Local bifurcations of critical periods in a generalized 2D LV system

In this paper, we investigate a generalized two-dimensional Lotka-Volterra system which has a center. We give an inductive algorithm to compute polynomials of periodic coefficients, find structures of solutions for systems of algebraic equation corresponding to isochronous centers and weak centers of finite order, and derive conditions on parameters under which the considered equilibrium is an isochronous center or a weak center of finite order. We show that with appropriate perturbations at most two critical periods bifurcate from the center.     

一类五次多项式系统原点等时中心的分析

研究了一类五次系统原点复等时中心的问题.先通过一种最新算法求出了这类五次系统原点的周期常数,从而得到复等时中心的必要条件,并利用一些有效途径证明它们的充分性.这实际上解决了这类五次系统的伴随系统原点等时中心问题与其自身为实系统时鞍点可线性化的问题.     

The center conditions and local bifurcation of critical periods for a Liénard system

In this paper we study the problems of centers and isochronous centers and the local bifurcation of critical periods for a Liénard system with forth damping. Calculating the singular point values and period constants, we find all center conditions and isochronous center conditions. Moreover, the numbers of local critical periods bifurcating from centers and isochronous centers is obtained by computing the orders of weak centers.     

Isochronicity of centers in a switching Bautin system

In this paper isochronicity of centers is discussed for a class of discontinuous differential system, simply called switching system. We give some sufficient conditions for the system to have a regular isochronous center at the origin and, on the other hand, construct a switching system with an irregular isochronous center at the origin. We give a computation method for periods of periodic orbits near the center and use the method to discuss a switching Bautin system for center conditions and isochronous center conditions. We further find all of those systems which have an irregular isochronous center.     

时间可逆系统的等时中心问题

若在中心附近的闭轨线都具有相同的周期,则此中心称为等时中心. 时间可逆多项式系统的等时中心问题是一类公开问题. 为了构造性地解决这一难题,讨论一类范围更广的时间可逆解析动力系统, 给出相应横截交换系统的递推公式,此公式可以用于等时中心条件的推导. 在递推公式的基础上,以吴特征集方法为工具,给出一类时间可逆三次系统具有横截交换系统的充要条件.     

Center and isochronous center at infinity for differential systems

In this article, the center conditions and isochronous center conditions at infinity for differential systems are investigated. We give a transformation by which infinity can be transferred into the origin. So we can study the properties of infinity with the methods of the origin. As an application of our method, we discuss the conditions of infinity to be a center and a isochronous center for a class of rational differential system. As far as we know, this is the first time that the isochronous center conditions of infinity are discussed.     

一类可逆三次系统的等时中心

对于一般多项式系统,给出可逆代数条件推导算法;对于一类可逆三次系统,提出周期系数改进算法,得到原点为等时中心的充要条件.     

一类五次多项式系统无穷远点等时中心条件与极限环分支

研究一类五次系统无穷远点的中心、拟等时中心条件与极限环分支问题.首先通过同胚变换将系统无穷远点转化成原点,然后求出该原点的前8个奇点量,从而导出无穷远点成为中心和最高阶细焦点的条件,在此基础上给出了五次多项式系统在无穷远点分支出8个极限环的实例.同时通过一种最新算法求出无穷远点为中心时的周期常数,得到了拟等时中心的必要条件,并利用一些有效途径一一证明了条件的充分性.     

Linearizability conditions of a time-reversible quartic-like system

In this paper we study the linearizability problem of polynomial-like complex differential systems. We give a reduction of linearizability problem of such non-polynomial systems to the problem of polynomial systems. Applying this reduction, we find some linearizability conditions for a time-reversible quartic-like complex system and derive from them conditions of isochronous center for the corresponding real system.     

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.     

A new method to determine isochronous center conditions for polynomial differential systems

The computation of period constants is a way to study isochronous center for polynomial differential systems. In this article, a new method to compute period constants is given. The algorithm is recursive and easy to realize with computer algebraic system. As an application, we discuss the center conditions and isochronous centers for a class of high-degree system.     

