一类生物流体力学连续系统的分岔研究

连续动力系统的非线性动力学研究,由于其应用的广泛性与问题的复杂性,近年来越来越受到重视。本文对一类生物流体力学中的连续系统-动脉局部狭窄时血液流动的分岔特性进行了研究,采用有限差分方法,将由偏微分方程组描述的边境动力系统约化为由常微分方程组描述的高维离散动力系统。求得了离散动力系统的平衡解并分析其稳定性,同时讨论了流场中变量空间分布的变化情况。求得了离散动力系统的前三个Lyapunov指数,以此作为系统是否发生混沌的判别条件。     

Subcritical Hopf Bifurcation in the Delay Equation Model for Machine Tool Vibrations

We show the existence of a subcritical Hopf bifurcation in thedelay-differential equation model of the so-called regenerative machine toolvibration. The calculation is based on the reduction of the infinite-dimensional problem to a two-dimensional center manifold. Due to the specialalgebraic structure of the delayed terms in the nonlinear part of the equation,the computation results in simple analytical formulas. Numerical simulationsgave excellent agreement with the results.     

An Energy Analysis of the Local Dynamics of a Delayed Oscillator Near a Hopf Bifurcation

Hopf bifurcation exists commonly in time-delay systems. The local dynamics of delayed systems near a Hopf bifurcation is usually investigated by using the center manifold reduction that involves a great deal of tedious symbolic and numerical computation. In this paper, the delayed oscillator of concern is considered as a system slightly perturbed from an undamped oscillator, then as a combination of the averaging technique and the method of Lyapunov's function, the energy analysis concludes that the local dynamics near the Hopf bifurcation can be justified by the averaged power function of the oscillator. The computation is very simple but gives considerable accurate prediction of the local dynamics. As an illustrative example, the local dynamics of a delayed Lienard oscillator is investigated via the present method.     

Bifurcation analysis of a double pendulum with internal resonance

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Stability and Bifurcation Analysis of a Modified Geometrically Nonlinear Orthotropic Jeffcott Model with Internal Damping

In this paper, a modified Jeffcott model is proposed and studied in order to shed light into the dynamics of a complex system, the Short Electrodynamic Tether (SET), which is similar to an unbalanced rotor. Due to the internal damping, a geometrically linear SET model appears to be unstable as predicted by the linear rotordynamics theory. Some studies in the field of rotordynamics suggest that this instability caused by internal damping do not appear if geometric nonlinearities are taken into account in the system equations of motion. Stability and bifurcation analysis have been carried out on the modified Jeffcott model, which accounts for geometric nonlinearities, orthotropy in the shaft's cross section, and a viscous damping-based internal damping model. The stability results analytically obtained have been compared with a nonlinear multibody model by means of time simulations and good agreement has been found.     

Hopf Bifurcation of the Generalized Lorenz Canonical Form

Hopf bifurcation of a unified chaotic system – the generalized Lorenz canonical form (GLCF) – is investigated. Based on rigorous mathematical analysis and symbolic computations, some conditions for stability and direction of the periodic obits from the Hopf bifurcation are derived.     

Bifurcation analysis in a time-delay model for prey–predator growth with stage-structure

A time-delay model for prey–predator growth with stage-structure is considered. At first, we investigate the stability and Hopf bifurcations by analyzing the distribution of the roots of associated characteristic equation. Then, an explicit formula for determining the stability and the direction of periodic solutions bifurcating from Hopf bifurcations is derived, using the normal form theory and center manifold argument. Finally, some numerical simulations are carried out for supporting the analytic results.     

Bifurcation of a shaft with hysteretic-type internal friction force of material

IntroductionItwasfoundlongtimeagothattheinternalfrictionofmaterialcancauseinstabilityofrotatingshaft.Soitisalwaysoneoftheimportantsubjectsinrotordynamics[1].Earlyinvestigationswerefocusedonthedynamicalstabilityproblemofrotorinfluencedbythelinearinternalfrictionofmaterial,aimingtoobtainthecriterionofstability[2～4 ].Asthedevelopmentofnonlineardynamics,moreandmoreattentionswerepaidtothestudyoftheself_excitedmotionofrotatingshaft,thatisthebifurcation .Thestabilityregionsandbifurcationsofbothanau…     

Stabilization of the Parametric Resonance of a Cantilever Beam by Bifurcation Control with a Piezoelectric Actuator

