On Approximations of First Integrals for Strongly Nonlinear Oscillators

In this paper strongly nonlinear oscillator equations will be studied.It will be shown that the recently developed perturbation method based onintegrating factors can be used to approximate first integrals. Not onlyapproximations of first integrals will be given, butit will also be shown how in a rather efficient way the existence and stability oftime-periodic solutions can be obtained from these approximations. In particularthe generalized Rayleigh oscillator equation will be studied in detail, and it will beshown that at least five limit cycles can occur.     

On Approximations of First Integrals for a Strongly Nonlinear Forced Oscillator

In this paper a strongly nonlinear forced oscillator will be studied. It will be shown that the recently developed perturbation method based on integrating factors can be used to approximate first integrals. Not onlyapproximations of first integrals will be given, butit will also be shown how, in a rather efficient way, the existence and stability oftime-periodic solutions can be obtained from these approximations. In additionphase portraits, Poincaré-return maps, and bifurcation diagrams for a set of values of the parameters will be presented. In particularthe strongly nonlinear forced oscillator equation will be studied in this paper. It will be shown that the presentedperturbation method not onlycan be applied to a weakly nonlinear oscillator problem (that is, when the parameter ) but also to a strongly nonlinear problem (that is, when ). The model equation as considered in this paper is related to the phenomenon of galloping ofoverhead power transmission lines on which ice has accreted.     

Approximate First Integrals of Weakly Nonlinear,Damped-Driven Oscillators with One Degree of Freedom

First-order approximate symmetries of weaklynonlinear, damped-driven oscillators have been determined.First-order approximate first integrals have been obtained byemploying an approximate version of Noether's theorem for the conservativecase. Furthermore, approximate first integrals of the damped case have been obtained based on the first integrals of the conservative case.Approximate first integrals enabled us to identify three types of genericbifurcations. Analytical results have been verified by numerical experiments.     

A Perturbation-Incremental Method for Strongly Nonlinear Autonomous Oscillators with Many Degrees of Freedom

The perturbation-incremental method is extended to determine thebifurcations and limit cycles of strongly nonlinear autonomousoscillators with many degrees of freedom. Coupled van der Poloscillators and coupled Rayleigh oscillators are taken as numericalexamples. Limit cycles of the oscillators can be calculated to anydesired degree of accuracy. The stabilities of limit cycles are alsodiscussed.     

On the Weakly Nonlinear,Transversal Vibrations of a Conveyor Belt with a Low and Time-Varying Velocity

In this paper the weakly nonlinear, transversal vibrations of aconveyor belt will be considered. The belt is assumed to move witha low and time-varying speed. Using Kirchhoff's approach a singleequation of motion will be derived from a coupled system ofpartial differential equations describing the longitudinal andtransversal vibrations of the belt. A two time-scalesperturbation method is then applied to approximate the solutionsof the problem. It will turn out that the frequencies of the belt speed fluctuations play an important role in the dynamic behaviourof the belt. It is well-known in linear systems that instabilitiescan occur if the frequency of the belt speed fluctuations is thesum of two natural frequencies. However, in the weakly nonlinearcase as considered in this paper this is no longer true. It turns out that the weak nonlinearity stabilizes the system.     

求解非线性动力系统周期解推广的打靶法

提出一种确定非线性系统周期轨道及周期的改进打靶算法。首先通过改变系统的时间尺度，将非线性系统周期轨道的周期显式地出现在非线性系统的系统方程中，然后对传统打靶法进行改造，将周期也作为一个参数一起参入打靶法的迭代过程，从而能迅速确定出系统的周期轨道及其周期。该方法对初始迭代参数没有苛刻要求，可以用于分析强非线性系统，而且对参数激励系统同样有效，对高维系统也能迅速、准确地求得周期解。文中应用该方法对三维Rǒssler系统和八维非线性柔性转子-轴承系统的周期轨道和周期进行了求解，通过与四阶Runge-Kutta数值积分结果比较，验证了方法的有效性。     

Nonlinear Vibrations of a Simply Supported Rectangular Metallic Plate Subjected to Transverse Harmonic Excitation in the Presence of a One-to-One Internal Resonance

The nonlinear response of rectangular and square metallic plates subjectto transverse harmonic excitations is studied. The nonlinearitiesoriginate from the use of Von Kármán strains. The method of multiplescales is used to solve the system of differential equationsapproximately. Frequency response curves are presented for both squareand rectangular plates for primary resonance of either mode in thepresence of a one-to-one internal resonance. Stability of steady statesolutions is investigated. Bifurcation points and their types arediscussed.     

On the Asymptotic Solutions of Boundary Value Problems for a Class of Systems of Nonlinear Differential Equations (I)

A new method is applied to study the asymptotic behavior of solutions of boundary value problems for a class of systems of nonlinear differential equations . The asymptotic expansions of solutions are constructed, the remainders are estimated. The former works are improved and generalized.