THE POINCARE BIFURCATION OF QUADRATIC SYSTEMS HAVING A REGION CONSISTING OF PERIODIC CYCLES BOUNDED BY A HYPERBOLA AND AN ARC OF EQUATOR

In this paper, we discuss the Poincaré bifurcation for a class of quadratic systems having a region consisting of periodic cycles bounded by a hyperbola and an arc of equator. We prove that the system can at most generate two limit cycles after a small perturbation.     

Cubic Lienard Equations with Quadratic Damping (II)

Abstract   Applying Hopf bifurcation theory and qualitative theory, we show that the general cubic Lienard equations with quadratic damping have at most three limit cycles. This implies that the guess in which the system has at most two limit cycles is false. We give the sufficient conditions for the system has at most three limit cycles or two limit cycles. We present two examples with three limit cycles or two limit cycles by using numerical simulation. Supported by the National Natural Science Foundation of China and National Key Basic Research Special Found (No. G1998020307).     

Algebraic limit cycles of degree 4 for quadratic systems

Yablonskii (Differential Equations 2 (1996) 335) and Filipstov (Differential Equations 9 (1973) 983) proved the existence of two different families of algebraic limit cycles of degree 4 in the class of quadratic systems. It was an open problem to know if these two algebraic limit cycles where all the algebraic limit cycles of degree 4 for quadratic systems. Chavarriga (A new example of a quartic algebraic limit cycle for quadratic sytems, Universitat de Lleida, Preprint 1999) found a third family of this kind of algebraic limit cycles. Here, we prove that quadratic systems have exactly four different families of algebraic limit cycles. The proof provides new tools based on the index theory for algebraic solutions of polynomial vector fields.     

Relationships between limit cycles and algebraic invariant curves for quadratic systems

Algebraic limit cycles for quadratic systems started to be studied in 1958. Up to now we know 7 families of quadratic systems having algebraic limit cycles of degree 2, 4, 5 and 6. Here we present some new results on the limit cycles and algebraic limit cycles of quadratic systems. These results provide sometimes necessary conditions and other times sufficient conditions on the cofactor of the invariant algebraic curve for the existence or nonexistence of limit cycles or algebraic limit cycles. In particular, it follows from them that for all known examples of algebraic limit cycles for quadratic systems those cycles are unique limit cycles of the system.     

ON THE NON-EXISTENCE OF LIMIT CYCLES OF CERTAIN QUADRATIC SYSTEMS

§１．Ｆｏｒｔｈｅｓｙｓｔｅｍｘ＝－ｙ＋δｘ＋ｌｘ２＋ｎｙ２＝Ｐ（ｘ，ｙ），ｙ＝ｘ（１＋ａｘ－ｙ）＝Ｑ（ｘ，ｙ），｛（１．１）ｗｅｃａｎｆｉｎｄｉｎ［１］ｔｈｅｆｏｌｏｗｉｎｇ：ＣｏｎｊｅｃｔｕｒｅＩ．Ａｓｓｕｍｅ１ａ＜０，ｎ＞１，ｎ＋ｌ＞０，ｎａ２...     

Bifurcation of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems of degree 5 with a cusp

This paper deals with small perturbations of a class of hyper-elliptic Hamiltonian system, which is a Li é nard system of the form \(\dot{x}=y,\)  \(\;\dot{y}=Q_1(x)+\varepsilon yQ_2(x)\) with \(Q_1\) and \(Q_2\) polynomials of degree 4 and 3, respectively. It is shown that this system can undergo degenerated Hopf bifurcation and Poincar é bifurcation, which emerge at most three limit cycles for \(\varepsilon\) sufficiently small.     

Ten limit cycles around a center-type singular point in a 3-d quadratic system with quadratic perturbation

In this paper, we show that perturbing a simple 3-d quadratic system with a center-type singular point can yield at least 10 small-amplitude limit cycles around a singular point. This result improves the 7 limit cycles obtained recently in a simple 3-d quadratic system around a Hopf singular point. Compared with Bautin’s result for quadratic planar vector fields, which can only have 3 small-amplitude limit cycles around an elementary center or focus, this result of 10 limit cycles is surprisingly high. The theory and methodology developed in this paper can be used to consider bifurcation of limit cycles in higher-dimensional systems.     

A study on Zoladek's example

In this paper, we consider an example of third-order polynomial planar system, proposed by Zoladek who claimed that this example had eleven small-amplitude limit cycles around a center. We use focus value computation to show that for this example there may exist maximal nine small-amplitude limit cycles around the center due to Hopf bifurcation.     

A criterion for the existence of four limit cycles in quadratic systems

A method for the asymptotic integration of the trajectories is proposed for the Liénard equation. The results obtained by this method are used to prove the existence of two “large” limit cycles in quadratic systems with a weak focus. The application of standard procedures of small perturbations of the parameters of quadratic systems enables one to find additionally two “small” limit cycles. It is shown that the criterion obtained for the existence of four limit cycles generalizes the well known Shi theorem.     