In this work, bifurcation control using a piezoelectric actuator isimplemented to stabilize the parametric resonance induced in acantilever beam. The piezoelectric actuator is attached to the surfaceof the beam to produce a bending moment in the beam. The dimensionlessequation of motion for the beam with the piezoelectric actuator on itssurface is derived and the modulation equations for the complexamplitude of an approximate solution are obtained using the method ofmultiple scales. We then acquire the bifurcation set that expresses theboundary of the stable and unstable regions. The bifurcation set ischaracterized by the modulation equations. Next, we determine the orderof feedback gains to modify these modulation equations. By actuating thepiezoelectric actuator under the appropriate feedback, bifurcationcontrol is carried out resulting in the shift of the bifurcation set andthe expansion of the stable region. The main characteristic of thestabilization method introduced above is that the work done by thepiezoelectric actuator is zero in the state where the parametricresonance is stabilized. Thus zero power control is realized in such astate. Experimental results show the validity of the proposedstabilization method for the parametric resonance induced in thecantilever beam.     

Bifurcation and stability analysis of laminar flow in curved ducts

The development of viscous flow in a curved duct under variation of the axial pressure gradient q is studied. We confine ourselves to two‐dimensional solutions of the Dean problem. Bifurcation diagrams are calculated for rectangular and elliptic cross sections of the duct. We detect a new branch of asymmetric solutions for the case of a rectangular cross section. Furthermore we compute paths of quadratic turning points and symmetry breaking bifurcation points under variation of the aspect ratio γ (γ=0.8…1.5). The computed diagrams extend the results presented by other authors. We succeed in finding two origins of the Hopf bifurcation. Making use of the Cayley transformation, we determine the stability of stationary laminar solutions in the case of a quadratic cross section. All the calculations were performed on a parallel computer with 32×32 processors. Copyright © 2009 John Wiley & Sons, Ltd.     

Bifurcation and stability analysis of a neural network model with distributed delays

In this paper, the dynamics of a generalized two-neuron model with self-connections and distributed delays are investigated, together with the stability of the equilibrium. In particular, the conditions under which the Hopf bifurcation occurs at the equilibrium are obtained for the weak kernel. This means that a family of periodic solutions bifurcates from the equilibrium when the bifurcation parameter exceeds a critical value. Explicit algorithms for determining the stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation are derived by using the theory of normal form and center manifold [20]. Some numerical simulations are given to illustrate the effectiveness of the results found. The obtained results are new and they complement previously known results.This work was supported by the National Natural Science Foundation of China under Grants 60574043 and 60373067, the Natural Science Foundation of Jiangsu Province, China under Grants BK2003053.     

Stability and Hopf Bifurcation of Four-Wheel-Steering Vehicles Involving Driver's Delay

A mathematical model is presented for four-wheel-steeringvehicles, with the time delay in driver's response and the nonlinearityin lateral tyre forces taken into account. It is proved that thevehicle-driver system has a trivial steady state motion, as well aseight non-trivial steady state motions due to the nonlinearity of tyreforces. The asymptotic stability and Hopf bifurcation of the trivialsteady state are analyzed for two control strategies ofrear-wheel-steering. It is shown through the numerical simulations thatthe four-wheel-steering technique based on the bilinear control strategyworks better when the driver's response involves time delay.     

Hopf Bifurcation and Stability of Periodic Solutions for van der Pol Equation with Distributed Delay

The van der Pol equation with a distributed time delay is analyzed. Itslinear stability is investigated by employing the Routh–Hurwitzcriteria. Moreover, the local asymptotic stability conditions are alsoderived. By using the mean time delay as a bifurcation parameter, themodel is found to undergo a sequence of Hopf bifurcations. The directionand the stability criteria of the bifurcating periodic solutions areobtained by the normal form theory and the center manifold theorem. Somenumerical simulation examples for justifying the theoretical analysisare also given.     

层合板分叉方程的Lyapunov—Schmidt约化分析

采用广义傅立叶级数法建立了具有弹性约束的复合材料矩形层板在面内载荷作用下的非线性稳定性控制方程,并简化为矩阵形式。利用分叉理论和泛函知识,对有限维的该分叉方程进行了Lyapunov-Schmidt约化,获得了三种典型的分叉图形式,同时指出当非齐次项等于零时必然发生分叉。数值计算结果表明了三种分叉图分别所对应的典型的力学模型,主要因素在于边界条件、铺层方式及初始缺陷三方面。     

Local Bifurcation Analysis of a Second Gradient Model for Deformations of a Rectangular Slab

