一类异宿环分支极限环的唯一性

本文研究平面上一类两点或三点异宿环附近极限环的分支,在一简洁条件下证明了异宿环分支极限环的唯一性,并给出了极限环唯一存在的充要条件.作为对三维余维2分支的应用,解决了所出现的两点异宿环产生唯一极限环的问题.     

三点粗异宿环分支

本文研究高维系统连接三个鞍点的粗异宿环的分支问题.在一些横截性条件和非扭曲条件下,获得了Γ附近的1-异宿三点环, 1-异宿两点环、 1-同宿环和1-周期轨的存在性,唯一性和不共存性.同时给出了分支曲面和存在域.上述结果被进一步推广到连接l个鞍点的异宿环的情况,其中l≥2.     

奇闭轨分支出极限环的唯一性(Ⅲ)

本文是文[1—2]的继续,讨论孤立或非孤立的异宿环在小扰动下分支出唯一极限环的条件.     

一类3维反转系统中的异维环分支

研究了一类3维反转系统中包含2个鞍点的对称异维环分支问题, 且仅限于研究系统的线性对合R的不变集维数为1的情形. 给出了R-对称异宿环与R-对称周期轨线存在和共存的条件, 同时也得到了R-对称的重周期轨线存在性. 其 次, 给出了异宿环、 同宿轨线、 重同宿轨线和单参数族周期轨线的存在性、 唯一性和共存性等结论, 并且发现不可数无穷条周期轨线聚集在某一同宿轨线的小邻域内. 最后给出了相应的分支图.     

同宿、异宿环分支出极限环的唯一性

本文在一简洁条件下证明任一含细鞍点的孤立同宿环及对称的双同宿环和两点异宿环至多产生一个极限环。     

非扭曲异宿环分支

考虑高维系统非扭曲细异宿环分支 ,给出了 1 同宿轨道和 1 周期轨道的存在性和存在域 ,并得到了 2 重周期轨道的分支曲面 .最后 ,这些分支结果被应用于平面系统细异宿环 ,获得了新的有趣的结论     

一类平面微分系统的极限环

研究一类平面微分系统的极限环,利用Hopf分支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ЧеркасЛА和ЖилевычЛИ的唯一性定理得到了极限环唯一性的若干充分条件.     

非扭曲异宿环分支

考虑高维系统非扭曲细异宿环分支,给出了1-同宿轨道和1-周期轨道的存在性和存在域,并得到了2-重周期轨道的分支曲面.最后,这些分支结果被应用于平面系统细异宿环,获得了新的有趣的结论.     

具有轨道翻转和倾斜翻转的退化异维环分支

本文研究4 维系统中一类具有轨道翻转和倾斜翻转的退化异维环分支问题. 通过在未扰异维环的小管状邻域内建立局部活动坐标系, 本文建立Poincaré 映射, 确定分支方程. 由对分支方程的分析,本文讨论在小扰动下, 异宿环、同宿环和周期轨的存在性、不存在性和共存性, 且给出它们的分支曲面以及共存区域, 推广了已有结果.     

一类平面微分系统极限环的存在唯一性

研究一类平面微分系统的极限环,利用Hopf分支理论得到了该系统极限环存在性的若干充分条件,利用Л.А.Чеpкас和Л..Иилевьтч的唯一性定理得到了极限环唯一性与稳定性的若干充分条件.     

关于一类平面n+2次系统极限环唯一性的一个注记

本文继续完善文[1]和[2]的工作,利用广义Lienard方程和张芷芬唯一性定 理证明了,当n≥3时一类n+2次生化反应系统极限环的唯一性.至此,该系统极 限环唯一性问题得到完整解决.     

Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response

The goal of this paper is to establish the uniqueness of limit cycles of the predator-prey systems with Beddington-DeAngelis functional response. Through a change of variables, the predator-prey system can be transformed into a better studied Gause-type predator-prey system. As a result, the uniqueness of limit cycles can be solved.     

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.     

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.     

一类Leslie模型的定性分析

对一类Leslie模型进行定性分析，研究了其极限环的存在性，不存在性和唯一性．证明了该系统在细焦点外围至多有一个极限环，以及如果系统有奇数个极限环，则它恰有一个极限环．     

Cubic Lienard equation with quadradic damping (I)

1. IntroductionLienard equationdZx dx~ f(.)g g(x) = 0 (l.0)dtZ dthas been extensively studied with particular emphasis on the ekistence and uniqueness oflimit cycles (see e.g. [l--4] and references there in). The number of limit cycles of (l.0) hasbeen also investigated by several authors (see e.g. [5--8]).In the present paper we study the general cubic Lienard equation, namelydx da~ = y ~ F(x), Z ~ ~g(x) (1.1)dt' dtwhereF(x) = ale a,x: a,x', (l.2)g(x) = blx b,x' b,x'. (1.3)Clea…     

一类平面多项式微分系统极限环的存在唯一性

本文研究一类平面多项式微分系统的极限环，得到了系统极限环存在、唯一的充分条件。     

Analysis and computation of limit cycles for systems with separable nonlinearities

The variational system obtained by linearizing a dynamical system along a limit cycle is always non-invertible. This follows because the limit cycle is only a unique modulo time translation. It is shown that questions such as uniqueness, robustness, and computation of limit cycles can be addressed using a right inverse of the variational system. Small gain arguments are used in the analysis.     

Analysis and computation of limit cycles for systems with separable nonlinearities

The variational system obtained by linearizing a dynamical system along a limit cycle is always non-invertible. This follows because the limit cycle is only a unique modulo time translation. It is shown that questions such as uniqueness, robustness, and computation of limit cycles can be addressed using a right inverse of the variational system. Small gain arguments are used in the analysis.