Double Hopf Bifurcation Analysis Using Frequency Domain Methods

The dynamic behavior close to a non-resonant double Hopf bifurcation is analyzed via a frequency-domain technique. Approximate expressions of the periodic solutions are computed using the higher order harmonic balance method while their accuracy and stability have been evaluated through the calculation of the multipliers of the monodromy matrix. Furthermore, the detection of secondary Hopf or torus bifurcations (Neimark–Sacker bifurcation for maps) close to the analyzed singularity has been obtained for a coupled electrical oscillatory circuit. Then, quasi-periodic solutions are likely to exist in certain regions of the parameter space. Extending this analysis to the unfolding of the 1:1 resonant double Hopf bifurcation, cyclic fold and torus bifurcations have also been detected in a controlled oscillatory coupled electrical circuit. The comparison of the results obtained with the suggested technique, and with continuation software packages, has been included.     

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.     

不可压气流中二元机翼的分叉分析

本文分析了不可压气流中带有非线性俯仰刚度二元机翼的分叉问题.分析采用了工程实用的等效线化法和作为比较标准的数值积分法,并借助计算机代数系统按Hassard渐近展开算法及平均化算法求得解析解进行比较.从而论证等效线化法的可用性.     

Regenerative Tool Chatter Near a Codimension 2 Hopf Point Using Multiple Scales

We study a well-known regenerative machine tool vibration model (a delay differential equation) near a codimension 2 Hopf bifurcation point. The method of multiple scales is used directly, bypassing a center manifold reduction. We use a nonstandard choice of expansion parameter that helps understand practically relevant aspects of the dynamics for not-too-small amplitudes. Analytical expressions are then obtained for the double Hopf points. Both sub- and supercritical bifurcations are predicted to occur near the reference point; and analytical conditions on the parameter variations for each type of bifurcation to occur are obtained as well. Analytical approximations are supported by numerics.     

Singularity Analysis on a Planar System with Multiple Delays

A planar model with multiple delays is studied. The singularities of the model and the corresponding bifurcations are investigated by using the standard dynamical results, center manifold theory and normal form method of retarded functional differential equations. It is shown that Bogdanov–Takens (BT) singularity for any time delays, and a serious of pitchfork and Hopf bifurcation can co-existent. The versal unfoldings of the normal forms at the BT singularity and the singularity of a pure imaginary and a zero eigenvalue are given, respectively. Numerical simulations have been provided to illustrate the theoretical predictions.     

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.     

The simplest parametrized normal forms of Hopf and generalized Hopf bifurcations

This paper considers the computation of the simplest parameterized normal forms (SPNF) of Hopf and generalized Hopf bifurcations. Although the notion of the simplest normal form has been studied for more than two decades, most of the efforts have been spent on the systems that do not involve perturbation parameters due to the restriction of the computational complexity. Very recently, two singularities – single zero and Hopf bifurcation – have been investigated, and the SPNFs for these two cases have been obtained. This paper extends a recently developed method for Hopf bifurcation to compute the SPNF of generalized Hopf bifurcations. The attention is focused on a codimension-2 generalized Hopf bifurcation. It is shown that the SPNF cannot be obtained by using only a near-identity transformation. Additional transformations such as time and parameter rescaling are further introduced. Moreover, an efficient recursive formula is presented for computing the SPNF. Examples are given to demonstrate the applicability of the new method.     

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.     

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.     

神经网络时滞系统非共振双Hopf分岔及其广义同步

本文建立了具有自连接和抑制-兴奋型他连接的两个同性神经元模型。其中自连接是由于兴奋型的突触产生,而他连接则分别对应于两神经元兴奋、抑制型的突触。发现如果有兴奋型自连接就会有双Hopf分岔,而没有时滞自连接时双Hopf分岔就会消失,因此自连接引起了双Hopf分岔。作为一个例子,通过变动连接中的时滞和他连接中的比重,1／√2双Hopf分岔得到了详细研究。通过中心流形约化,分岔点邻域内各种不同的动力学行为得到了分类,并以解析形式表出。神经元活动的分岔路径得以表明。从得到的解析近似解可以发现,本文所研究的具有兴奋一抑制型他连接的两相同神经元的节律不能完全同步而只能广义同步。时滞也可以使其节律消失,两神经元变为非活动的。这些结果在控制神经网络关联记忆和设计人工神经网络方面有着潜在的应用。     

Double Hopf Bifurcations and Chaos of a Nonlinear Vibration System

A double pendulum system is studied for analyzing the dynamic behaviour near a critical point characterized by nonsemisimple 1:1 resonance. Based on normal form theory, it is shown that two phase-locked periodic solutions may bifurcate from an initial equilibrium, one of them is unstable and the other may be stable for certain values of parameters. A secondary bifurcation from the stable periodic solution yields a family of quasi-periodic solutions lying on a two-dimensional torus. Further cascading bifurcations from the quasi-periodic motions lead to two chaoses via a period-doubling route. It is shown that all the solutions and chaotic motions are obtained under positive damping.     