In this paper we carry out a derivation of the equilibrium equations of nonlinear elasticity with an added second-gradient term proportional to a small parameter . These equations are given by a fourth order semilinear system of pdes. We discuss different types of possible boundary conditions for these equations. We then specialize the equations to a rectangular slab and study the linearized problem about a homogenous deformation. We show that these equations admit solutions representable as Fourier series in one of the independent variables. Furthermore, we obtain the characteristic equation for the eigenvalues (possible bifurcation points) for the linear problem and derive asymptotic representations for this equation for small . We used these expressions to show that in the limit as the characteristic equation for converges uniformly (in certain regions of the parameter space) to the corresponding characteristic equation for . When the base material () is that of a Blatz–Ko type, we get conditions for the existence of eigenvalues of the linear problem with and small. Our numerical results in this case indicate that the number of bifurcation points is finite when and that this number monotonically increases as . For the problem with we get conditions for the existence of local branches of non-trivial solutions.      

Hopf bifurcation for a ecological mathematical model on microbe populations

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Multiple Scales without Center Manifold Reductions for Delay Differential Equations near Hopf Bifurcations

We study small perturbations of three linear Delay DifferentialEquations (DDEs) close to Hopf bifurcation points. In analytical treatments of such equations, many authors recommend a center manifold reductionas a first step. We demonstrate that the method of multiple scales, onsimply discarding the infinitely many exponentially decaying components of the complementary solutionsobtained at each stage of the approximation,can bypass the explicit center manifold calculation.Analytical approximations obtained for the DDEs studied closely matchnumerical solutions.     

Estimating Critical Hopf Bifurcation Parameters for a Second-Order Delay Differential Equation with Application to Machine Tool Chatter

Nonlinear time delay differential equations are well known to havearisen in models in physiology, biology and population dynamics. Theyhave also arisen in models of metal cutting processes. Machine toolchatter, from a process called regenerative chatter, has been identifiedas self-sustained oscillations for nonlinear delay differentialequations. The actual chatter occurs when the machine tool shifts from astable fixed point to a limit cycle and has been identified as arealized Hopf bifurcation. This paper demonstrates first that a class ofnonlinear delay differential equations used to model regenerativechatter satisfies the Hopf conditions. It then gives a precisecharacterization of the critical eigenvalues on the stability boundaryand continues with a complete development of the Hopf parameter, theperiod of the bifurcating solution and associated Floquet exponents.Several cases are simulated in order to show the Hopf bifurcationoccurring at the stability boundary. A discussion of a method ofintegrating delay differential equations is also given.     

Global Bifurcation Analysis of a Controlled Underactuated Mechanical System

In this paper, bifurcation theory is employed to classify different dynamical behaviors arising in an underactuated mechanical system subject to bounded controls. The methodology is applied to an inertia wheel pendulum consisting of a simple pendulum with a rotating disk at the end. Restricting the magnitude of the control action places an important obstacle to the design of a continuous controller capable of swinging-up and stabilize the pendulum at the inverted position: the arm only can reach that position by means of oscillations of increasing amplitude. The controller is derived from a simple nonlinear state-feedback law, followed by a saturating device that limits the maximum amplitude of the control action applied to the system. This bound gives birth to a rich dynamical behavior, including pitchfork and Hopf bifurcations of equilibria, saddle-node bifurcations of periodic orbits, homoclinic and heteroclinic bifurcations. The global dynamics is analyzed in terms of certain control gains and a two-parameter bifurcation diagram is derived. It is shown that the dynamics on this bifurcation diagram is organized in a pair of codimension-two rotationally symmetric bifurcation points. Finally, it is found out that when the control gains lie on a certain region in the parameter space simultaneous stabilization of the upright position together with a large basin of attraction is obtained. Simulation results show that almost global stabilization of the system can be achieved.     

超静定连续梁弹塑性过程分析的单位荷载法

讨论了分析超静定连续梁弹塑性受力和变形全过程的单位荷载法,运用该方法分析了集中荷载作用下一次超静定两跨连续梁的弹塑性加载和变形全过程．根据受力变形的特点,集中荷载作用下两跨连续梁的弹塑性加载过程可分为四个阶段,分别是弹性阶段、集中荷载作用点附近塑性区扩展阶段、集中荷载作用点保持为塑性铰而附近区域线性卸载阶段、两跨连接点附近塑性区扩展直至形成第二个塑性铰阶段．给出了加载过程中各阶段的弯矩内力和竖向位移随外荷载而变化的解析公式．研究结果表明：在相同的单跨荷载工况下,连续梁的变形过程不同于单跨一次超静定梁,其塑性铰形成顺序不同,静定结构形成顺序不同,但塑性极限破坏荷载相同．