Local Hopf bifurcation and global existence of periodic solutions in TCP system

This paper investigates the dynamics of a TCP system described by a first- order nonlinear delay differential equation. By analyzing the associated characteristic transcendental equation, it is shown that a Hopf bifurcation sequence occurs at the pos- itive equilibrium as the delay passes through a sequence of critical values. The explicit algorithms for determining the Hopf bifurcation direction and the stability of the bifur- cating periodic solutions are derived with the normal form theory and the center manifold theory. The global existence of periodic solutions is also established with the method of Wu （Wu, J. H. Symmetric functional differential equations and neural networks with memory. Transactions of the American Mathematical Society 350（12）, 4799-4838 （1998））.     

Cell mapping methods-beyond global analysis of nonlinear dynamic systems

在20世纪80年代由徐皆苏教授创建的胞映射方法一直受非线性科学界同仁的欢迎.近几年胞映射方法有了许多新的应用和算法.本文介绍了一些控制应用和算法的文献.另外,还介绍和讨论胞映射方法应用与多目标优化问题的研究和方法,多目标优化控制设计和非线性代数方程找零解.文中指出胞映射方法在并行计算的帮助下,现在可以解决中等高维空间中的各类问题,新的应用还会不断出现.     

Global analysis of secondary bifurcation of an elastic bar

In a three dimensional framework of finite deformation configurations, this paper investigates the secondary bifurcation of a uniform, isotropic and linearly elastic bar under compression in a large range of parameters. The governing differential equations and finite dimensional equations of this problem are discussed. It is found that, for a bar with two ends hinged, usually many secondary bifurcation points appear on the primary branches which correspond to the maximum bending stiffness. Results are shown on parameter charts. Secondary modes and branches are also calculated with numerical methods. The project supported in part by the National Natural Science Foundation of China     

Homoclinic bifurcation at resonant eigenvalues

We consider a bifurcation of homoclinic orbits, which is an analogue of period doubling in the limit of infinite period. This bifurcation can occur in generic two parameter vector fields when a homoclinic orbit is attached to a stationary point with resonant eigenvalues. The resonance condition requires the eigenvalues with positive/negative real part closest to zero to be real, simple, and equidistant to zero. Under an additional global twist condition, an exponentially flat bifurcation of double homoclinic orbits from the primary homoclinic branch is established rigorously. Moreover, associated period doublings of periodic orbits with almost infinite period are detected. If the global twist condition is violated, a resonant side switching occurs. This corresponds to an exponentially flat bifurcation of periodic saddle-node orbits from the homoclinic branch.Partially supported by DARPA and NSF.Partially supported by the Deutsche Forschungsgemeinschaft and by Konrad-Zuse-Zentrum für Informationstechnik Berlin.     

四维超混沌系统Hopf分岔分析与反控制

对超混沌系统进行分岔反控制的研究已成为当前一个重要研究方向,常采用线性控制器实现反控制。首先,对一个四维超混沌系统的Hopf分岔特性进行了分析,利用高维分岔理论推导出分岔特性与参数之间的关系式,以此判断系统的分岔类型。然后,设计一个由线性与非线性组合成的混合控制器对系统进行分岔反控制,控制参数取值不同时,系统会呈现出不同的分岔特性。通过分析得出,调控线性控制器参数可以使系统Hopf分岔提前或延迟发生;同时,调控混合控制器的两个控制参数,可以改变系统Hopf分岔特性,实现分岔反控制。     

Duffing-van der Pol系统的随机分岔

应用广义胞映射图论方法(GCMD)研究了在谐和激励与随机噪声共同作用下的Duffing-van der Pol系统的随机分岔现象. 系统参数选择在多个吸引子与混沌鞍共存的范围内. 研究发现, 随着随机激励强度的增大,该系统存在两种分岔现象: 一种为随机吸引子与吸引域边界上的鞍碰撞, 此时随机吸引子突然消失; 另一种为随机吸引子与吸引域内部的鞍碰撞, 此时随机吸引子突然增大. 研究证实, 当随机激励强度达到某一临界值时, 该系统还会发生D-分岔(基于Lyapunov指数符号的改变而定义), 此类分岔点不同于上述基于系统拓扑性质改变所得的分岔点.     