Classification of the centers and isochronicity for a class of quartic polynomial differential systems

In this paper, the conditions of center and isochronous center at the origin for a class of planar quartic differential systems are studied. At first, a constructive theorem of singular point quantities is presented, which plays an important role in simplifying periodic constants. The sufficient and necessary conditions for the origin of the systems being a center are obtained. Then a complete classification of the sufficient and necessary conditions are given for the origin of the systems being an isochronous center.     

Non-isochronicity of the center at the origin in polynomial Hamiltonian systems with even degree nonlinearities

In 2002 X. Jarque and J. Villadelprat proved that no center in a planar polynomial Hamiltonian system of degree 4 is isochronous and raised a question: Is there a planar polynomial Hamiltonian system of even degree which has an isochronous center? In this paper we give a criterion for non-isochronicity of the center at the origin of planar polynomial Hamiltonian systems. Moreover, the orders of weak centers are determined. Our results answer a weak version of the question, proving that there is no planar polynomial Hamiltonian system with only even degree nonlinearities having an isochronous center at the origin.     

Bifurcation of limit cycles and isochronous center at infinity for a class of differential systems

In this paper, we study a seventh degree polynomial differential system with full linear terms and cubic terms. The conditions of infinity to be a center and to be a fine focus of the highestorder are given and it is proved that this system has eight limit cycles in the neighborhood of infinity. Moreover, the conditions of infinity to be an isochronous center for a rational system associated the seventh degree polynomial differential system are obtained.     

Solution of Center-Focus Problem for a Class of Cubic Systems

For a class of cubic systems, the authors give a representation of the $n${\rm th} order Liapunov constant through a chain of pseudo-divisions. As an application, the center problem and the isochronous center problem of a particular system are considered. They show that the system has a center at the origin if and only if the first seven Liapunov constants vanish, and cannot have an isochronous center at the origin.     

Sufficient center conditions for some classes of polynomial uniformly isochronous systems

A method is proposed for deriving center conditions for uniformly isochronous systems of a particular form. The method is based on reducing a system to the Abel ordinary differential equation.     

一类五次微分系统的定性分析

对一类五次平面多项式微分系统进行了定性分析.给出原点的中心与等时中心条件及极限环的存在性.研究了此系统无穷远点的性态,该无穷远点是高次奇点,并运用把大角域分为若干小角域的方法对此高次奇点在不定号情形下轨线的分布情况进行讨论.     

中心条件推导的间接方法

对于一类多项式系统,给出两类对称条件的推导算法,具体讨论了一类三次系统的中心条件;对于Poincare型系统,给出一类等时中心的充分条件．     

Center, limit cycles and isochronous center of a Z 4-equivariant quintic system

In this paper, we study the limit cycles bifurcations of four fine focuses in Z4-equivariant vector fields and the problems that its four singular points can be centers and isochronous centers at the same time. By computing the Liapunov constants and periodic constants carefully, we show that for a certain Z4-equivariant quintic systems, there are four fine focuses of five order and five limit cycles can bifurcate from each, we also find conditions of center and isochronous center for this system. The process of proof is algebraic and symbolic by using common computer algebra soft such as Mathematica, the expressions after being simplified in this paper are simple relatively. Moreover, what is worth mentioning is that the result of 20 small limit cycles bifurcating from several fine focuses is good for Z4-equivariant quintic system and the results where multiple singular points become isochronous centers at the same time are less in published references.     

Persistent centers of complex systems

On the basis of some works on persistent centers and weakly persistent centers, in this paper we discuss a generalized version of persistent center and weakly persistent center for complex planar differential systems, in which conjugacy of variables may not be required. We give some complex systems which have a persistent center or weakly persistent center at the origin. Then, we find all conditions of persistent center for cubic systems and all conditions of weakly persistent center for complex cubic Lotka–Volterra system. Relations between complex systems and real ones are given concerning persistent centers and weakly persistent centers.