A new type of four-wing chaotic attractors in 3-D quadratic autonomous systems

In this paper, several smooth canonical 3-D continuous autonomous systems are proposed in terms of the coefficients of nonlinear terms. These systems are derived from the existing 3-D four-wing smooth continuous autonomous chaotic systems. These new systems are the simplest chaotic attractor systems which can exhibit four wings. They have the basic structure of the existing 3-D four-wing systems, which means they can be extended to the existing 3-D four-wing chaotic systems by adding some linear and/or quadratic terms. Two of these systems are analyzed. Although the two systems are similar to each other in structure, they are different in dynamics. One is sensitive to the initializations and sampling time, but another is not, which is shown by comparing Lyapunov exponents, bifurcation diagrams, and Poincaré maps.     

双频内共振系统的Normal Form及其简化

讨论双频内共振系统的 Normal Form及其降维问题.利用发展的 Normal Form直接方法,导出了任意双频内共振系统 Normal Form的一般形式.指出 Poincare共振项分为内共振项和非内共振性两类,并定义了内共振项的阶.提出了一种普遍适用的降维变换,并证明了该变换可将任意双频内共振系统的 Normal Form方程降到3维.应用举例表明,该变换不仅适用于半单问题,也适于非半单问题（即强:1:1内共振系统）.     

系统科学方法论探索

众所周知,系统科学是适应科学方法论的变革而产生的新学科。系统科学研究自然界和人类社会各类复杂系统的共同特性,探索系统的生成、演化和涌现的普遍规律。系统科学与经典科学在方法论上有着本质的区别。经典科学注重还原分析,而系统科学强调整体把握。本文首先回顾我国在系统科学方法论上所取得的重要成果,接着简介作者从运筹学角度对描述方法所做的研究,然后着重探讨网络科学的研究方法对系统科学的借鉴作用。      

Skew-gradient representations of constrained mechanical systems

The characteristics of stationary and non-stationary skew-gradient systems are studied. The skew-gradient representations of holonomic systems, Birkhoffian systems, generalized Birkhoffian systems, and generalized Hamiltonian systems are given. The characteristics of skew-gradient systems are used to study integration and stability of the solution of constrained mechanical systems. Examples are given to illustrate applications of the result.     

Projective synchronization between two different time-delayed chaotic systems using active control approach

In this paper, we investigate the projective synchronization between two different time-delayed chaotic systems. A suitable controller is chosen using the active control approach. We relax some limitations of previous work, where projective synchronization of different chaotic systems can be achieved only in finite dimensional chaotic systems, so we can achieve projective synchronization of different chaotic systems in infinite dimensional chaotic systems. Based on the Lyapunov stability theory, we suggest a generic method to achieve the projective synchronization between two different time-delayed chaotic systems. The validity of the proposed method is demonstrated and verified by observing the projective synchronization between two well-known time-delayed chaotic systems; the Ikeda system and Mackey–Glass system. Numerical simulations fully support the analytical approach.     

Bridging reductionism and holism — multi-scale methodology with applications to particle-fluid systems

While science continues to extend to two extremes — micro-scale towards dimensions even smaller than elemental particles and mega-scale even beyond the universe, one recognizes that reductionism is not sufficient to solve many problems we encounter in engineering, which are likely characterized by nonlinearity, nonequilibrium and dissipative multi-scale structures. On the other hand, the common features of these nonlinear systems, such as bifurcation, state multiplicity and self-organization, have attracted much attention, leading to the approaches of the so-called complexity science which has become a focus not only in natural science and engineering science, but also in social science.However, no effective methodology has been established to understand these complex systems, though noticeable progress has been achieved in studying these systems, such as particle-fluid multi-phase systems. Multi-scale methodology has been considered as a promising methodology to tackle complex systems due to its capability in correlating phenomena at different scales. In this presentation, we shall review the development of the multi-scale methodology and its applications to particle-fluid systems, elucidating the essential relevance of complex systems and the challenging problems in chemical engineering.Multi-scale structure is considered to be the focus in studying complex systems, particularly, correlation between phenomena at different scales, compromise between different dominant mechanisms, coupling between spatial and temporal structural changes and critical phenomena occurring in these systems — these are the four critical issues in understanding complex systems. We first propose that by analyzing particle-fluid systems complex systems can be formulated as a multi-objective variational problem. Such an analytical multi-scale method will be reviewed in particular by analyzing the above four critical issues and by showing its 20-year development at IPE from a rough idea to modeling approaches, softwares and finally to industrial applications as well as its extension to different chemical and physical systems. The strategy of “from the particular to the general” in developing this multi-scale methodology is emphasized and challenges to mathematicians and physicists are identified to show the necessity of transdisciplinary cooperation. This presentation will be concluded by prospects and suggestions.     

