be about 1.8 nm. Even if the non-linear high amplitude motions disappear at solidification, the remnants of structural irregularity remain and the disordered structure of glass is formed.     

Set controllability of Markov jump switching Boolean control networks and its applications

In this article, a novel method is proposed for investigating the set controllability of Markov jump switching Boolean control networks (MJSBCNs). Specifically, the switching signal is described as a discrete-time homogeneous Markov chain. By resorting to the expectation and switching indicator function, an expectation system is constructed. Based on the expectation system, a novel verifiable condition is established for solving the set reachability of MJSBCNs. With the newly obtained results on set reachability, a necessary and sufficient condition is further derived for the set controllability of MJSBCNs. The obtained results are applied to Boolean control networks with Markov jump time delays. Examples are demonstrated to justify the theoretical results.     

Dynamic complexity of optimal paths and discount factors for strongly concave problems

We study the relationship between the dynamical complexity of optimal paths and the discount factor in general infinite-horizon discrete-time concave problems. Given a dynamic systemx _t+1=h(x _t ), defined on the state space, we find two discount factors 0 < ^{* **} < 1 having the following properties. For any fixed discount factor 0 < < ^*, the dynamic system is the solution to some concave problem. For any discount factor ^** < < 1, the dynamic system is not the solution to any strongly concave problem. We prove that the upper bound ^** is a decreasing function of the topological entropy of the dynamic system. Different upper bounds are also discussed.This research was partially supported by MURST, National Group on Nonlinear dynamics in Economics and Social Sciences. The author would like to thank two anonymous referees for helpful comments and suggestions.     

Probabilistic approach to homoclinic chaos

Three-dimensional systems possessing a homoclinic orbit associated to a saddle focus with eigenvalues ±i, – and giving rise to homoclinic chaos when the Shil'nikov condition < is satisfied are studied. The 2D Poincaré map and its 1D contractions capturing the essential features of the flow are given. At homoclinicity, these 1D maps are found to be piecewise linear. This property allows one to reduce the Frobenius—Perron equation to a master equation whose solution is analytically known. The probabilistic properties such as the time autocorrelation function of the state variablex are explicitly derived.     

改进海鸥算法的多阈值图像分割算法北大核心CSCD

为进一步提高图像分割精度,改善传统多阈值图像分割方法计算量大、分割慢的问题,提出了改进海鸥算法(improved seagull optimization algorithm,ISOA)的多阈值图像分割方案。针对原始海鸥算法(seagull optimization algorithm,SOA)存在早熟、寻优精度不足的问题,首先,采用cubic混沌映射优化初始解,提高搜索效率;其次,引入鹰栖息优化算法(eagle perching optmizer,EPO)的缩放因子和疯狂算子进行扰动,并与麻雀搜索算法(sparow search algorithm,SSA)警戒者的位置更新相结合,优化寻优精度和收敛速度,避免陷入局部最优。利用6种基准测试函数对ISOA进行寻优性能测试。最后,将ISOA与图像分割的最优阈值选取相结合,进行基于Otsu的多阈值图像分割,并与现有分割算法进行对比。仿真结果表明,ISOA在基于Otsu的图像分割中,100%取得了最优值,且80.9%的结果优于其余算法,使图像的分割精度和质量均得到了优化。     

The Boolean Prime Ideal Theorem Plus Countable Choice Do Not Imply Dependent Choice

Two Fraenkel-Mostowski models are constructed in which the Boolean Prime Ideal Theorem is true. In both models, AC for countable sets is true, but AC for sets of cardinality 2 and the 2m = m principle are both false. The Principle of Dependent Choices is true in the first model, but false in the second. Mathematics Subject Classification: 03E25, 03E35, 04A25.     

On fluctuations in μ-space

A master equation is derived microscopically to describe the fluctuating motion of the particle density in . space. This equation accounts for the drift motion of particles and is valid for any inhomogeneous gas. The Boltzmann equation is obtained from the first moment of this equation by neglecting the second cumulant (the pair correlation function). The successive moments form coarse-grained BBGKY-like hierarchy equations, in which small spatial regions with r_ij < the force range are smeared out. These hierarchy equations are convenient for investigating the nonequilibrium long-range pair correlation function, which arises mainly from sequences of isolated binary collisions and gives rise to the much-discussed long-time tail and the logarithmic term in the density expansion of transport coefficients. It is shown to have a spatial long tail, like the Coulombic potential, in a steady laminar flow. The stochastic nature of the nonlinear Boltzmann-Langevin equation is also investigated; the random source term is found to be expressed as a linear superposition of Poisson random variables and to become Gaussian in special cases.