LIMIT CYCLES FOR A CLASS OF DIFFERENTIAL SYSTEM WITH POSITIVE DEFINITE POLYNOMIAL

In this paper we consider a class of differential systems with positive definite polynomial having exactly one and two limit cycles.Such a system is more extensive than paper[1,2].     

二次系统极限环的相对位置与个数

<正> 中的P_2(x,y)与Q_2(x,y)为x,y的二次多项式.文[1].曾指出,系统(1)最多有三个指标为+1的奇点,且极限环只可能在两个指标为+1的奇点附近同时出现.如果方程(1)的极限环只可能分布在一个奇点外围,我们就说此系统的极限环是集中分布的.本文主要研究具非粗焦点的方程(1)的极限环的集中分布问题,和极限环的最多个数问题.文[2]-[5]曾证明,当方程(1)有非粗焦点与直线解或有两个非粗焦点或有非粗焦点与具特征根模相等的鞍点时。方程(1)无极限环.本文给出方程(1)具非粗焦点时,极限环集     

The limit cycles of a second-order system of differential equations: The method of small forms

In this paper we investigate the existence of limit cycles of a system of the second-order differential equations with a vector parameter.We propose a method for representing a solution as a sum of forms with respect to the initial value and the parameter; we call this technique the method of small forms. We establish the conditions under which a sufficiently small neighborhood of the equilibrium point contains no limit cycles. We construct a polynomial, whose positive roots of odd multiplicity define the lower bound for the number of cycles, and simple positive roots (other positive roots do not exist) define the number of limit cycles in a sufficiently small neighborhood of the equilibrium point.We prove theorems, whose conditions guarantee that a positive root of odd multiplicity defines a unique limit cycle, but a positive root of even multiplicity defines exactly two limit cycles.We propose a method for defining the type of the stability of limit cycles.     

(En)系统极限环的相对位置

本文证明了，对任意正整数K，存在平面n次系统，它具有一串不少于K个大极限环．这些大极限环两两之间各有若干小极限环．     

一类三次Kolmogorov系统的极限环分支

本文研究了一类三次Kolmogorov系统，得出了该系统可分支出三个极限环，且其中有两个是稳定的，同时给出了其中心条件.     

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。     

Some Conditions for the Nonexistence of Limit Cycles in a Predator-Prey System

In this paper, we study the problem of the existence of limit cycles for a predator-prey system with a functional response. It is assumed that the functional response is positive, increasing, concave down, and its third derivative has a unique root. A necessary condition for the nonexistence of limit cycles is presented. Some conditions are given under which the necessary condition is also the sufficient condition for the nonexistence of limit cycles.     

THE PROBLEM OF LIMIT CYCLES FOR A LIENARD TYPE CUBIC SYSTEM

The purpose of this paper is to study a general Lienard type cubic system with one antisaddle and two saddles. We give some results of the existence and uniqueness of limit cycles as well as the evolution of limit cycles around the antisaddle for system (2) in the following when parameter a1 changes.     

QUALITATIVE ANALYSIS OF A MULTIMOLECULEBIOCHEMICAL SYSTEM

ＱＵＡＬＩＴＡＴＩＶＥＡＮＡＬＹＳＩＳＯＦＡＭＵＬＴＩＭＯＬＥＣＵＬＥＢＩＯＣＨＥＭＩＣＡＬＳＹＳＴＥＭＧｕＤａｏｘｉｕ（ＨｕｎａｎＥｄｕｃａｔｉｏｎａｌｉｎｓｔｉｔｕｔｅ）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐａｐｅｒｔｈｅａｕｔｈｏｒｐｒｏｖｅｓｓｏｍ...     

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     

一类极限环唯一性的充分条件

包围多个奇点的极限环的唯一性给出一组简洁的充分条件,并将它应用于几类非线性振动方程及多项式微分系统.最后通过系统(1)我们指出证明极限环唯一性中的几种常用方法之间的内在联系,并指出对形如(1)的系统,作 Dulac 函数的一般规律.假设(1)中函数对一切变元连续且满足初值解的唯一性条件.若涉及它们的导数或     

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Z_2对称五次微分系统的中心条件和极限环分支

本文研究了一类Z2对称五次微分系统的中心条件和小振幅极限环分支.通过前6阶焦点量的计算,获得了原点为中心的充要条件,并证明系统从原点分支出的小振幅极限环的个数至多为6.最后通过构造后继函数,给出系统具有6个围绕原点的小振幅极限环的实例.     

ANALYSIS OF LIMIT CYCLES TO A PERTURBED INTEGRABLE NON-HAMILTONIAN SYSTEM

Bifurcation of limit cycles to a perturbed integrable non-Hamiltonian system is investigated using both qualitative analysis and numerical exploration.The investigation is based on detection functions which are particularly effective for the perturbed integrable non-Hamiltonian system.The study reveals that the system has 3 limit cycles.By the method of numerical simulation,the distributed orderliness of the 3 limitcycles is observed,and their nicety places are determined.The study also indicates that each ...     

THE LIMIT CYCLES OF A KIND OF EXPLOITED PREDATOR-PREY SYSTEM

In this paper, a kind of exploited predator-prey system is studied. By using the qualitative theory, we obtain some sufficient conditions for the existence and nonexistence of limit cycles of the system.     

The planar discontinuous piecewise linear refracting systems have at most one limit cycle

In this paper we investigate the limit cycles of planar piecewise linear differential systems with two zones separated by a straight line. It is well known that when these systems are continuous they can exhibit at most one limit cycle, while when they are discontinuous the question about maximum number of limit cycles that they can exhibit is still open. For these last systems there are examples exhibiting three limit cycles.The aim of this paper is to study the number of limit cycles for a special kind of planar discontinuous piecewise linear differential systems with two zones separated by a straight line which are known as refracting systems. First we obtain the existence and uniqueness of limit cycles for refracting systems of focus-node type. Second we prove that refracting systems of focus–focus type have at most one limit cycle, thus we give a positive answer to a conjecture on the uniqueness of limit cycle stated by Freire, Ponce and Torres in Freire et al. (2013). These two results complete the proof that any refracting system has at most one limit cycle.     

二次微分系统极限环的分布

在假设文中命题Ａ成立的条件下证明了一般二次微分系统的极限环所有可能的分布为（３,１）,（１,３）,（３,０）,（０,３）,（２,１）,（１,２）,（２,０）,（１,１）,（０,２）,（１,０）,（０,１）和（０,０）。     

一类7次微分自治系统的极限环分支

研究一类平面7次微分系统,通过作两个适当的变换以及焦点量的仔细计算,得出了系统的无穷远点与2个初等焦点能够同时成为广义细焦点的条件,进一步得出在一定条件下该系统能够分支出15个极限环的结论,其中5个大振幅极限环来自无穷远点,10个小振幅极限环来自2个初等焦点.     

BIFURCATION THEORY OF QUADRATIC DIFFERENTIAL SYSTEMS

This paper deals with the number of limit cycles and bifurcation problem of quadratic differential systems. Under conditions $a<0,b+2l>0,l+1<0$, the author draws three bifurcation diagrams of the system (1.18) below in the (\delta,m) plane, which show that the maximum number of limit cycles around a focus is two in this case.