Spatio-temporal symmetries and bifurcations via bi-orthogonal decompositions

Summary A tool for analyzing spatio-temporal complex physical phenomena was recently proposed by the authors, Aubry et al. [5]. This tool consists in decomposing a spatially and temporally evolving signal into orthogonal temporal modes (temporal “structures”) and orthogonal spatial modes (spatial “structures”) which are coupled. This allows the introduction of a temporal configuration space and a spatial one which are related to each other by an isomorphism. In this paper, we show how such a tool can be used to analyze space-time bifurcations, that is, qualitative changes in both space and time as a parameter varies. The Hopf bifurcation and various spatio-temporal symmetry related bifurcations, such as bifurcations to traveling waves, are studied in detail. In particular, it is shown that symmetry-breaking bifurcations can be detected by analyzing the temporal and spatial eigenspaces of the decomposition which then lose their degeneracy, namely their invariance under the symmetry. Furthermore, we show how an extension of the theory to “quasi-symmetries” permits the treatment of nondegenerate signals and leads to an exponentially decreasing law of the energy spectrum. Examples extracted from numerically obtained solutions of the Kuramoto-Sivashinsky equation, a coupled map lattice, and fully developed turbulence are given to illustrate the theory.     

随机性参数自适应的混沌同步

对两个不同参数的混沌系统进行随机性参数自适应控制，选取合适的控制律和反馈系数，导致其同步.以Henon映射为例进行数值模拟，结果表明，由于控制周期和反馈系数的随机变化，具有一定的实用意义. 关键词： Henon映射 混沌同步 随机性自适应控制     

A study of the chaotic behaviour of the elastic-plastic Euler arch

This paper deals with the dynamics of a truss structure, the Euler arch. The bars are made of elastic-plastic material, and the structure can exhibit large displacements. The aim of this paper is to give evidence of the possible chaotic behavior of this structure, even in the presence of a hardening plastic branch. The tools used are the diagrams of bifurcation, the measure of the dimension of the attractor, the Kolmogorov entropy, and the maximum Lyapunov exponent. This study emphasizes the sensitivity to the initial conditions by means of generalized basins of attraction.     

Probabilistic models of multidimensional piecewise expanding mappings

Strongly chaotic systems (e.g., piecewise expanding mappings) exhibit diffusion-like behavior in the sense of central limit theorems. To find more precise statements about the similarity to probabilistic diffusion, we study how the evolution of probability densities underd-dimensional piecewise expanding mappings can be modeled by Markov processes with smooth transition probabilities (such as diffusion processes). Our results can be viewed as a special type of local limit theorem.     

机械动态系统混沌状态特征与判据应用研究

本基于混沌动力学与分行几何学理论分析，通过研究机械动态系统的Lypunov指数、关联维数、Kolmogorov熵以及它们对系统吸引子特征的描述，结合递次振幅图方法，提供了一种证明系统状态混沌性的方法。并以变速箱中齿轮运行状态为例，通过对其动态信号混沌特征量的研究，证实了该系统混沌性，进而得出对系统故障进行进一步混沌预测的可行性。     

漩涡阻止器水力性能的PIV试验分析及其机理研究

漩涡阻止器是用来防止或减少漩涡发生的特殊装置,它具有特殊的几何形状,能够优化水力机械内部的流动。在水泵吸水池内部,尤其在吸入口的周围存在着较大的漩涡,在吸入口的下方,安装一个“T”型的漩涡阻止器,应用PIV技术测出安装漩涡阻止器前后吸水池内部的流场分布,并对其流动特性进行深入分析,证明了该阻止器能够有效的减小漩涡发生。对这种“T”型漩涡阻止器的作用过程进行深入的研究发现,能够用摩阻理论来解释它的作用机理。     

非线性非均衡蛛网模型的动态分析

描述单商品市场价格波动的蛛网模型是动态经济分析中的经典模型 ,证明了由模型 ( )所生成的价格序列在一定的条件下不会出现 3以上周期运动和混沌现象 ,并对其进行稳定性分析 ,得到模型关于均衡价格稳定的六个判别定理 .     

Local structure theory: Calculation on hexagonal arrays,and interaction of rule and lattice

Local structure theory calculations⁷ are applied to the study of cellular automata on the two-dimensional hexagonal lattice. A particular hexagonal lattice rule denoted (3422) is considered in detail. This rule has many features in common with Conway'sLife. The local structure theory captures many of the statistical properties of this rule; this supports hypotheses raised by a study ofLife itself⁽⁶⁾. As inLife, the state of a cell under (3422) depends only on the state of the cell itself and the sum of states in its neighborhood at the previous time step. This property implies that evolution rules which operate in the same way can be studied on different lattices. The differences between the behavior of these rules on different lattices are dramatic. The mean field theory cannot reflect these differences. However, a generalization of the mean field theory, the local structure theory, does account for the rule-lattice interaction.     

一种改进的谐振参数扰动法及其在DC/DC变换器混沌控制中的应用

种改进的谐振参数扰动法来控制混沌，该混沌控制方法引入了基于非线性映射的调制，其控制目标可以是任意周期的轨道，而不是原混沌吸引子中的嵌入轨道。作为一个应用实例，文中描述了该方法在电流模式控制dc/dc变换器中的应用结果。