Codimension two bifurcation of a vibro-bounce system

A three-degree-of-freedom vibro-bounce system is considered. The disturbed map of period one single-impact motion is derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map to a three-dimensional one, and the normal form map associated with Hopf-flip bifurcation is obtained. Dynamical behavior of the system, near the point of codimension two bifurcation, is investigated by using qualitative analysis and numerical simulation. It is found that near the point of Hopf-flip bifurcation there exists not only Hopf bifurcation of period one single-impact motion, but also Hopf bifurcation of period two double-impact motion. The results from simulation show that there exists an interesting torus doubling bifurcation near the codimension two bifurcation. The torus doubling bifurcation makes the quasi-periodic attractor associated with period one single-impact motion transform to the other quasi-periodic attractor represented by two attracting closed circles. The torus bifurcation is qualitatively different from the typical torus doubling bifurcation occurring in the vibro-impact systems. Different routes from period one single-impact motion to chaos are observed by numerical simulation.The project supported by the National Natural Science Foundation of China (10172042, 50475109) and the Natural Science Foundation of Gansu Province Government of China (ZS-031-A25-007-Z (key item))     

Singularity structure of the Hopf bifurcation surface of a differential equation with two delays

We investigated the structure of the so-called first Hopf bifurcation surface associated to a differential equation with two time delays. A geometrical approach leading naturally to a number theoretic approach provides rigourous results which are corroborated by previous numerical and experimental (optical compound resonator) results.     

The mechanical condition of shear band bifurcation

Based on the understanding of the role played by the strain-softening effect in the formation of shear band bifurcation, this paper investigates (a) What is the most favourable condition that stimulates the occurrence of shear band? (b) With what model and characterizing parameters can the curved type of band bifurcation be simulated?     

ROBUST CONTROL OF PERIODIC BIFURCATION SOLUTIONS

The topological bifurcation diagrams and the coefficients of bifurcation equation were obtained by C-L method. According to obtained bifurcation diagrams and combining control theory, the method of robust control of periodic bifurcation was presented, which differs from generic methods of bifurcation control. It can make the existing motion pattern into the goal motion pattern. Because the method does not make strict requirement about parametric values of the controller, it is convenient to design and make it. Numerical simulations verify validity of the method.     

峰后岩石非Darcy渗流的分岔行为研究

煤矿采动围岩大多处于峰后应力状态或破碎状态，其渗流一般不符合Darcy定律，为非Darcy渗流系统．峰后岩石非Darcy渗流系统的失稳和分岔是煤矿突水和煤与瓦斯突出动力灾害发生的根源．文中用谱截断方法建立了Ahmed-Sunada型非Darcy渗流系统的降阶动力学方程，再由变量代换得到以无量纲变量表示的平衡态附近的演化方程，分析了系统的分岔条件，给出了系统的各种吸引子图案，并结合采矿工程实际，用非线性数学的观点揭示了煤矿突水和煤与瓦斯突出的机理．研究表明：当非Darcy渗流系统渗流特性和边界压力的初始值满足一定条件时，系统由平衡转向不稳定，即存在跨临界Hopf分岔和切分岔，并且，系统的动力学响应不随渗透特性连续变化，即该系统存在突变性．     

The global bifurcation and chaos for the coupling between longitudinal and transverse vibration of a thin elastic plate in large overall motion

This paper presents the global bifurcation and chaotic behavior for the coupling of longitudinal and transverse vibration of a thin elastic plate in large overall motion. First the parametric equations of the homoclinic orbits of such a system is obtained. Then, by using the Melnikov method and digital computer simulation. the behavior of bifurcation and chaos of this vibration system is investigated in the cases of different resonances. The obvious difference between the transverse vibration and the coupling of transverse and longitudinal vibration is also shown.The project supported by the National Natural Science Foundation of China.     

Duffing系统随机分岔的全局分析

应用广义胞映射方法研究了在谐和与随机噪声联合作用下的Dnmng系统的随机分岔现象．对于随机Dnmng系统，以吸引子形态的突然变化，描述一类随机分岔现象．数值结果表明，随着随机激励强度的逐渐增大，当随机激励强度通过临界值时，随机系统的吸引子与其吸引域边界(吸引域)上的鞍碰撞，发生分岔现象．比较结果表明，在同样的参数区域内，Lyapunov指数均为负值，也就是说，在Lyapunov指数意义下，无法发现这种随机分岔现象．     

An experimental study of bifurcation,chaos, and dimensionality in a system forced through a bifurcation parameter

An experimental study of a system that is parametrically excited through a bifurcation parameter is presented. The system consits of a lightly-damped, flexible beam which is buckled and unbuckled magnetically: it is parametrically excited by driving an electromagnet with a low-frequency sine wave. For voltage amplitudes in excess of the static bifurcation value, the beam slowly switches between the one-and two-well configurations. Experimental static and dynamic bifurcation results are presented. Static bifurcatons for the system are shown to involve a butterfly catastrophe. The dynamic bifurcation diagram, obtained with an automated data acquisition system, shows several period-doubling sequences, jump phenomena, and a chaotic region. Poincaré sections of a chaotic steady-state are obtained for various values of the driving phase, and the correlation dimension of the chaotic attractor is estimated over a large scaling region. Singular system analysis is used to demonstrate the effect of delay time on the noise level in delay-reconstructions, and to provide an independent check on the dimension estimate by directly estimating the number of independent coordinates from time series data. The correlation dimension is also estimated using the delay-reconstructed data and shown to be in good agrement with the value obtained from the Poincaré sections. The bifurcation and dimension results are used together with physical sonsiderations to derive the general form of a single-degree-of-freedom model for the experimental system.     

