统计物理的微观动力学基础

统计的基本出发点是研究系统具有的随机性,不同系统在不同情形下的宏观热力学性质起源于系统内部随机性的差异,通过对宏观热力学系统的微观非线性动力学进行研究探索，我们可以进一步更为深入地理解物态方程、相变等诸多的宏观热力学现象。本文通过哈密顿系统的非线性动力学研究，以及遍历性理论的动力学随机性研究对此问题进行了分析，研究表明，动力学系统的全局性混沌是系统统计成立的根本要素，系统的无限大自由度（热力学极限）已不是决定性的因素，人们可以在此基础上建立少自由度系统的统计力学及热力学。     

一类动力学系统通过函数耦合实现混沌同步

非线性动力学系统的混沌同步, 一般采用单向线性耦合的控制方式, 对于函数耦合方式研究的比较少. 这就存在一个问题, 对于非线性动力学系统, 在线性耦合实现混沌同步后, 是否其他函数的耦合方式都可以实现混沌同步? 本文对于一类非线性动力学系统, 研究了其线性耦合同步与函数耦合同步的关系, 证明当线性耦合实现同步后, 函数在满足一定的条件下, 可以通过函数耦合实现系统的混沌同步. 最后对于Duffing系统采用两种函数耦合进行了仿真计算, 证明了结论的正确性.     

立方非线性Schr?dinger方程的动力学行为分析

本文采用辛算法，研究了立方非线性Schr?dinger方程在不同的空间离散下的动力学行为，并讨论了其在不同的非线性参数下的动力学性质。 我们指出，系统的守恒量在长时间的演化下保持不变，但对于强非线性，不同的空间离散精度，将导致不同的动力学行为。     

固体破坏的损伤演化诱致突变现象

固体破坏问题在理论上及实际上均极为重要，是涉及力学，物理学及非线性科学等学科的一个十分复杂基本问题，文章介绍了基于细观非线性动力学模型的研究所取得的进展，发现系统显示一种共性特征，称为演化诱致突变，即演化模式从整体稳定向灾变性模式转变，宏观破坏的样本个性行为，以及宏观性质对细观无序性的敏感性。     

非线性Schroedinger方程的动力学行为分析

采用辛算法数值求解非线性Schroedinger方程的周期初值问题，建立不同的相空间来分忻其动力学持性．首先比较分析了不同的相空间中立方非线性Sehroedinger方程在不同立方非线性参数下的长时间演化的动力学特性，然后讨论了相空间中立方一五次方非线性Sehroedinger方程在不同立方和五次方非线性参数下的长时间演化的动力学特性，数值结果显示，对于不同的立方非线性参数，随着五次方非线性参数的增加，动力学行为的演化路径是不一样的．     

一个非线性力学问题的讨论

利用非线性动力学的原理和方法，通过对不动点类型的分析，并利用数值计算给出所研究系统的相图，从而得出质点在不同条件下的运动情况。     

基于Matlab的非线性动力学系统分析

利用Matlab软件对洛仑兹和达芬2个典型的非线性动力学系统进行了较为详细的分析．在使学生初步掌握了如何使用Matlab软件解决数值分析中的一些计算问题和绘图问题的同时．也使学生对非线性动力学系统的运动本质有一定的认识．     

离子迁移忆阻混沌电路及其在语音保密通信中的应用

忆阻器是一种具有记忆功能和纳米级尺寸的非线性元件, 作为混沌系统的非线性部分, 能够使系统的物理尺寸大大减小, 同时可以得到各种丰富的非线性曲线, 提高混沌系统的复杂度和信号的随机性. 因此, 本文采用离子迁移忆阻器的磁控模型设计了一个新的混沌系统. 通过理论推导、数值仿真、Lyapunov指数谱、分岔图和Poincaré截面图研究了系统的基本动力学特性, 并分析了改变不同参数时系统动力学行为的变化. 同时, 建立了模拟该系统的SPICE电路, SPICE仿真结果与数值分析相符, 从而验证该混沌系统的混沌产生能力. 最后, 利用线性反馈同步控制方法实现了新构造的离子迁移忆阻混沌系统的同步, 并且采用该同步方法有效实现了语音信号的保密通信. 数值仿真证实了新混沌系统的存在性以及同步控制应用的可行性.     

二端面转轴相对转动非线性动力学系统的稳定性与近似解

建立了二端面转轴相对转动系统的非线性动力学方程.对于等力矩的动力学方程进行了定性分析，得到了方程的稳定性等性质.用平均方法求得方程在一定条件下的近似解. 关键词： 非线性动力学方程 稳定性 极限环 近似解     

一个非线性力学问题的讨论

利用非线性动力学的原理和方法，通过对不动点类型的分析，并利用数值计算给出所研究系统的相图，从而得出质点在不同条件下的运动情况。     

What is life? Active particles tools towards behavioral dynamics in social-biology and economics

This review paper is devoted to study the conceptual difficulties that mathematics meets when attempting to describe the complexity of living matter focusing on the challenging perspective of developing a mathematical theory for living systems including mutations and selection. The quest starts with the identification of a number of common complexity features of living systems. Then, mathematical structures are derived to include these features, while mathematical models are derived by inserting in the structures models of individual based interactions. Three applications are examined by active particles methods, i.e., models of SARS2-CoV-2 pandemics, models of idiosyncratic learning in open markets and of the dynamics of prices accounting for human behaviors. A critical study, which pervades the whole paper, shows that also economics can be viewed as a behavioral science thus accounting for specific aspects typical of living systems.     

