一类三次系统极限环的个数与分布

本文研究一类三次系统的极限环,利用分支理论与定性分析技巧发现这类系统有四个极限环,并给出了他们的分布。     

DISTRIBUTIONS OF LIMIT CYCLES FOR POLYNOMIAL VECTOR FIELDS (P_2, Q_3)

The distributions of limit cycles of cubic vector fields (P2, Q3) are considered in this paper, where P2 and Q3 are polynomials of x and y of order two and three, respectively. It is possibly seven different distributions of limit cycles given in [1]. We now prove that in which three kinds of distributions are impossible and other four kinds all can be realized by concrete vector fields of (P2,Q3). Some related results are also given.     

Perturbation from an asymmetric cubic Hamiltonian

This paper concerns with the number of limit cycles from an asymmetric Hamiltonian of degree three under cubic perturbation. Eleven limit cycles are found and three different distributions are given by using the methods of bifurcation theory and qualitative analysis, two of which are new.     

On the Bifurcations of a Hamiltonian Having Three Homoclinic Loops under Z3 Invariant Quintic Perturbations

A cubic system having three homoclinic loops perturbed by Z3 invariant quintic polynomials is considered. By applying the qualitative method of differential equations and the numeric computing method, the Hopf bifurcation, homoclinic loop bifurcation and heteroclinic loop bifurcation of the above perturbed system are studied. It is found that the above system has at least 12 limit cycles and the distributions of limit cycles are also given.     

Bifurcations of limit cycles for a cubic Hamiltonian system under quartic perturbations

This paper is concerned with the number of limit cycles of a cubic system with quartic perturbations. Fifteen limit cycles are found and their distributions are studied by using the methods of bifurcation theory and qualitative analysis. It gives rise to the conclusion: H(4)15, where H(n) is the Hilbert number for the second part of Hilbert's 16th problem.     

BIFURCATIONS OF A CUBIC SYSTEM PERTURBED BY DEGREE FIVE

In this article, using multi-parameter perturbation theory and qualitative analysis, the authors studied a kind of cubic system perturbed by degree five and ob-tained the system that can have 17 limit cycles giving their two kinds of distributions (see Fig.5).     

The Same Distribution of Limit Cycles in a Hamiltonian System with Nine Seven-order Perturbed Terms

Using qualitative analysis and numerical simulation, we investigate the number and distribution of limit cycles for a cubic Hamiltonian system with nine different seven-order perturbed terms. It is showed that these perturbed systems have the same distribution of limit cycles. Furthermore, these systems have 13, 11 and 9 limit cycles for some parameters, respectively. The accurate positions of the 13, 11 and 9 limit cycles are obtained by numerical exploration, respectively. Our results imply that these perturbed systems are equivalent in the sense of distribution of limit cycles. This is useful for studying limit cycles of perturbed systems.     

Bifurcation and Isochronicity at Infinity in a Class of Cubic Polynomial Vector Fields

In this paper, we study the appearance of limit cycles from the equator and isochronicity of infinity in polynomial vector fields with no singular points at infinity. We give a recursive formula to compute the singular point quantities of a class of cubic polynomial systems, which is used to calculate the first seven singular point quantities. Further, we prove that such a cubic vector field can have maximal seven limit cycles in the neighborhood of infinity. We actually and construct a system that has seven limit cycles. The positions of these limit cycles can be given exactly without constructing the Poincare cycle fields. The technique employed in this work is essentially different from the previously widely used ones. Finally, the isochronous center conditions at infinity are given.     

The center conditions and bifurcation of limit cycles at the infinity for a cubic polynomial system

In this paper, the problem of center conditions and bifurcation of limit cycles at the infinity for a class of cubic systems are investigated. The method is based on a homeomorphic transformation of the infinity into the origin, the first 21 singular point quantities are obtained by computer algebra system Mathematica, the conditions of the origin to be a center and a 21st order fine focus are derived, respectively. Correspondingly, we construct a cubic system which can bifurcate seven limit cycles from the infinity by a small perturbation of parameters. At the end, we study the isochronous center conditions at the infinity for the cubic system.     

具有三次曲线解的中心对称三次系统的极限环的存在性问题

本文证明了具有三次曲线解y＝αx3的中心对称三次系统可以存在极限环，从而纠正了文[1]认为具有三次曲线解的中心对称三次系统不可能存在极限环的错误结论     

Center conditions and bifurcation of limit cycles at three-order nilpotent critical point in a cubic Lyapunov system

In the present paper, for the three-order nilpotent critical point of a cubic Lyapunov system, the center problem and bifurcation of limit cycles are investigated. With the help of computer algebra system-MATHEMATICA, the first 7 quasi-Lyapunov constants are deduced. As a result, sufficient and necessary conditions in order to have a center are obtained. The fact of there exist 7 small amplitude limit cycles created from the three-order nilpotent critical point is also proved. Henceforth we give a lower bound of cyclicity of three-order nilpotent critical point for cubic Lyapunov systems.     

