强非线性自激振子同宿轨道的摄动分析方法

在双曲函数摄动法的基础上,推广双曲函数Lindstedt-Poincaré (L-P)法的适用范围,使之适用于定量分析一类含五次强非线性项的自激振子的同宿分岔和同宿解问题。以双曲函数系为基础推导出适用于高次非线性系统的摄动步骤,对极限环的同宿分岔参数进行摄动展开,给出同宿摄动解奇异项的定义,以消除同宿摄动解奇异项作为确定极限环同宿分岔点的条件,给出能够严格满足同宿条件的同宿轨道摄动解。算例表明,在相平面内该方法的结果与Runge-Kutta法数值周期轨道的逼近结果比较吻合。     

余维4的Duffing-Van der Pol方程全局分岔分析

本文采用级数展开形式的Melnikov函数解决高余维分岔问题。通过研究一类5次项和3次项共存,具有异宿轨的Duffing-Van der Pol方程的余维4全局分岔问题,得到了该系统的分岔方程及全局拓扑结构,说明了该方法的可行性。研究结果表明,该系统有单个极限环、单个异宿轨、异宿轨和极限环共存、两个极限环共存等情况。最后通过数值模拟验正了理论分析结果的正确性。     

非线性弹性直杆纵振时的混沌行为

研究了S形本构关系的弹性直杆纵振时的混沌行为.用Galerkin原理将杆纵振时的动力控制方程转化为二阶三次非线性微分动力系统;给出了其产生同宿轨道和异宿轨道的条件,得到了同宿轨道的参数方程;借助Melnikov函数给出了系统发生混沌的临界条件;数值计算给出了混沌运动区域随β和γ的变化规律,用分岔图、位移时程曲线、相平面图和Poincaré映射判断了系统的运动行为即定常还是混沌.进一步的研究还表明本构关系中的二次非线性项对系统的动力响应具有很大的影响.     

Furuta倒立摆的分岔与混沌动力学分析

利用Hopf定理和规范形理论,讨论了Furuta旋转倒立摆非线性数学模型的Hopf分岔特性。给出系统存在Hopf分岔的条件,讨论了周期轨道的稳定性,利用数值模拟,得到系统的相轨迹图,进一步验证分析过程的正确性。利用Silnikov定理,讨论了旋转倒立摆的混沌动力学特征。利用卡尔达诺公式和微分方程级数解讨论了该系统的特征值和同宿轨道的存在性,比较严格地证明了系统存在Smale马蹄意义下的混沌现象,并给出发生Silnikov型Smale混沌的条件。     

非线性振动系统极限环振幅与频率的控制

研究非线性振动系统极限环振幅与频率的控制.利用多尺度法,计算了动力系统Hopf分岔点附近的正规型,获得极限环振幅及频率与反馈系数的近似解析关系.通过选择适当的反馈系数,可对极限环的振幅与频率进行控制.选取一类三维系统作为算例,讨论并比较了几类线性、非线性反馈控制器的控制效果,所得的解析结果与数值模拟的结果基本吻合.     

确定前轮非线性摆振极限环振幅的一种方法

用摄动法给出了当前轮减摆阻尼器含有速度平方阻尼时前轮摆振的极限环振幅表达式，并讨论了极限环的稳定性。在确定极限环振幅时，利用非线性系统所对应的线性系统在临界稳定时特征矩阵的特点，通过对特征矩阵进行三角分解，得到了消除摄动解中奇异项的条件，利用此条件，很方便地获得了飞机前轮非线性摆振的摄动解，并改正了有关文献中的错误。     

带三次恢复力项频率依赖于速度的非线性振子研究

带三次恢复力项频率依赖于速度（Velocity-Dependent-Frequency, VDF）的非线性振子 的周期及其性质目前没有文献讨论,且使用传统的摄动法或谐波平衡法求解这类振子一阶近似解时往往失效。特别的,其频率在有限的幅值范围内奇异。首先求得了该振子周期的积分表达式,基于积分表达式可积性条件采用谐波平衡法得到了该振子一类初始条件下的精确解;研究了该振子的周期性质,给出了由第一类完全椭圆积分表示的周期-振幅的近似解析表达式,分析了振子的方波现象及产生原因。研究表明,振子周期最终随着幅值的增大衰减到0;振子方波现象产生原因是由于系统参数 ,随着幅值的增大,方波现象更明显。最后提出使用一种Hermite插值法求解该振子的周期解,该方法将时间变量转换为新的谐振时间变量,其频率为振子频率的一半,对应的控制微分方程转变为适合于Hermite插值分析的形式,其解与数值解的对比证明了该方法的有效性。     