Poincare型胞映射分析方法及其应用

本文用Poincare型胞映射方法对平衡及不平．衡轴承转子非线性动力系统的全局特性进行了分析研究,同时求得了一定状态空间内系统存在的周期解及其在各不同Poincare截面上的吸引域,得到了一些新的现象和规律,并通过对平衡及不平衡轴承转子系统的全局特性异同的比较,说明了要建立既适用于平衡轴承转子系统又适用于不平衡轴承转子系统的非线性稳定性准则应注意的几个问题     

辛体系下含对称破缺因素动力学系统的近似守恒律

自冯康先生创立Hamilton系统辛几何算法以来,诸如辛结构和能量守恒等守恒律逐渐成为动力学系统数值分析方法有效性的检验标准之一。然而,诸如阻尼耗散、外部激励与控制和变参数等对称破缺因素是实际力学系统本质特征,影响着系统的对称性与守恒量。因此,本文在辛体系下讨论含有对称破缺因素的动力学系统的近似守恒律。针对有限维随机激励Hamilton系统,讨论其辛结构;针对无限维非保守动力学系统、无限维变参数动力学系统、Hamilton函数时空依赖的无限维动力学系统和无限维随机激励动力学系统,重点讨论了对称破缺因素对系统局部动量耗散的影响。上述结果为含有对称破缺因素的动力学系统的辛分析方法奠定数学基础。     

Constrained mechanical systems and gradient systems with strong Lyapunov functions

The characteristics of stationary and non-stationary gradient systems with strong Lyapunov functions are studied. The conditions under which holonomic mechanical systems, generalized Birkhoff systems and generalized Hamilton systems can be considered as gradient systems are given. The characteristics of the gradient systems can be used to study the stability of the mechanical systems. Some examples are given to illustrate the application of the results.     

时滞系统动力学近期研究进展与展望

综述了1999年以来时滞系统动力学在力学、机械工程、航空航天、生态学、生物学、神经网络、激光、电子和信息技术、保密通讯和经济学等领域的研究进展, 总结了其中的研究方法. 通过本文可以看出时滞系统普遍存在于自然和工程实际中, 即使对已经非常熟悉的简单振子,考虑到时滞的影响, 仍有许多问题有待作更深入的理论研究和新现象的发现.针对以往研究中出现的问题, 提出今后几年的发展方向、建议和展望, 同时指出了在理论上急需解决的一些科学问题,例如以时滞反馈控制为中心的控制策略、非线性因素和时滞联合作用的影响、时滞导致的多级分岔使系统呈现出复杂动力学行为、以时滞状态变量耦合为中心构成的网络系统计算模型对系统的影响等问题都是非线性动力学系统所遇到的科学基础问题.      

Experimental and numerical study on the self-stress design of tensegrity systems

Tensegrity systems as kinematically and statically indeterminate pin-jointed systems are characterized by mechanisms and self-stress states. Unlike the other reticulated systems, in tensegrity systems, unilateral behavior of cables causes some problems in determining the basis of compatible self-stress states. At the present study, self-stress design of tensegrity systems is presented. Experimental study on two 3×3×0.7 m tensegrity grids was conducted to verify the accuracy and validity of the numerical method. Using supporting constraints, an effective method for the implementation of self-stress states in a much reduced number of stages is proposed and calibrated. Considering the results of the present study, the self-stress design of these systems can be improved to obtain specific desired behavior.     

Averaging Theory of Arbitrary Order for Piecewise Smooth Differential Systems and Its Application

The averaging theory for studying periodic orbits of smooth differential systems has a long history. Whereas the averaging theory for piecewise smooth differential systems appeared only in recent years, where the unperturbed systems are smooth. When the unperturbed systems are only piecewise smooth, there is not an existing averaging theory to study existence of periodic orbits of their perturbed systems. Here we establish such a theory for one dimensional perturbed piecewise smooth periodic differential equations. Then we show how to transform planar perturbed piecewise smooth differential systems to one dimensional piecewise smooth periodic differential equations when the unperturbed planar piecewise smooth differential systems have a family of periodic orbits. Finally as application of our theory we study limit cycle bifurcation of planar piecewise differential systems which are perturbation of a \(\Sigma \)-center.     

Finite Dimensional State Representation of Linear and Nonlinear Delay Systems

We consider the question of when delay systems, which are intrinsically infinite dimensional, can be represented by finite dimensional systems. Specifically, we give conditions for when all the information about the solutions of the delay system can be obtained from the solutions of a finite system of ordinary differential equations. For linear autonomous systems and linear systems with time-dependent input we give necessary and sufficient conditions and in the nonlinear case we give sufficient conditions. Most of our results for linear renewal and delay differential equations are known in different guises. The novelty lies in the approach which is tailored for applications to models of physiologically structured populations. Our results on linear systems with input and nonlinear systems are new.     