Limit Cycle Bifurcations Of A Kind Of Lienard System With A Hypobolic Saddle And A Nilpotent Cusp

This paper gives a general theorem on the number of limit cycles of a near Hamiltonian system with a heteroclinic loop passing through a hyperbolic saddle and a nilpotent cusp. Then we study a kind of Lienard systems of type (n,4) for 3<=n<=27 and obtain the lower bound of the maximal number of limit cycles for this kind of system.     

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.     

一类不连续平面二次微分系统的极限环

本文考虑了一类不连续平面二次可积非Hamilton微分系统在二次扰动下的极限环个数问题.利用一阶平均法,我们得到了从该系统中心的周期环域至少可以分支出5个极限环的结论.该结果表明不连续二次微分系统比其相应光滑微分系统至少可以多分支出2个极限环.     

Limit cycles of quadratic systems

In this paper, the global qualitative analysis of planar quadratic dynamical systems is established and a new geometric approach to solving Hilbert’s Sixteenth Problem in this special case of polynomial systems is suggested. Using geometric properties of four field rotation parameters of a new canonical system which is constructed in this paper, we present a proof of our earlier conjecture that the maximum number of limit cycles in a quadratic system is equal to four and their only possible distribution is (3:1) [V.A. Gaiko, Global Bifurcation Theory and Hilbert’s Sixteenth Problem, Kluwer, Boston, 2003]. Besides, applying the Wintner–Perko termination principle for multiple limit cycles to our canonical system, we prove in a different way that a quadratic system has at most three limit cycles around a singular point (focus) and give another proof of the same conjecture.     

Invariant algebraic curves of large degree for quadratic system

In this paper we present for the first time examples of algebraic limit cycles and saddle loops of degree greater than 4 for planar quadratic systems. In particular, we give examples of algebraic limit cycles of degree 5 and 6, and algebraic saddle loops of degree 3 and 5 surrounding a strong focus. We also give an example of an invariant algebraic curve of degree 12 for which the quadratic system has no Darboux integrating factors or first integrals.     

Cyclicity of several quadratic reversible systems with center of genus one

By using the Chebyshev criterion to study the number of zeros of Abelian integrals, developed by M. Grau, F. Ma\(\~n\)osas and J. Villadelprat in [2], we prove that the cyclicity of period annulus of the quadratic reversible systems with center of genus one, classified as (r8), (r13) and (r16) by S. Gautier, L. Gavrilov and I. D. Iliev in [1], under quadratic perturbations is two. These results partially give a positive answer to the conjecture 1 in [1].     

Lyapunov quantities and limit cycles of two-dimensional dynamical systems. Analytical methods and symbolic computation

In the present work the methods of computation of Lyapunov quantities and localization of limit cycles are demonstrated. These methods are applied to investigation of quadratic systems with small and large limit cycles. The expressions for the first five Lyapunov quantities for general Lienard system are obtained. By the transformation of quadratic system to Lienard system and the method of asymptotical integration, quadratic systems with large limit cycles are investigated. The domain of parameters of quadratic systems, for which four limit cycles can be obtained, is determined.     

具有退化三次曲线解的Hamilton二次系统的Poincare分支

具有退化三次曲线解的Hamilton二次系统,经二次微扰后的Poincare分支,是否存在两个极限环?这是一个长期受到困扰的问题.本文证明了在特定条件下,可以分支出两个极限环.     

Bifurcation of limit cycles for the quadratic differential system (III)l=n=0

In this paper we will prove that limit cycles for the quadratic differential system (III)l=n=0 in Chinese classification are concentratedly distributed, and that the maximum number of limit cycles around O (0,0) is at least two. This project is supported by the Natural Science Foundation of Educational Committee of Jiangsu Province.     

关于二次微分系统Ⅰ类方程的极限环的存在性

本文讨论二次微分系统Ⅰ类方程ｄｘｄｔ＝ － ｙ＋ δｘ＋ ｌｘ２＋ ｍ ｘｙ＋ ｎｙ２,ｄｙｄｔ＝ ｘ 的极限环的存在性问题,纠正了文［１］中讨论当ｌ＝ ０ 时大范围内存在极限环的缺陷     

On the Limit Cycles Bifurcating from a Quadratic Reversible Center of Genus One

The paper is concerned with the bifurcation of limit cycles in perturbations of a quadratic reversible system with a center of genus one. By studying the properties of the auxiliary curve and centroid curve defined by the Abelian integrals, we have proved that under small quadratic perturbations, at most two limit cycles arise from the period annulus surrounding the quadratic reversible center, and the bound is sharp. This partially verifies Conjecture 1 given in Gautier et al. (Discrete Contin Dyn Syst 25:511–535, 2009).