EMS磁浮列车车/轨耦合系统的分岔现象研究

通过分析EMS磁浮列车系统的Hopf分岔, 研究车轨耦合系统对应的轨道固有频率稳定范围. 采用串级控制悬浮控制器，给出控制器参数与特征频率的关系，在控制器稳定的前提下利用 数值算法计算在系统参数一定、轨道固有频率变化条件下所对应的Hopf分岔点，结合仿真方法 得出轨道固有频率的稳定范围，定性说明了系统载荷、控制器频率、二次系刚度与轨道固有 频率之间的变化关系.     

On the appearance of a Hopf bifurcation in a non-ideal mechanical problem

In this paper we studied a non-ideal system with two degrees of freedom consisting of a dumped nonlinear oscillator coupled to a rotatory part. We investigated the stability of the equilibrium point of the system and we obtain, in the critical case, sufficient conditions in order to obtain an appropriate Normal Form. From this, we get conditions for the appearance of Hopf Bifurcation when the difference between the driving torque and the resisting torque is small. It was necessary to use the Bezout Theorem, a classical result of Algebraic Geometry, in the obtaining of the foregoing results.     

Cyclic motions near a Hopf bifurcation of a four-dimensional system

We study motions near a Hopf bifurcation of a representative nonconservative four-dimensional autonomous system with quadratic nonlinearities. Special cases of the four-dimensional system represent the envelope equations that govern the amplitudes and phases of the modes of an internally resonant structure subjected to resonant excitations. Using the method of multiple scales, we reduce the Hopf bifurcation problem to two differential equations for the amplitude and phase of the bifurcating cyclic solutions. Constant solutions of these equations provide asymptotic expansions for the frequency and amplitude of the bifurcating limit cycle. The stability of the constant solutions determines the nature of the bifurcation (i.e., subcritical or supercritical). For different choices of the control parameter, the range of validity of the analytical approximation is ascertained using numerical simulations. The perturbation analysis and discussions are also pertinent to other autonomous systems.     

Numerical analysis of shear-band bifurcation

A numerical analysis of bifurcation in shear band pattern is presented to help understand the distributions of the velocity variation in the shear band. Comparison with the results of analytic method indicates that: (1) the critical strain is irrelevant to the relative width of the shear band; (2) the variations along the direction normal to the band have indeed the controlling effect whilst the effect of variations along the tangential direction is negligible. The project is supported by the National Natural Science Foundation of China.     

Theoretical study on the bifurcation of vortexes structure for flow in curved tube

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｃｕｒｖｅｄｔｕｂｅｉｓｎｏｔｏｎｌｙｃｏｍｍｏｎｉｎｉｎｄｕｓｔｒｙ ,ｂｕｔａｌｓｏａｔｔｒａｃｔｓｓｐｅｃｉａｌａｔｔｅｎｔｉｏｎｏｆｒｅｓｅａｒｃｈｅｒｓｉｎｂｉｏｍｅｃｈａｎｉｃｓｂｅｃａｕｓｅａｏｒｔｉｃａｒｃｈｉｓｔｈｅｐｌａｃｅｗｈｅｒｅａｔｈｅｒｏｓｃｌｅｒｏｓｉｓｈａｐｐｅｎｓｍｏｓｔｆｒｅｑｕｅｎｔｌｙ[1～ 3].Ｓｉｎｃｅｔｈｅｐｈｙｓｉｏｌｏｇｉｃａｌｆｅａｔｕｒｅｏｆｒｅａｌｂｌｏｏｄｖｅｓｓｅｌｉｓｖｅｒｙｃｏｍｐｌｉｃａｔｅｄ ,ｔｈｅｆｌｏ…     

Stability and bifurcation analysis of a nonlinear transversally loaded rotating shaft

The dynamic stability and self-excited posteritical whirling of rotating transversally loaded shaft made of a standard material with elastic and viscous nonlinearities are analyzed in this paper using the theory of bifurcations as a mathematical tool. Partial differential equations of motion are derived under assumption that von Karman's nonlinearity is absent but geometric curvature nonlinearity is included. Galerkin's first-mode discretization procedure is then applied and the equations of motion are transformed to two third-order nonlinear equations that are analyzed using the theory of bifurcation. Condition for nontrivial equilibrium stability is determined and a bifurcating periodic solution of the second-order approximation is derived. The effects of dimensionless stress relaxation time and cubic elastic and viscous nonlinearities as well as the role of the transverse load are studied in the exemplary numerical calculations. A strongly stabilizing influence of the relaxation time is found that may eliminate self-excited vibration at all. Transition from super- to subcritical bifurcation is observed as a result of interaction between system nonlinearities and the transverse load.