电动力绳索系统的周期解求解及其稳定性分析

摘 要：对运行在倾斜圆轨道上的电动力绳索系统的动力学特性进行了分析研究。首先建立了系统的动力学模型,分别采用摄动法及推广后的数值算法求得系统的基本周期解,并运用所给数值算法中的稳定性判据分析了周期解的稳定性,得出该系统周期运动不稳定的结论。最后进行仿真验证,结果表明在摄动量较小时,两种求解算法得到的周期解基本相同,但当摄动量较大时,摄动法求得的周期解发生了畸变,不理想此时通常借助数值算法加以求解;仿真结果同样证实了所得周期解的不稳定特性。     

基于模态耦合的摩擦自激振荡系统稳定性研究

模态耦合是摩擦引起系统自激振荡的主要不稳定性机理之一.针对此类问题,构建了两自由度非线性质量一传动带系统.首先,通过劳斯判据分析系统的稳定性,给出了计算Hopf分岔点的数学表达式以及系统参数改变时分岔点和特征值分岔图的变化状态.其次,将扩展谐波平衡法加以利用和延伸,得到了自激振荡系统极限环幅值的解析解,进而研究了系统外界参数和结构参数变化对极限环幅值的影响.外界参数(带速和摩擦系数)变化可造成极限环幅值的规律性变化,而结构参数(阻尼比和固有频率)的改变会引起极限环幅值复杂的动力学行为.研究方法及结果可为机械系统结构设计和减振等方面提供理论分析参考.     

非自治旋转机械系统自同步与异结构同步的研究

摘要:研究一类非自治旋转机械系统的复杂动力学行为.通过系统运动的拉格朗日方程和牛顿第二定律,建立了机械式离心调速é器系统的动力学方程.通过系统的分岔图和Lyapunov指数研究系统的混沌行为,通过仿真Poincaré截面分析系统通向混凝沌的道路,并且验证该系统的分岔图与Lyapunov指数谱是完全吻合的.基于Lyapunov稳定性理论,采用非线性控制方法进行一类不同阶非自治混沌系统之间的同步控制的研究．通过构造合适的控制函数,成功地实现两个不同阶混沌系统之间的同步控制,并用数值的仿真进一步证明该方法的有效性．     

Bounded solutions for non-autonomous parabolic equations

The existence of bounded solutions (including in particular homoclinic and heteroclinic solutions) is studied for non-autonomous perturbed parabolic partial differential equations, without the restriction that the linear variational equation has a unique non-trivial bounded solution. Specifically, an idea applied to ordinary differential equations by Hale (1984) and by Battelli and Laari (1990) is realised in an infinite-dimensional setting. Like other work on related problems, the main technique is Lyapunov?Schmidt reduction; we use that technique here in the context of bounded solutions, rather than the more usual setting of periodic or homoclinic solutions. Moreover, several technical obstacles are circumvented in the infinite-dimensional setting?in particular in the proof of the existence of a solution to the reduced bifurcation equation. Non-uniqueness is shown to occur for the Kuramoto-Sivashinsky equation, demonstrating the need to remove the uniqueness restriction     

Dynamics near the product of planar heteroclinic attractors

Motivated by problems in equivariant dynamics and connection selection in heteroclinic networks, Ashwin and Field investigated the product of planar dynamics where at least one of the factors was a planar homoclinic attractor. However, they were only able to obtain partial results in the case of a product of two planar homoclinic attractors. We give general results for the product of planar homoclinic and heteroclinic attractors. We show that the likely limit set of the basin of attraction of the product of two planar heteroclinic attractors is always the unique one-dimensional heteroclinic network which covers the heteroclinic attractors in the factors. The method we use is general and likely to apply to products of higher dimensional heteroclinic attractors as well as to situations where the product structure is broken but the cycles are preserved.     

Partial classification of heteroclinic behaviour associated with the perturbation of hexagonal planforms

Physical systems often exhibit pattern-forming instabilities. Equivariant bifurcation theory is often used to investigate the existence and stability of spatially doubly periodic solutions with respect to the hexagonal lattice. Previous studies have focused on the six- and twelve-dimensional representation of the hexagonal lattice where the symmetry of the model is perfect. Here, perturbation of group orbits of translation-free axial planforms in the six- and twelve-dimensional representations is considered. This problem is studied via the abstract action of the symmetry group of the perturbation on the group orbit of the planform. A partial classification for the behaviour of the group orbits is obtained, showing the existence of homoclinic and heteroclinic cycles between equilibria.     

Extending Melnikov theory to invariant manifolds on non-compact domains

Melnikov theory is usually examined under the hypothesis of hyperbolicity of critical points. We are extending this theory for regular perturbation problems of autonomous systems in arbitrary dimensions to the case of non-hyperbolic critical points that are located at infinity. The heteroclinic orbit of the unperturbed problem connecting these non-hyperbolic equilibria is of at most algebraic growth. By using a weaker dichotomy property than in the classical approach we obtain by a Lyapunov-Schmidt reduction a bifurcation equation that describes the existence of solutions of at most algebraic growth under small perturbations. We apply this extended Melnikov theory to problems arising in singularly perturbed systems to prove the existence of 'canard solutions'.     

