共查询到20条相似文献,搜索用时 31 毫秒
1.
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. 相似文献
2.
Maoan Han 《中国科学A辑(英文版)》1997,40(12):1247-1258
A general method for a homoclinic loop of planar Hamiltonian systems to bifurcate two or three limit cycles under perturbations
is established. Certain conditions are given under which the cyclicity of a homoclinic loop equals 1 or 2. As an application
to quadratic systems, it is proved that the cyclicity of homoclinic loops of quadratic integrable and non-Hamiltonian systems
equals 2 except for one case.
Project supported by the National Natural Science Foundation of China. 相似文献
3.
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. 相似文献
4.
This paper concerns with the bifurcation of limit cycles from a double homoclinic loop under multiple parameter perturbations for general planar systems. The existence conditions of 4 homoclinic bifurcation curves and small and large limit cycles are especially investigated. 相似文献
5.
On the number of limit cycles in double homoclinic bifurcations 总被引:7,自引:0,他引:7
LetL be a double homoclinic loop of a Hamiltonian system on the plane. We obtain a condition under whichL generates at most two large limit cycles by perturbations. We also give conditions for the existence of at most five or six
limit cycles which appear nearL under perturbations. 相似文献
6.
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. 相似文献
7.
Lin Ping Peng 《数学学报(英文版)》2002,18(4):737-754
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 相似文献
8.
In this paper, we investigate the cyclicity of the period annulus of two classes of cubic isochronous systems.By using the Chebyshev criterion, we prove that the two systems have respectively at most three and four limit cycles produced fromthe period annulus around the isochronous center under cubic perturbations. 相似文献
9.
Zhao Yong Han Maoan Yang Junmin 《Annals of Differential Equations》2007,23(4):593-602
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. 相似文献
10.
The stability and bifurcations of a homoclinic loop for planar vector fields are closely related to the limit cycles. For a homoclinic loop of a given planar vector field, a sequence of quantities, the homoclinic loop quantities were defined to study the stability and bifurcations of the loop. Among the sequence of the loop quantities, the first nonzero one determines the stability of the homoclinic loop. There are formulas for the first three and the fifth loop quantities. In this paper we will establish the formula for the fourth loop quantity for both the single and double homoclinic loops. As applications, we present examples of planar polynomial vector fields which can have five or twelve limit cycles respectively in the case of a single or double homoclinic loop by using the method of stability-switching. 相似文献
11.
Transverse homoclinic orbit bifurcated from a homoclinic manifold by the higher order melnikov integrals
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Consider an autonomous ordinary differential equation in $\mathbb{R}^n$ that has a $d$ dimensional homoclinic solution manifold $W^H$. Suppose the homoclinic manifold can be locally parametrized by $(\alpha,\theta) \in \mathbb{R}^{d-1}\times \mathbb{R}$. We study the bifurcation of the homoclinic solution manifold $W^H$ under periodic perturbations. Using exponential dichotomies and Lyapunov-Schmidt reduction, we obtain the higher order Melnikov function. For a fixed $(\alpha_0,\theta_0)$ on $W^H$, if the Melnikov function have a simple zeros,
then the perturbed system can have transverse homoclinic solutions near $W^H$. 相似文献
12.
Junmin Yang Valery G. Romanovski 《Journal of Mathematical Analysis and Applications》2010,366(1):242-255
In this paper we first give some general theorems on the limit cycle bifurcation for near-Hamiltonian systems near a double homoclinic loop or a center as a preliminary. Then we use these theorems to study some polynomial Liénard systems with perturbations and give new lower bounds for the maximal number of limit cycles of these systems. 相似文献
13.
本文应用有限光滑正规形理论,对可积系统的多角环S^(2),在小扰动下,当满足一定的非退化条件时,证明其环性为2。 相似文献
14.
Jianhua Sun 《数学学报(英文版)》1995,11(2):128-136
We consider perturbations which may or may not depend explicitly on time for the three-dimensional homoclinic systems. We obtain the existence and bifurcation theorems for transversal homoclinic points and homoclinic orbits, and illustrate our results with two examples. 相似文献
15.
This paper is concerned with limit cycles which bifurcate from a period annulus of a quadratic reversible Lotka-Volterra system with sextic orbits. The authors apply the property of an extended complete Chebyshev system and prove that the cyclicity of the period annulus under quadratic perturbations is equal to two. 相似文献
16.
研究了一类3维反转系统中包含2个鞍点的对称异维环分支问题, 且仅限于研究系统的线性对合R的不变集维数为1的情形.
给出了R-对称异宿环与R-对称周期轨线存在和共存的条件, 同时也得到了R-对称的重周期轨线存在性. 其
次, 给出了异宿环、 同宿轨线、 重同宿轨线和单参数族周期轨线的存在性、 唯一性和共存性等结论,
并且发现不可数无穷条周期轨线聚集在某一同宿轨线的小邻域内. 最后给出了相应的分支图. 相似文献
17.
Linping PENG 《Frontiers of Mathematics in China》2011,6(5):911-930
In this paper, we study a reversible and non-Hamitonian system with a period annulus bounded by a hemicycle in the Poincaré
disk. It is proved that the cyclicity of the period annulus under quadratic perturbations is equal to two. This verifies some
results of the conjecture given by Gautier et al. 相似文献
18.
The main aims of this paper are to study the persistence of homoclinic and heteroclinic orbits of the reduced systems on normally hyperbolic critical manifolds, and also the limit cycle bifurcations either from the homoclinic loop of the reduced systems or from a family of periodic orbits of the layer systems. For the persistence of homoclinic and heteroclinic orbits, and the limit cycles bifurcating from a homolinic loop of the reduced systems, we provide a new and readily detectable method to characterize them compared with the usual Melnikov method when the reduced system forms a generalized rotated vector field. To determine the limit cycles bifurcating from the families of periodic orbits of the layer systems, we apply the averaging methods.We also provide two four-dimensional singularly perturbed differential systems, which have either heteroclinic or homoclinic orbits located on the slow manifolds and also three limit cycles bifurcating from the periodic orbits of the layer system. 相似文献
19.
Changrong Zhu Guangping Luo Kunquan Lan 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2010
The homoclinic bifurcations of ordinary differential equation under singular perturbations are considered. We use exponential dichotomy, Fredholm alternative and scales of Banach spaces to obtain various bifurcation manifolds with finite codimension in an appropriate infinite-dimensional space. When the perturbative term is taken from these bifurcation manifolds, the perturbed system has various coexistence of homoclinic solutions which are linearly independent. 相似文献
20.
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. 相似文献