Lyapunov stability for continuous-time multidimensional nonlinear systems

This paper deals with the stability of continuous-time multidimensional nonlinear systems in the Roesser form. The concepts from 1D Lyapunov stability theory are first extended to 2D nonlinear systems and then to general continuous-time multidimensional nonlinear systems. To check the stability, a direct Lyapunov method is developed. While the direct Lyapunov method has been recently proposed for discrete-time 2D nonlinear systems, to the best of our knowledge what is proposed in this paper are the first results of this kind on stability of continuous-time multidimensional nonlinear systems. Analogous to 1D systems, a sufficient condition for the stability is the existence of a certain type of the Lyapunov function. A new technique for constructing Lyapunov functions for 2D nonlinear systems and general multidimensional systems is proposed. The proposed method is based on the sum of squares (SOS) decomposition, therefore, it formulates the Lyapunov function search algorithmically. In this way, polynomial nonlinearities can be handled exactly and a large class of other nonlinearities can be treated introducing some auxiliary variables and constrains.     

Configuration controllability for non-zero potential mechanical control systems with dissipation

Introduction Mechanicalcontrolsystemspresentachallengingandpromisingresearchareabetweenthe studyofclassicalmechanicsandmodernnonlineargeometriccontroltheory.Forthestudyof mechanicalcontrolsystems,therichresultshavebeenproducedwithintheHamiltonian frameworkandLagrangianframework,respectively[1,2].FromtheLagrangianperspective,itis worthwhiletomentiontheconfigurationcontrollabilitynotions,recentlyintroducedbyLewisandMurray[3,4]forsimplemechanicalcontrolsystemswhoseLagrangianiskineticenergy minu…     

Crystal-plasticity finite-element analysis of inelastic behavior during unloading in a magnesium alloy sheet

A crystal-plasticity finite-element analysis of the loading-unloading process under uniaxial tension of a rolled magnesium alloy sheet was carried out, and the mechanism of the inelastic response during unloading was examined, focusing on the effects of basal and nonbasal slip systems. The prismatic and basal slip systems were mainly activated during loading, but the activation of the prismatic slip systems was more dominant. Thus the overall stress level during loading was determined primarily by the prismatic slip systems. The prismatic slip systems were hardly activated during unloading because the stress level was of course lower than that during loading. On the other hand, because the strength of the basal slip systems was much lower than that of the prismatic slip systems, the basal slip systems would be easily activated under the stress level during unloading in the opposite direction when their Schmid’s resolved shear stresses changed signs because of the inhomogeneity of the material. These results indicated that one explanation for the inelastic behavior during unloading was that the basal slip systems were primarily activated owing to their low strengths compared to that of the prismatic slip systems. Numerical tests using the sheets with random orientations and with the more pronounced texture were conducted to further examine the mechanism.     

CONTROLLABILITY OF A CLASS OF HYBRID DYNAMIC SYSTEMS (Ⅱ)—SINGLE TIME_DELAY CASE

IntroductionConsideraswitchedlinearsystemwithsingletime_delayinthecontrolfunctiongivenby x(t) =Ar(t) x(t) +Br(t) u(t) +Dr(t) u(t -τ) ,(1)wherex(t) ∈Rnisthestate,u(t) ∈R mtheinput,thepiecewiseconstantscalarfunctionr(t) :R+ → 1,2 ,… ,Nistheswitchingpath ,(Ai,Bi,Di) ｜i =1,… ,Nisafinitefamilyofsystemrealizations.Moreover,r(t) =iimpliesthat(Ai,Bi,Di)ischosenasthesystemrealizationattimet.τ >0isthefixedtimedelayincontrol.Inthispaper ,wewillstudythecontrollabilityofsystem (1) .Theone_per…     

LAGRANGE'S THEOREM FOR A CLASS OF NONHOLONOMICSYSTEMS AND ITS APPLICATION

I.IntroductionSinceE.T.Whittaker.proposedfoestabilit}'problellll'lofnonholononlicsystemsin1904forthefirsttime,thescholarsathomeandabroad11a\'emadealotofresearchesontheequilibriunlstabilityoflinearand11olllinearnonllolollolnicsystems,andhaveobtainedaseriesofimportantresultslZ--7].Hobbled'er,theexpositionandapplicationrelatedtoLagrange'stheorenlinthestabilityanalysisfornonholonomicsystemsisseldonlseenuptonow.Althoughitwasmentionedinreference[3].aspecialdiscussionhasnotbeencarriedoutyet.Asafam…