The cyclicity of the period annulus of the cubic isochronous center

This paper is concerned with limit cycles which bifurcate from periodic orbits of the cubic isochronous center. It is proved that in this situation, the cyclicity of the period annulus under cubic perturbations is equal to four. Moreover, for each k?=?0,1, . . .,4, there are perturbations that give rise to exactly k limit cycles bifurcating from the period annulus.     

Twelve limit cycles around a singular point in a planar cubic-degree polynomial system

In this paper, we prove the existence of 12 small-amplitude limit cycles around a singular point in a planar cubic-degree polynomial system. Based on two previously developed cubic systems in the literature, which have been proved to exhibit 11 small-amplitude limit cycles, we applied a different method to show 11 limit cycles. Moreover, we show that one of the systems can actually have 12 small-amplitude limit cycles around a singular point. This is the best result so far obtained in cubic planar vector fields around a singular point.     

Quadratic systems with maximum number of limit cycles

We suggest a method for obtaining quadratic systems with a given distribution of limit cycles. We use it to obtain a set of quadratic systems with the distributions (3, 1), (3, 0), and 3 of limit cycles and with different configurations of singular points. The distributions are justified with the use of a modified Dulac function in a natural domain of existence of limit cycles.     

THE POINCAR BIFURCATION OF CUBIC HAMILTONIAN SYSTEMS WITH DOUBLE CENTERS

In this paper, we discuss the Poincare bifurcation of cubic Hamiltonian systems with double centers and prove that the systems may at least generate two limit cycles and at most generate three limit cycles outside the lemniscate after a small cubic perturbation.     

GLOBAL BIFURCATIONS OF LIMIT CYCLES FOR A CUBIC SYSTEM

1MainResultsConsiderthecubicsystemwheree>Oissmall,parametersa=(a,,a,)EDCR2withDboundedandcl)cz)bl?bzareconstants.LetWesupposep'(y)2ofory/oorequivalentlyoscf53c2.(1.3)Assumethatg(x)hasonlysimplezeroswith6152/o.Withoutlossofgene-ralitywecansuppose610,5z<0(onecenterandtwosaddles);(3)5l-46z<0,bz>0(twocentersandonesaddle),Inthispaperweonlyconsiderthefirsttwocases.Obviously,(1'1).=ohasafamily0fperiodicorbitswhe…     

Cubic Lienard equation with quadradic damping (I)

1. IntroductionLienard equationdZx dx~ f(.)g g(x) = 0 (l.0)dtZ dthas been extensively studied with particular emphasis on the ekistence and uniqueness oflimit cycles (see e.g. [l--4] and references there in). The number of limit cycles of (l.0) hasbeen also investigated by several authors (see e.g. [5--8]).In the present paper we study the general cubic Lienard equation, namelydx da~ = y ~ F(x), Z ~ ~g(x) (1.1)dt' dtwhereF(x) = ale a,x: a,x', (l.2)g(x) = blx b,x' b,x'. (1.3)Clea…     

LIMIT CYCLES OF SOME Z_3-EQUIVARIANT NEAR-HAMILTONIAN SYSTEMS OF DEGREES 3 AND 4

This paper studies the number of limit cycles of some Z3-equivariant near-Hamiltonian systems of degrees 3 and 4,which are a perturbation of a cubic Hamiltonian system. By the Melnikov function method,we obtain 5 and 6 limit cycles respectively.     

Limit cycles of a lienard cubic system with quadratic friction function

We consider a Lienard cubic system with quadratic friction function and suggest a method for constructing such systems with the following distributions of limit cycles around the singular points: ((2, 0), 0), ((0, 2), 0), ((1, 1), 1), ((1, 1), 0), ((1, 0), 1), ((0, 1), 1), ((0, 0), 2), ((0, 1), 2), and ((1, 0), 2).     

Cubic Lienard Equations with Quadratic Damping (Ⅱ)

Abstract Applying Hopf bifurcation theory and qualitative theory, we show that the general cubic Lienardequations with quadratic damping have at most three limit cycles. This implies that the guess in which thesystem has at most two limit cycles is false. We give the sufficient conditions for the system has at most threelimit cycles or two limit cycles. We present two examples with three limit cycles or two limit cycles by usingnumerical simulation.