Hopf Bifurcation on a Two-Neuron System with Distributed Delays: A Frequency Domain Approach

In this paper, a more general two-neuron model with distributed delays and weak kernel is investigated. By applying the frequency domain approach and analyzing the associated characteristic equation, the existence of bifurcation parameter point is determined. Furthermore, we found that if the mean delay is used as a bifurcation parameter, Hopf bifurcation occurs for the weak kernel. This means that a family of periodic solutions bifurcates from the equilibrium when the bifurcation parameter exceeds a critical value. The direction and stability of the bifurcating periodic solutions are determine by the Nyquist criterion and the graphical Hopf bifurcation theorem. Some numerical simulations for justifying the theoretical analysis are also given.     

Normal form of the nonsemi-simple bifurcation problem

I.IntroductionBynow,agreatamountoftheoreticalresultshavebeenachievedinnonlinearscience.Tofindthenewmotivationforthefurtherdevelopmentinnonlinearscience,moreattentionarepayedtoapplicationsoftheseachievementandnewbreakthroughhasbeenexpectedsince1990s.Thesit…     

糖尿病治疗模型中技术时滞诱发的双Hopf分岔

本文研究了利用外部辅助设备来治疗糖尿病的生理模型,其中存在着两个时滞;辅助设备的技术时滞τ1和肝脏的生理时滞τ2。发现由于技术时滞τ1的出现,使模型存在着共振和非共振的双Hopf分岔。应用非线性动力学理论,对由此产生的非共振分岔的动力学行为进行了分类。结果表明,随着技术时滞τ1和糖尿病人患病程度α的变化,利用该模型可以预测不同的医疗结果：血糖稳定（康复）、简单的和复杂的血糖波动。结果对分析、预测、优化糖尿病治疗方案的医疗结果、评估该方案的医疗风险和可行性等有着潜在的应用价值。本文结果的意义在于针对糖尿病患者患病的不同程度,可以定性的调节辅助设备的技术时滞τ1,以达到更好的治疗效果。     

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。     

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.     

Hopf bifurcation of an oscillator with quadratic and cubic nonlinearities and with delayed velocity feedback

This paper studies the local dynamics of an SDOF system with quadratic and cubic stiffness terms, and with linear delayed velocity feedback. The analysis indicates that for a sufficiently large velocity feedback gain, the equilibrium of the system may undergo a number of stability switches with an increase of time delay, and then becomes unstable forever. At each critical value of time delay for which the system changes its stability, a generic Hopf bifurcation occurs and a periodic motion emerges in a one-sided neighbourhood of the critical time delay. The method of Fredholm alternative is applied to determine the bifurcating periodic motions and their stability. It stresses on the effect of the system parameters on the stable regions and the amplitudes of the bifurcating periodic solutions. The project supported by the National Natural Science Foundation of China (19972025)     

QUALITATIVE ANALYSIS OF SPHERICAL CAVITY NUCLEATION AND GROWTH FOR INCOMPRESSIBLE GENERALIZED VALANIS-LANDEL HYPERELASTIC MATERIALS

I. INTRODUCTION In practice, cavity formulation in materials is recognized as precursors to failure. Thus void nucleationand growth in solid materials have a great in?uence on failure mechanism. Gent and Lindley[1] have observed experimentally the phe…     

Qualitative study of cavitated bifurcation for a class of incompressible generalized neo-Hookean spheres

IntroductionIn 1 958,GentandLindleyobservedthephenomenonofsuddenvoidnucleationinsolidsexperimentallyintensioningahomogenousclose_grainedvulcanizedrubbercylinderforthefirsttime.ButthemathematicalmodelonvoidnucleationandgrowthhasnotbeendescribedasabifurcationproblembasedonthetheoryofnonlinearelasticmechanicsbyBall[1]until1 982 .Inrecentyears,manyinvestigationshavebeenmadeonthisaspect.Theproblemofcavitatedbifurcationforincompressibleisotropichyperelasticmaterialswithpower_lawtypehasbeeninvestig…