耦合相振子系统同步的序参量理论

节律行为,即系统行为呈现随时间的周期变化,在我们的周围随处可见.不同节律之间可以通过相互影响、相互作用产生自组织,其中同步是最典型、最直接的有序行为,它也是非线性波、斑图、集群行为等的物理内在机制.不同的节律可以用具有不同频率的振子(极限环)来刻画,它们之间的同步可以用耦合极限环系统的动力学来加以研究.微观动力学表明,随着耦合强度增强,振子同步伴随着动力学状态空间降维到一个低维子空间,该空间由序参量来描述.序参量的涌现及其所描述的宏观动力学行为可借助于协同学与流形理论等降维思想来进行.本文从统计物理学的角度讨论了耦合振子系统序参量涌现的几种降维方案,并对它们进行了对比分析.序参量理论可有效应用于耦合振子系统的同步自组织与相变现象的分析,通过进一步研究序参量的动力学及其分岔行为,可以对复杂系统的涌现动力学有更为深刻的理解.     

介观物理系统的统计物理和持续电流

本文讨论介观系统的统计物理一些基本概念，在一般教科书和文献教很少有解释到的。我们讨论自平均效应，它在介观与宏观系统之不同，对杂质作系统平均后热力学量在介观与宏观系之不同。我们较详细引介用格林函数和图解法计算介观持续电流。同样，我们强调宏观系统和介观系统不同之处，对介观物理系统从计算热力学量的角度来看需要特别考虑因素。     

远离平衡问题及其描述的理论和方法概述

本文简要阐述了耗散结构、混沌、外场驱动下的耗散系统非线性动力学等远离平衡和非线性问题,及描述这些问题的各种宏观、半宏观及非平衡统计物理的理论和方法。     

远离平衡问题及其描述的理论和方法概述

本文简要阐述了耗散结构、混沌、外场驱动下的耗散系统非线性动力学等远离平衡和非线性问题,及描述这些问题的各种宏观、半宏观及非平衡统计物理的理论和方法。     

生物系统中的非线性现象

文章通过用非线性科学的基本概念与系统方法分析了芽殖酵母细胞周期网络的动力学行为，说明非线性科学在定量研究生命过程所可能起到的重要作用，同时说明非线性科学现有的分析手段在研究生命系统中的局限性．这样利用非线性动力学研究生物系统的动力学问题，不但可以为系统生物学的研究提供强大的数学工具，同时也可以推动非线性科学本身的发展．这项研究从另一个角度显示了交叉学科的生命力．     

介观物理系统的统计物理和持续电流

本文讨论介观系统的统计物理一些基本概念 ,在一般教科书和文献都很少有解释到的。我们讨论自平均效应 ,它在介观与宏观系统之不同 ,对杂质作系统平均后热力学量在介观与宏观系之不同。我们较详细引介用格林函数和图解法计算介观持续电流。同样 ,我们强调宏观系统和介观系统不同之处 ,对介观物理系统从计算热力学量的角度来看需要特别考虑因素。     

强迫耗散系统的有序结构和系统的发展(Ⅰ)，最小熵产生原理和有序结构

一个系统的发展总是由不可逆热力过程和非线性动力过程所驱动.将大气动力学方程组同考虑了动能变化的Gibbs关系结合起来构建的熵平衡方程，才能更好地描述大气系统的不可逆热力过程和非线性动力过程.至今非平衡态热力学仅利用Onsager线性唯象关系证明了最小熵产生原理.利用新建立的熵平衡方程和大气动力学方程的性质证明，最小熵产生原理在热力学线性区和非线性区都是普遍成立的.且当热量输送平衡、水汽输送平衡和动量输送平衡时，系统达到不可逆过程最弱的最小熵产生热力学状态.当系统又是动力平衡且无平流时，这种最小熵产生态就是 关键词： 非线性热力学 熵产生 最小熵产生原理 有序结构     

非线性传送带系统的复杂分岔

讨论了一类单自由度非线性传送带系统. 首先通过分段光滑动力系统理论得出系统滑动区域的解析分析和平衡点存在性条件; 其次利用数值方法, 对系统几种类型的周期轨道进行单参数和双参数延拓, 得到系统的余维一滑动分岔曲线和若干余维二滑动分岔点, 以及系统在参数空间中的全局分岔图. 通过对系统分岔行为的研究, 反映出传送带速度和摩擦力振幅对系统动力学行为有较大影响, 揭示了非线性传送带系统的复杂动力学现象. 关键词： 传送带系统 滑动分岔 周期运动     

