一类三次系统极限环的惟一性

讨论三次系统x=x(A0+A1x+A2y+A3xy-A4y2)y=y(x-1)的极限环问题.得到了该系统不存在极限环和存在惟一极限环的条件.     

一类三次系统极限环的唯一性和唯二性

讨论了一类三次系统x=-y(1-βx2)-(a1x a2x2 a3x3),y=b1x b2x2 b3x3的极限环问题.对包含一个奇点或多个奇点的极限环的唯一性和唯二性给出了若干充分条件.     

微分方程=φ(y)-F(x),=-g(x)的极限环的存在唯一性和唯二性

本文§1讨论方程组 (?)=(?)(y)-F(x),(?)=-g(x)极限环的存在性,推广了作者的结果和方法. §2建立了各种类型的极限环存在唯一性定理.包括(E)的一切轨线是否绕原点打转,积分integral from 0 to ±∞(g(x)dx)和integral from 0 to ±∞(F′(x)dx)是否发散,奇点为一个及两个等情况;包括(E)的一切异于零的轨线当t→+∞时都趋于此唯一的极限环,以及可用以确定极限环的位置     

Liénard型系统解的有界性与极限环的存在性

本文讨论比Lienard系统更广的一类非线性系统 x=h(y)-F(x) y=-g(x)解的有界性与极限环的存在性,得到了解正向有界的充要条件和存在极限环的充分条件,推广和改进了文[1-5]的工作.     

一类三次系统的极限环

本文讨论了一类三次系统 x=-y(1-αx~2)+δx-ιx~3,y=x(1-βx~2)和 x=-y(1-ax)(1-bx)+δx-ιx~3,y=x(1-cx)(1-bx)的极限环问题。     

非线性方程极限环存在性的另一类情况

本文讨论了非线性方程(dx)/(dt)=h(y)-F(x),(dy)/(dt)=-g(x)的极限环存在性的另一类情况. 即去掉了G(±∞)的假设,该方程的极限环的存在性问题.     

一类E13系统极限环的惟一性

研究一类E13系统x=y,y=-x+δy+nx2+mxy+ly2+bxy2,求出奇点O的焦点量W0=δ,W1=m(n+l),W2=-mnb.证明了W0=W1=W2=0时O为中心.其次证明了W0=0,W1W2≥0时系统无极限环;W0=0,W1W2＜0时系统至多有一个极限环.最后讨论了n=0,b＞0的情况.证明了存在δ0,0＜δ0≤-l/m,当0＜δ＜δ0时系统存在惟一极限环,δ=δ0时系统存在无穷远分界线环,δ≤0或δ＞δ0时系统无闭轨与奇闭轨.     

一类非线性微分系统极限环的存在性

研究了非线性微分系统 (dx)/(dt)=p(y),(dy)/(dt)=-q(y)f(x)-g(x)极限环的存在性,获得了该系统包围多个奇点的极限环存在的两个充分条件,所获结果改进和推广了文[1,2,3]中的相应结果,并且指出了文[2,3,4,5]中的疏漏.     

Periodic solutions of a class of superquadratic second order Hamiltonian systems

§1　IntroductionInthispaperwediscusstheexistenceofthesolutionforthefollowingsecondorderHamiltoniansystemx¨ Ax ΔF(x)=0,(1.1)whereAisann×nrealsymmetricmatrixandisnon-definite,F∈C1(Rn,R),andΔF(x)denotesthegradientofF.WhileworksforsecondorderHamiltonsystemshavemostlybeendoneundertheconditionA=0,westudythecasewhereA≠0andisnon-definiteinthepapers[1,2].DefineH=H1,2T([0,T],Rn)={x:R→Rn|xisabsolutelycontinuous,x∈L2([0,T],Rn),x(0)=x(T),x(0)=x(T)}and〈x,y〉=∫T0[(x(t),y(t)) (x…     

广义Liénard方程零解的全局渐近稳定性和极限环的存在性

研究了一类广义Liénard方程x。=(y),y。=-f(x)(y)-g(x),式中,F,g:R→R连续且保证系统初值解惟一,给出零解全局渐近稳定性条件,并讨论极限环的存在性.     

Cyclical strategies in two-dimensional optimal control models: Necessary conditions and existence

This paper derives necessary conditions such that cyclical policies may be optimal in concave, two state variable (economic) control problems. These conditions identify four different routes. One major implication is that two of these four conditions may be met by separable models. This possibility has been overlooked so far. Therefore, even separable and structurally very simple models may be characterized by optimal cyclical policies. Indeed, it will be shown that stable limit cycles exist for concave and separable control problems.     

具有一类三次曲线解的Kolmogorov三次系统的极限环的存在性

证明了具有三次曲线解y=-x(x-1)2 4/24的Kolmogorov三次系统是有存在极限环可能的.     

一类平面三次微分系统极限环的存在性与唯一性

蛤出三次系统{dx/dt=-y(ax^2 bx 1) δx-lx^2 dy/dt=x(ax^2 bx 1)极限环的不存在性，存在性及唯一性的一些充分条件．     

关于Filippov变换的一个注记

本文利用 Filippov变换导出来一个研究 Liénard方程极限环不存在性的判据 ,它为证明某些Liénard方程不存在极限环提供了一个简捷有效的思想方法 .     

Counterexample to a Conjecture on the Algebraic Limit Cycles of Polynomial Vector Fields

In Geometriae Dedicata 79 (2000), 101–108, Rudolf Winkel conjectured: for a given algebraic curve f=0 of degree m 4 there is in general no polynomial vector field of degree less than 2m -1 leaving invariant f=0 and having exactly the ovals of f=0 as limit cycles. Here we show that this conjecture is not true. Mathematics Subject Classifications (2000). Primary 14P25, 34C05, 34A34.     

THE MAXIMAL NUMBER OF LIMIT CYCLES FOR QUADRATIC DIFFERENTIAL SYSTEM  = -y + lx~2 + xy,  = x(l + ax + by)

In this paper, we investigate the maximal number of limit cycles surrounding a first order weak focus for the quadratic differential system. And proved such a system has at most two limit cycles under some certain conditions.     

SAME DISTRIBUTION OF LIMIT CYCLES TO PERTURBED CUBIC HAMILTONIAN SYSTEMS WITH DIFFERENT PERTURBATIONS

Using qualitative analysis, we study perturbed Hamiltonian systems with different n-th order polynomial as perturbation terms. By numerical simulation, we show that these perturbed systems have the same distribution of limit cycles. Our results imply that these perturbed systems are equivalent in the sense of distribution of limit cycles. This is useful for studying limit cycles of perturbed systems.     

On the Detection of Limit Cycles by the Variational Velocity Method

This paper summarizes, clarifies, and corrects some aspects of the variational velocity methodfor the detection of limit cycles. After definitions and statements of the most important theoremsassociated with this method, some aspects of the proof of the main theorem are corrected andreworked. An example from the original paper in Acta Appl. Math. 48 (1997),13–32, is then discussed and criticized. Finally, the limitations of this method are discussed,especially as it applies to systems involving multiple limit cycles (and therefore as it applies toHilberts XVIth Problem).