Squeezed states in spontaneous Raman and in stimulated Raman processes

The Hamiltonian and the equations of motion for various modes involving spontaneous and stimulated Raman processes are derived. The coupled differential equations involving these modes are solved analytically by using an intuitive approach (IA). The solution of the present paper does not require any short time approximation. As a matter of fact, the present solution for field operators and hence the squeezing for various modes are valid for all interaction time t. In this way, the present paper is more general compared to the earlier investigations where short-time approximations were found a must. By exploiting the solutions for the field operators, we obtain the squeezing effects of input coherent light for pure and for mixed field modes. The IA employs the perturbation theory and hence we came across the secular terms in the solution. Of course, for small coupling constants these secular terms could be summed for all orders. To establish our claim, we remove the secular terms at least for one occasion.     

Essentially asymptotically stable homoclinic networks

Melbourne [An example of a nonasymptotically stable attractor, Nonlinearity 4(3) (1991), pp. 835–844] discusses an example of a robust heteroclinic network that is not asymptotically stable but which has the strong attracting property called essential asymptotic stability. We establish that this phenomenon is possible for homoclinic networks, where all heteroclinic trajectories are symmetry related. Moreover, we study a transverse bifurcation from an asymptotically stable to an essentially asymptotically stable homoclinic network. The essentially asymptotically stable homoclinic network turns out to attract all nearby points except those on codimension-one stable manifolds of equilibria outside the homoclinic network.     

Cascades of homoclinic orbits to a saddle-centre for reversible and perturbed Hamiltonian systems

The bifurcation of double-pulse homoclinic orbits under parameter perturbation is analysed for reversible systems having a homoclinic solution that is biasymptotic to a saddle-centre equilibrium. This is a non-hyperbolic equilibrium with two real and two purely imaginary eigenvalues. Reversibility enforces that small perturbations will not change this eigenvalue configuration. Using a Shil'nikov-type analysis, it is found that (generically) an infinite sequence of parameter values exists, on one side of that of the primary homoclinic, for which there are double-pulse homoclinic orbits. Mielke, Holmes and O'Reilly considered the same situation with the additional assumption of Hamiltonian structure. There, double pulses exist on either both or neither side, depending on a sign condition which also determines whether there can be any recurrent dynamics. It is shown how this sign condition occurs in the purely reversible case, via the breaking of a non-degeneracy assumption. Two possible two-parameter bifurcation diagrams are constructed under the addition of a perturbation that keeps reversibility but destroys Hamiltonian structure. The results can be stated rigorously only under a technical hypothesis on the validity of a normal form reduction. Even if this hypothesis fails to be strictly true, then the analysis is shown to be qualitatively and quantitatively correct by careful comparison with two numerical examples. The examples are also of interest in their own right; one of them a generalization of the classical Massive Thirring Model for optical spatial solitons in the presence of linear and nonlinear dispersion, the other is a perturbation to a continuum model of a discrete lattice. The computations agree perfectly with the theory including the prediction of different rates at which double pulses accumulate in the Hamiltonian and nonHamiltonian cases.     

Isolas, cusps and global bifurcations in an electronic oscillator

The aim of the present work is to describe the bifurcation behaviour of a class of asymmetric periodic orbits, in an electronic oscillator. The first time we detected them they were organized in a closed branch: that is, their bifurcation diagram showed an eight-shaped isola, with a nice structure of secondary branches emerging from period-doubling bifurcations. In a two-parameter bifurcation set, the isola structure persists. We find the regions of its existence, and describe its destruction in an isola centre with a cusp of periodic orbits. Finally, the introduction of a third parameter allows us to find the relation of our orbits to symmetric periodic orbits (via a symmetry-breaking bifurcation) and to homoclinic connections of the non-trivial equilibria. The isolas are successively created by collision of two adjacent limbs of the wiggly bifurcation curve. The Shil?nikov homoclinic and heteroclinic connections, related to the symmetric and asymmetric periodic orbits, emerge from T-points and end at Shil?nikov-Hopf singularities     

Damped oscillations modeled by an n-th order time dependent quasi-linear differential system

Summary. A general formula based on the extended (by Popov [4]) Krylov-Bogoliubov-Mitropolskii method [1], [2] is presented for obtaining asymptotic solution of an n-th order time dependent quasi-linear differential equation with damping. The method of determination of the solution is simple and easier than the classical formulae developed by several authors as well as the technique initiated by the original contributors [1], [2]. The general solution can be used arbitrarily for different values of n = 2, 3. The method can be used not only for periodic forcing terms, but also for some non-periodic (bounded) forces. All the solutions can be determined from a single trial solution. On the contrary, at least two trial solutions are needed to investigate time-dependent differential equations; one is for the resonance case and the other for the non-resonance case. The later solution is sometimes used in the case of non-periodic external forces. However, the resonance cases (including damped forced vibrations [7]) are mainly considered in this paper, since these are important in vibration problems.