力与耦合系统的交互作用和随机能量共振

外界对系统作功的过程就是系统状态发生变化的过程.根据Langevin方程的随机动力学特性,采用微观动力学和宏观热力学方法,建立了基于单一随机轨线的耦合双稳系统的热力学关系.通过力与耦合系统的功交互作用定量地刻画了能量的传递和转换关系,揭示了耦合系统中存在着的随机能量共振现象.从作功与能量的关系进行分析,进一步揭示了随机共振产生的物理本质. 关键词： 耦合双稳系统 随机热力学 随机能量共振 功     

夹具动力学效应的工程分析

振动夹具附加约束的客观存在，使得在研究振动试验模拟的有效性时必须充分考虑夹具附加约束的动力学效应。通过边界的动力学设计，可以使得改变了边界的试验系统能够满足给定的动力学特性要求，但这种分析的工程应用需要依靠复杂的工程与工具。文中提出了利用二自由度模型评估夹具动力学效应的工程方法，并通过数字模拟进行了方法的验证。     

Chaotic behavior in nonlinear Hamiltonian systems and equilibrium statistical mechanics

The relation between chaotic dynamics of nonlinear Hamiltonian systems and equilibrium statistical mechanics in its canonical ensemble formulation has been investigated for two different nonlinear Hamiltonian systems. We have compared time averages obtained by means of numerical simulations of molecular dynamics type with analytically computed ensemble averages. The numerical simulation of the dynamic counterpart of the canonical ensemble is obtained by considering the behavior of a small part of a given system, described by a microcanonical ensemble, in order to have fluctuations of the energy of the subsystem. The results for the Fermi-Pasta-Ulam model (i.e., a one-dimensional anharmonic solid) show a substantial agreement between time and ensemble averages independent of the degree of stochasticity of the dynamics. On the other hand, a very different behavior is observed for a chain of weakly coupled rotators, where linear exchange effects are absent. In the high-temperature limit (weak coupling) we have a strong disagreement between time and ensemble averages for the specific heat even if the dynamics is chaotic. This behavior is related to the presence of spatially localized chaos, which prevents the complete filling of the accessible phase space of the system. Localized chaos is detected by the distribution of all the characteristic Liapunov exponents.     

Stochasticity in many-dimensional nonlinear oscillating systems

Qualitative analytic estimates of the stochasticity limit of one- and many-dimensional nonlinear oscillating systems are derived using the overlapping of first-order resonances as a criterion for stochasticity. Computational results obtained with several very simple transformations are compared with the analytic estimates. Numerical studies significantly below the stochasticity limit of a many-dimensional nonlinear system reveal an example of a very slow instability, the first, to the best of our knowledge.     

Controlling global stochasticity of Hamiltonian systems by nonlinear perturbation

A method of controlling global stochasticity in Hamiltonian systems by applying nonlinear perturbation is proposed. With the well-known standard map we demonstrate that this control method can convert global stochasticity into regular motion in a wide chaotic region for arbitrary initial condition, in which the control signal remains very weak after a few kicks. The system in which chaos has been controlled approximates to the original Hamiltonian system, and this approach appears robust against small external noise. The mechanism underlying this high control efficiency is intuitively explained. Received 15 January 2002 Published online 6 June 2002     

Evolutionary game theory: Theoretical concepts and applications to microbial communities

Ecological systems are complex assemblies of large numbers of individuals, interacting competitively under multifaceted environmental conditions. Recent studies using microbial laboratory communities have revealed some of the self-organization principles underneath the complexity of these systems. A major role of the inherent stochasticity of its dynamics and the spatial segregation of different interacting species into distinct patterns has thereby been established. It ensures the viability of microbial colonies by allowing for species diversity, cooperative behavior and other kinds of “social” behavior.A synthesis of evolutionary game theory, nonlinear dynamics, and the theory of stochastic processes provides the mathematical tools and a conceptual framework for a deeper understanding of these ecological systems. We give an introduction into the modern formulation of these theories and illustrate their effectiveness focussing on selected examples of microbial systems. Intrinsic fluctuations, stemming from the discreteness of individuals, are ubiquitous, and can have an important impact on the stability of ecosystems. In the absence of speciation, extinction of species is unavoidable. It may, however, take very long times. We provide a general concept for defining survival and extinction on ecological time-scales. Spatial degrees of freedom come with a certain mobility of individuals. When the latter is sufficiently high, bacterial community structures can be understood through mapping individual-based models, in a continuum approach, onto stochastic partial differential equations. These allow progress using methods of nonlinear dynamics such as bifurcation analysis and invariant manifolds. We conclude with a perspective on the current challenges in quantifying bacterial pattern formation, and how this might have an impact on fundamental research in non-equilibrium physics.     

