A cubic differential system with nine limit cycles

Advances in Computer Algebra software have made calculations possible that were previously intractable. Our particular interest is in the investigation of limit cycles of nonlinear differential equations. We describe some recent developments in handling very large computations involving resultants and present an example of a nonlinear differential system of degree three with nine small amplitude limit cycles surrounding a focus. We know of no examples of cubic systems with more than this number bifurcating from a fine focus, as opposed to a centre. Our example appears to be the first to have been obtained without recourse to some numerical calculation.     

生物化学反应中一类非线性系统极限环的存在惟一性

研究了生物化学反应中一类非线性系统，得到了该系统的环绕正奇点极限环的充分必要条件，并且证明了如果存在极限环，则必惟一。     

Existence and uniqueness results for a nonlinear differential system

We study the existence, uniqueness and asymptotic behavior of the strong and weak solutions of a nonlinear differential system with 2N equations in a real Hilbert space H, subject to a boundary condition and initial data. This problem is a discrete version with respect to spatial variable x of some partial differential problems (with H = ℝⁿ ), which have applications in integrated circuits modelling     

A cubic system with thirteen limit cycles

We construct a planar cubic system and demonstrate that it has at least 13 limit cycles. The construction is essentially based on counting the number of zeros of some Abelian integrals.     

一类E13系统极限环的惟一性

研究一类E13系统x=y,y=-x+δy+nx2+mxy+ly2+bxy2,求出奇点O的焦点量W0=δ,W1=m(n+l),W2=-mnb.证明了W0=W1=W2=0时O为中心.其次证明了W0=0,W1W2≥0时系统无极限环;W0=0,W1W2＜0时系统至多有一个极限环.最后讨论了n=0,b＞0的情况.证明了存在δ0,0＜δ0≤-l/m,当0＜δ＜δ0时系统存在惟一极限环,δ=δ0时系统存在无穷远分界线环,δ≤0或δ＞δ0时系统无闭轨与奇闭轨.     

Existence of multiple limit cycles in Chen system

In this paper, the existence of multiple limit cycles for Chen system are investigated. By using the method of computing the singular point quantities, the simple and explicit parametric conditions can be determined to the number and stability of multiple limit cycles from Hopf bifurcation. Especially, at least 4 limit cycles can be obtained for the Chen system as a three-dimensional perturbed system.     

Fourteen limit cycles in a cubic Hamiltonian system with nine-order perturbed term

We study limit cycles of the following system:

with a>c>0, ac>1, 0<1, m,l,λ are real parameters and n is a positive integer. When n=2, J.B. Li and Z.R. Liu [Publ. Math. 35 (1991) 487] showed that the system has 11 limit cycles. When n=6, H.J. Cao, Z.R. Liu and Z.J. Jing [Chaos, Solitons & Fractals 11 (2000) 2293] showed the system has 13 limit cycles. Using the same method of detection function, we first show that the system and others four systems have the same bifurcation diagrams of limit cycle. Then we demonstrate that any one of the five systems has 14 limit cycles for n=8. The positions of the 14 limit cycles are given by numerical exploration.     

One and three limit cycles in a cubic predator–prey system

A cubic differential system is proposed, which can be considered a generalization of the predator–prey models, studied recently by many authors. The properties of the equilibrium points, the existence of a uniqueness limit cycle, and the conditions for three limit cycles are investigated. The criterion is easy to apply in applications. Copyright © 2006 John Wiley & Sons, Ltd.     

一类平面多项式系统极限环的存在唯一性

研究一类平面2n 1次多项式微分系统的极限环问题,利用Hopf分枝理论得到了该系统极限环存在性与稳定性的若干充分条件,利用Cherkas和Zheilevych的唯一性定理得到了极限环唯一性的若干充分条件.     

The uniqueness of limit cycles for Liénard system

In monographs [Theory of Limit Cycles, 1984] and [Qualitative Theory of Differential Equations, 1985], eleven propositions by several mathematicians are listed on the uniqueness of limit cycles for equations of type (I), (II), and (III) of the quadratic ordinary differential systems. In this paper, we first point out that all these propositions were not completely proved since the equations under consideration do not satisfy the conditions of the theorems used to guarantee the uniqueness of limit cycles. Then we give a new set of theorems that guarantee the uniqueness of limit cycles for the Liénard systems, which not only can be applied to complete the proof of the propositions mentioned above but generalize many other uniqueness theorems as well. The conditions in these uniqueness theorems, which are independent and were obtained by different methods, can be combined into one improved general theorem that is easy to apply. Thus many of the most frequently used theorems on the uniqueness of limit cycles are corollaries of the results in this paper.     

A cubic system with twelve small amplitude limit cycles

In this paper, the bifurcation of limit cycles for a cubic polynomial system is investigated. By the computation of the singular point values, we prove that the system has 12 small amplitude limit cycles. The process of the proof is algebraic and symbolic.     

具有一类三次曲线解的Kolmogorov三次系统的极限环的存在性

证明了具有三次曲线解y=-x(x-1)2 4/24的Kolmogorov三次系统是有存在极限环可能的.     

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.     

Z 2-equivariant cubic system which yields 13 limit cycles

For the planar Z_2-equivariant cubic systems having two elementary focuses,the characterization of a bi-center problem and shortened expressions of the first six Lyapunov constants are completely solved.The necessary and sufficient conditions for the existence of the bi-center are obtained.On the basis of this work,in this paper,we show that under small Z_2-equivariant cubic perturbations,this cubic system has at least 13 limit cycles with the scheme 16∪6.     

The number of limit cycles of cubic Hamiltonian system with perturbation

In this paper, by using qualitative analysis, we investigate the number of limit cycles of perturbed cubic Hamiltonian system with perturbation in the form of (2n+2m) or (2n+2m+1)th degree polynomials . We show that the perturbed systems has at most (n+m) limit cycles, and has at most n limit cycles if m=1. If m=1, n=1 and m=1, n=2, the general conditions for the number of existing limit cycles and the stability of the limit cycles will be established, respectively. Such conditions depend on the coefficients of the perturbed terms. In order to illustrate our results, two numerical examples on the location and stability of the limit cycles 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.