A generalized Padé approximation method of solving homoclinic and heteroclinic orbits of strongly nonlinear autonomous oscillators

An intrinsic extension of Pad′e approximation method, called the generalized Pad′e approximation method, is proposed based on the classic Pad′e approximation theorem. According to the proposed method, the numerator and denominator of Pad′e approximant are extended from polynomial functions to a series composed of any kind of function, which means that the generalized Pad′e approximant is not limited to some forms, but can be constructed in different forms in solving different problems. Thus, many existing modifications of Pad′e approximation method can be considered to be the special cases of the proposed method. For solving homoclinic and heteroclinic orbits of strongly nonlinear autonomous oscillators, two novel kinds of generalized Pad′e approximants are constructed. Then, some examples are given to show the validity of the present method. To show the accuracy of the method, all solutions obtained in this paper are compared with those of the Runge–Kutta method.     

Lauwerier吸引子结构和SBR测度

Abstract. In this paper,the Lauwerier map     

Homoclinic orbits for a perturbed quintic-cubic nonlinear Schrdinger equation

IntroductionIn recent years, there have been extensive studies on the existence of homoclinic orbit5 fOrnear integrable Hamiltonbo partial fferential equations, which are closely related to chaosI1--7].In this work, we consider a perturbed quintic-cubic nonlinear Schr5dinger (NLS) equationwhere q is 27-Periodic and even in x, D is a bounded dissipative operator and is assumed totake the formDq = --aq + jBqfor posititre constants cr and J. Here B is a Fourier truncation of the differentia…     

Global Bifurcations in Parametrically Excited Systems with Zero-to-One Internal Resonance

In this work we study the existence of Silnikov homoclinicorbits in the averaged equations representing the modal interactionsbetween two modes with zero-to-one internal resonance. The fast mode isparametrically excited near its resonance frequency by a periodicforcing. The slow mode is coupled to the fast mode when the amplitude ofthe fast mode reaches a critical value so that the equilibrium of theslow mode loses stability. Using the analytical solutions of anunperturbed integrable Hamiltonian system, we evaluate a generalizedMelnikov function which measures the separation of the stable and theunstable manifolds of an annulus containing the resonance band of thefast mode. This Melnikov function is used together with the informationof the resonances of the fast mode to predict the region of physicalparameters for the existence of Silnikov homoclinic orbits.     

Solitary Wave and Wave Front as Viewed From Curvature

The solitary wave and wave front are two important behaviors of nonlinear evolution equations. Geometrically, solitary wave and wave front are all plane curve. In this paper, they can be represented in terms of curvature c(s), which varies with arc length s. For solitary wave when s → ±∞, then its curvature c(s) approaches zero, and when s=0, the curvature c(s) reaches its maximum. For wave front, when s → ±∞, then its curvature c(s) approaches zero, and when s=0, the curvature c(s) is still zero, but c'(s) ≠ 0. That is, s=0 is a turning point. When c(s) is given, the variance at some point (x,y) in stream line with arc length s satisfies a 2-order linear variable-coefficient ordinary differential equation. From this equation, it can be determined qualitatively whether the given curvature is a solitary wave or wave front.     

一类共振条件下两个自由度系统的同宿轨道

研究一类在参数和强迫激励下发生共振时的两个自由度系统,利用多重尺度法证明存在锁频于Ω的周期解.在一定条件下可变换成Wiggins的系统,给出了判断这类系统同宿轨道存在的计算公式.     

New bifurcations of basin boundaries involving Wada and a smooth Wada basin boundary

This paper demonstrates and analyses double heteroclinic tangency in a three-well potential model, which can produce three new types of bifurcations of basin boundaries including from smooth to Wada basin boundaries, from fractal to Wada basin boundaries in which no changes of accessible periodic orbits happen, and from Wada to Wada basin boundaries. In a model of mechanical oscillator, it shows that a Wada basin boundary can be smooth.     

The abundance of strange attractor in the families of nearby singular diffeomorphism

In this paper, we consider the families of nearby singular diffeomorphism and the measure of a set in the parameter space, such that for each point of the set the corresponding diffeomorphism possesses strange attractor. For some families of one-dimensional mapping satisfying certain transversality condition, we prove that there is a positive measure set in the parameter space, such that the system in the corresponding families of nearly singular diffeomorphism has strange attractor. Furthermore, we study the dynamics of this type of strange attractor. Project Supported by Fund of National Science of China     

Bifurcations of three-dimensional diffeomorphisms with non-simple quadratic homoclinic tangencies and generalized Hénon maps

We study bifurcations of periodic orbits in two parameter general unfoldings of a certain type homoclinic tangency (called a generalized homoclinic tangency) to a saddle fixed point. We apply the rescaling technique to first return (Poincaré) maps and show that the rescaled maps can be brought to so-called generalized Hénon maps which have non-degenerate bifurcations of fixed points including those with multipliers e ^±iϕ . On the basis of this, we prove the existence of infinite cascades of periodic sinks and periodic stable invariant circles.      

Cyclicity of homoclinic loops and degenerate cubic Hamiltonians

New conditions for a planar homoclinic loop to have cyclicity two under multiple parameter perturbations have been obtained. As an application it is proved that a homoclinic loop of a nongeneric cubic Hamiltonian has cyclicity two under arbitrary quadratic perturbations. Project supported by the National Natural Science Foundation of China (Grant Nos. 19531070 and 19771037).