Geometry and nonlinear evolution equations

We briefly review the nonlinear dynamics of diverse physical systems which can be described in terms of moving curves and surfaces. The interesting connections that exist between the underlying differential geometry of these systems and the corresponding nonlinear partial differential equations are highlighted by considering classic examples such as the motion of a vortex filament in a fluid and the dynamics of a spin chain. The association of the dynamics of a non-stretching curve with a hierarchy of completely integrable soliton-supporting equations is discussed. The application of the surface embeddability approach is shown to be useful in obtaining such connections as well as exact solutions of some nonlinear systems such as the Belavin-Polyakov equation and the inhomogeneous Heisenberg chain.     

Nonlinear dynamics: Challenges and perspectives

The study of nonlinear dynamics has been an active area of research since 1960s, after certain path-breaking discoveries, leading to the concepts of solitons, integrability, bifurcations, chaos and spatio-temporal patterns, to name a few. Several new techniques and methods have been developed to understand nonlinear systems at different levels. Along with these, a multitude of potential applications of nonlinear dynamics have also been enunciated. In spite of these developments, several challenges, some of them fundamental and others on the efficacy of these methods in developing cutting edge technologies, remain to be tackled. In this article, a brief personal perspective of these issues is presented.     

Methodology for the solutions of some reduced Fokker‐Planck equations in high dimensions

In this paper, a new methodology is formulated for solving the reduced Fokker‐Planck (FP) equations in high dimensions based on the idea that the state space of large‐scale nonlinear stochastic dynamic system is split into two subspaces. The FP equation relevant to the nonlinear stochastic dynamic system is then integrated over one of the subspaces. The FP equation for the joint probability density function of the state variables in another subspace is formulated with some techniques. Therefore, the FP equation in high‐dimensional state space is reduced to some FP equations in low‐dimensional state spaces, which are solvable with exponential polynomial closure method. Numerical results are presented and compared with the results from Monte Carlo simulation and those from equivalent linearization to show the effectiveness of the presented solution procedure. It attempts to provide an analytical tool for the probabilistic solutions of the nonlinear stochastic dynamics systems arising from statistical mechanics and other areas of science and engineering.     

Dynamic characteristics of nonlinear Bragg gratings in photonic crystal fibres

We study numerically the temporal and spatial dynamics of light in Bragg gratings in highly nonlinear photonic crystal fibre for a CW input signal. Our numerical model is based on the plane wave mode solver and a set of nonlinear coupled-mode equations which we solve using a variation of implicit fourth order Runge-Kutta method. We observe not only bistability of the intensity versus transmitted and reflected light but also complex dynamics. We demonstrate that for values of input intensity above the bistable region the steady state may undergo a supercritical Hopf bifurcation. For some ranges of the input intensity we also observe a coexistence of two periodic attractors. The dynamics found, in particular the features in the bifurcation diagram, strongly depend on the parameters of the fibre. Consequently, we suggest that by proper design of the photonic crystal in the cladding we can adjust such nonlinear features of the Bragg gratings as the width of the bistable region, the intensity at which the bifurcation occurs and also the characteristics of the dynamics at high values of input intensity.     

Modeling and Analysis of Social Phenomena: Challenges and Possible Research Directions

This opening editorial aims to interest researchers and encourage novel research in the closely related fields of sociophysics and computational social science. We briefly discuss challenges and possible research directions in the study of social phenomena, with a particular focus on opinion dynamics. The aim of this Special Issue is to allow physicists, mathematicians, engineers and social scientists to show their current research interests in social dynamics, as well as to collect recent advances and new techniques in the analysis of social systems.     

Investigations on dynamics of interacting cavitation bubbles in strong acoustic fields

Given its importance to the dynamics of cavitation bubbles, the mutual interaction between bubbles was carefully investigated in this work. The cavitation noises emitted in different sonication conditions were recorded to study the dynamical behavior of the bubbles. The frequency spectra of the noises suggest that the dispersing state of the bubbles severely influence the oscillations of bubbles, and that the nonlinear feature of the dynamics of cavitation bubbles, imposed by the mutual bubble-bubble interaction, gradually develops with the decrease of the dispersing height. Theoretical analysis shows that the size difference between the interacting bubbles should be responsible for the increase of nonlinearity of the oscillation, and that the decrease of the distance between them could effectively enhance the nonlinear feature of the oscillation of the bubble, both of which agree well with the experimental observation.