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.     

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).     

Bifurcation of limit cycles from generalized homoclinic loops in planar piecewise smooth systems

The generalized homoclinic loop appears in the study of dynamics on piecewise smooth differential systems during the past two decades. For planar piecewise smooth differential systems, there are concrete examples showing that under suitable perturbations of a generalized homoclinic loop one or two limit cycles can appear. But up to now there is no a general theory to study the cyclicity of a generalized homoclinic loop, that is, the maximal number of limit cycles which are bifurcated from it.     

A GENERAL PROPERTY OF THE QUADRATIC DIFFERENTIAL SYSTEMS

In this paper, it is proved that the quadratic differential systems with a weak saddle of order 2 or 3 have no closed or singular closed orbit. Then by the results of [3], it follows that the greatest order of the homoclinic loop bifurcation of a quadratic differential system is between 2 and 3 It means a homoclinic loop can be split into at most three limit cycles.     

LIMIT CYCLES OF A QUADRATIC HAMILTONIAN SYSTEM BY CUBIC PERTURBATIONS

In this paper,we are concerned with a cubic near-Hamiltonian system,whose unperturbed system is quadratic and has a symmetric homoclinic loop.By using the method developed in [12],we find that the system can have 4 limit cycles with 3 of them being near the homoclinic loop.Further,we give a condition under which there exist 4 limit cycles.     

On the cyclicity of a 2-polycycle for quadratic systems

It has been already known that the maximum number of limit cycles near a homoclinic loop of a quadratic Hamiltonian system under quadratic perturbations is two. However, the problem of finding the maximum number of limit cycles in the 2-polycycle case is still open. This paper addresses the problem in some detail and solves it partially.     

Unfolding of a Quadratic Integrable System with a Homoclinic Loop

In this paper, we make a complete study of the unfolding of a quadratic integrable system with a homoclinic loop. Making a Poincaré transformation and using some new techniques to estimate the number of zeros of Abelian integrals, we obtain the complete bifurcation diagram and all phase portraits of systems corresponding to different regions in the parameter space. In particular, we prove that two is the maximal number of limit cycles bifurcation from the system under quadratic non-conservative perturbations. Received July 16, 1999, Revised March 15, 2001, Accepted May 25, 2001     

高维同宿环扭曲分支与稳定性

利用沿同宿环的线性变分方程的线性独立解作为在同宿环的小管状邻域内的局部坐标系来建立Poincaré映射,研究了高维系统扭曲同宿环的分支问题．在非共振条件和共振条件下,获得了1-同宿环、 1-周期轨道、 2-同宿环、 2-周期轨道和两重2-同期轨道的存在性、 存在个数和存在区域．给出了相关的分支曲面的近似表示．同时,研究了高维系统同宿环和平面系统非扭曲同宿环的稳定性．     

Limit cycles near generalized homoclinic and double homoclinic loops in piecewise smooth systems

In this paper, we study the bifurcation of limit cycles in piecewise smooth systems by perturbing a piecewise Hamiltonian system with a generalized homoclinic or generalized double homoclinic loop. We first obtain the form of the expansion of the first Melnikov function. Then by using the first coefficients in the expansion, we give some new results on the number of limit cycles bifurcated from a periodic annulus near the generalized (double) homoclinic loop. As applications, we study the number of limit cycles of a piecewise near-Hamiltonian systems with a generalized homoclinic loop and a central symmetric piecewise smooth system with a generalized double homoclinic loop.     

Homoclinic orbits for a class of first-order nonperiodic asymptotically quadratic Hamiltonian systems with spectrum point zero

In this paper, we study the existence and multiplicity of homoclinic orbits for a class of first-order nonperiodic Hamiltonian systems. By applying two recent critical point theorems for strongly indefinite functionals, we give some new criteria to guarantee that Hamiltonian systems with asymptotically quadratic terms and spectrum point zero have at least one and a finite number of pairs of homoclinic orbits under some adequate conditions, respectively.