Seasonally forced disease dynamics explored as switching between attractors

Biological phenomena offer a rich diversity of problems that can be understood using mathematical techniques. Three key features common to many biological systems are temporal forcing, stochasticity and nonlinearity. Here, using simple disease models compared to data, we examine how these three factors interact to produce a range of complicated dynamics. The study of disease dynamics has been amongst the most theoretically developed areas of mathematical biology; simple models have been highly successful in explaining the dynamics of a wide variety of diseases. Models of childhood diseases incorporate seasonal variation in contact rates due to the increased mixing during school terms compared to school holidays. This ‘binary’ nature of the seasonal forcing results in dynamics that can be explained as switching between two nonlinear spiral sinks. Finally, we consider the stability of the attractors to understand the interaction between the deterministic dynamics and demographic and environmental stochasticity. Throughout attention is focused on the behaviour of measles, whooping cough and rubella.     

Domain wall dynamics: Growth laws,localized structures and stable droplets

We discuss the relation between the dynamics of walls separating two equivalent domains and the existence of different kinds of localized structures in systems far from thermodynamic equilibrium. In particular we focus in systems displaying a modulational instability of a flat front where an amplitude equation for the dynamics of the curvature allows to characterize different growth regimes and to predict the existence of stable droplets, localized structures whose stability comes from nonlinear curvature effects.     

Weak and strong chaos in Fermi-Pasta-Ulam models and beyond

We briefly review some of the most relevant results that our group obtained in the past, while investigating the dynamics of the Fermi-Pasta-Ulam (FPU) models. The first result is the numerical evidence of the existence of two different kinds of transitions in the dynamics of the FPU models: (i) A stochasticity threshold (ST), characterized by a value of the energy per degree of freedom below which the overwhelming majority of the phase space trajectories are regular (vanishing Lyapunov exponents). It tends to vanish as the number N of degrees of freedom is increased. (ii) A strong stochasticity threshold (SST), characterized by a value of the energy per degree of freedom at which a crossover appears between two different power laws of the energy dependence of the largest Lyapunov exponent, which phenomenologically corresponds to the transition between weak and strong chaotic regimes. It is stable with N. The second result is the development of a Riemannian geometric theory to explain the origin of Hamiltonian chaos. Starting this theory has been motivated by the inadequacy of the approach based on homoclinic intersections to explain the origin of chaos in systems of arbitrarily large N, or arbitrarily far from quasi-integrability, or displaying a transition between weak and strong chaos. Finally, the third result stems from the search for the transition between weak and strong chaos in systems other than FPU. Actually, we found that a very sharp SST appears as the dynamical counterpart of a thermodynamic phase transition, which in turn has led, in the light of the Riemannian theory of chaos, to the development of a topological theory of phase transitions.     

A priori tests of a stochastic mode reduction strategy

Several a priori tests of a systematic stochastic mode reduction procedure recently devised by the authors [Proc. Natl. Acad. Sci. 96 (1999) 14687; Commun. Pure Appl. Math. 54 (2001) 891] are developed here. In this procedure, reduced stochastic equations for a smaller collections of resolved variables are derived systematically for complex nonlinear systems with many degrees of freedom and a large collection of unresolved variables. While the above approach is mathematically rigorous in the limit when the ratio of correlation times between the resolved and the unresolved variables is arbitrary small, it is shown here on a systematic hierarchy of models that this ratio can be surprisingly big. Typically, the systematic reduced stochastic modeling yields quantitatively realistic dynamics for ratios as large as 1/2. The examples studied here vary from instructive stochastic triad models to prototype complex systems with many degrees of freedom utilizing the truncated Burgers–Hopf equations as a nonlinear heat bath. Systematic quantitative tests for the stochastic modeling procedure are developed here which involve the stationary distribution and the two-time correlations for the second and fourth moments including the resolved variables and the energy in the resolved variables. In an important illustrative example presented here, the nonlinear original system involves 102 degrees of freedom and the reduced stochastic model predicted by the theory for two resolved variables involves both nonlinear interaction and multiplicative noises. Even for large value of the correlation time ratio of the order of 1/2, the reduced stochastic model with two degrees of freedom captures the essentially nonlinear and non-Gaussian statistics of the original nonlinear systems with 102 modes extremely well. Furthermore, it is shown here that the standard regression fitting of the second-order correlations alone fails to reproduce the nonlinear stochastic dynamics in this example.     

连续动力系统时间序列的非线性检验

用相位随机化的替代数据方法，研究了连续混沌动力系统时间序列的非线性检验判定.研究发现，在不同的采样情况下，混沌时间序列的非线性特性检验有所差异，尤其对于过采样时间序列，往往会出现虚假的判断结果.针对过采样时间序列，最好采用能够反映非线性特征的参数，作为检验统计量进行检验判断. 关键词： 替代数据 关联维数 连续时间序列 非线性检验