The cyclicity of the period annulus of two classes of cubic isochronous systems

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.     

Decomposition of algebraic sets and applications to weak centers of cubic systems

There are many methods such as Gröbner basis, characteristic set and resultant, in computing an algebraic set of a system of multivariate polynomials. The common difficulties come from the complexity of computation, singularity of the corresponding matrices and some unnecessary factors in successive computation. In this paper, we decompose algebraic sets, stratum by stratum, into a union of constructible sets with Sylvester resultants, so as to simplify the procedure of elimination. Applying this decomposition to systems of multivariate polynomials resulted from period constants of reversible cubic differential systems which possess a quadratic isochronous center, we determine the order of weak centers and discuss the bifurcation of critical periods.     

Local bifurcation of critical periods in quadratic-like cubic systems

In this paper, we investigate quadratic-like cubic systems having a center at $O$ for the local bifurcation of critical periods. We provide an inductive algorithm to compute polynomials of periodic coefficients, find structures of solutions for systems of algebraic equations corresponding to weak centers of finite order, and derive conditions on parameters under which the considered equilibrium is a weak center of order $k$, $k=0,1,2,3,4$. Furthermore, we show that with appropriate perturbations, at most four critical periods bifurcate from the weak center of finite order, and we give conditions under which exactly $k$ critical periods bifurcate from the center $O$ for each integer $k=1,2,3,4$.     

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.     

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.     

Linearizability conditions of time-reversible cubic systems

In this paper we present the necessary and sufficient conditions for linearizability of the planar time-reversible cubic complex system , . From these conditions, the necessary and sufficient conditions for the origin to be an isochronous center of the time-reversible cubic real system , can be obtained. Thus, the isochronous center problem of time-reversible cubic systems is solved completely.     

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.     

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.     

Weak centers and local bifurcations of critical periods at infinity for a class of rational systems

We describe an approach to studying the center problem and local bifurcations of critical periods at infinity for a class of differential systems. We then solve the problem and investigate the bifurcations for a class of rational differential systems with a cubic polynomial as its numerator.     

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.     

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．     

Critical periods of perturbations of reversible rigidly isochronous centers

In this paper we discuss bifurcation of critical periods in an m-th degree time-reversible system, which is a perturbation of an n-th degree homogeneous vector field with a rigidly isochronous center at the origin. We present period-bifurcation functions as integrals of analytic functions which depend on perturbation coefficients and reduce the problem of critical periods to finding zeros of a judging function. This procedure gives not only the number of critical periods bifurcating from the period annulus but also the location of these critical periods. Applying our procedure to the case n=m=2 we determine the maximum number of critical periods and their location; to the case n=m=3 we investigate the bifurcation of critical periods up to the first order in ε and obtain the expression of the second period-bifurcation function when the first one vanishes.     

Generalized isochronous centers for complex systems

In this paper, the definition of generalized isochronous center is given in order to study unitedly real isochronous center and linearizability of polynomial differential systems. An algorithm to compute generalized period constants is obtained, which is a good method to find the necessary conditions of generalized isochronous center for any rational resonance ratio. Its two linear recursive formulas are symbolic and easy to realize with computer algebraic system. The function of time-angle difference is introduced to prove the sufficient conditions. As the application, a class of real cubic Kolmogorov system is investigated and the generalized isochronous center conditions of the origin are obtained.     

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.     

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.     

Improved approximations for cubic bipartite and cubic TSP

We show improved approximation guarantees for the traveling salesman problem on cubic bipartite graphs and cubic graphs. For connected cubic bipartite graphs with n nodes, we improve on recent results of Karp and Ravi by giving a “local improvement” algorithm that finds a tour of length at most \(5/4n-2\). For 2-connected cubic graphs, we show that the techniques of Mömke and Svensson can be combined with the techniques of Correa, Larré and Soto, to obtain a tour of length at most \((4/3-1/8754)n\).     

Isochronous Centers and Isochronous Functions

Abstract We study isochronous centers of two classes of planar systems of ordinary differential equations.Forthe first class which is the Linard systems of the form =y-F(x),=-g(x) with a center at the origin, we provethat if g is isochronous(see Definiton 1.1),then the center is isochronous if and only if F≡0.For the secondclass which is the Hamiltonian systems of the form =-g(y),=f(x) with a center at the origin,we prove thatif f or g is isochronous,then the center is isochronous if and only if the other is also isochronous.     

Centers of quasi-homogeneous polynomial differential equations of degree three

We characterize the centers of the quasi-homogeneous planar polynomial differential systems of degree three. Such systems do not admit isochronous centers. At most one limit cycle can bifurcate from the periodic orbits of a center of a cubic homogeneous polynomial system using the averaging theory of first order.     

The number of limit cycles bifurcating from the periodic orbits of an isochronous center

This paper concerns the bifurcation of limit cycles for a quartic system with an isochronous center. By using the averaging theory, it shows that under any small quintic homogeneous perturbations, at most 14 limit cycles birfucate from the period annulus of the considered system.     

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.