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 real ellipse 实椭圆(0)
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 To be and to be real 实在与真实 短句来源 The Real Courage 真正的勇气 短句来源 On Philosophy of Ellipse 论大平正芳的“椭圆哲学” 短句来源 ON THE ELLIPSE AREA THEORY 论椭圆区理论——山地居住区公建服务范围的研究 短句来源 The Analysis for Real-Time Performance and Realization of CNC System on Middle-Convex and Varying Ellipse Piston Machining 中凸变椭圆活塞车削数控系统的实时性分析及实现 短句来源

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 Consider the real differential system dx/dt=P_3(x,y), dy/dt=Q_3(x,y)where P_3(x,y) and Q_3(x,y) are polynomials of the 3-rd degree. Suppose that the system has at lease one elementary critical point of index+1, and it can be written in the form dx/dt=-y+xF_1(x,y)+gy~2+hy~2 (1) dy/dt-x+yF_2(x,y)+rx~2+sx~3 where F_1(x,y)and F_2(x,y) are polynomials of the 2-nd degree. If g=h=r=s=0 and F_1(x,y)≡F_2(x,y), then systen(1)takes the form dx/dt=-y+xF(x,y),dy/dt=x+yF(x,y) (2)for which we have the following: Theorem. If... Consider the real differential system dx/dt=P_3(x,y), dy/dt=Q_3(x,y)where P_3(x,y) and Q_3(x,y) are polynomials of the 3-rd degree. Suppose that the system has at lease one elementary critical point of index+1, and it can be written in the form dx/dt=-y+xF_1(x,y)+gy~2+hy~2 (1) dy/dt-x+yF_2(x,y)+rx~2+sx~3 where F_1(x,y)and F_2(x,y) are polynomials of the 2-nd degree. If g=h=r=s=0 and F_1(x,y)≡F_2(x,y), then systen(1)takes the form dx/dt=-y+xF(x,y),dy/dt=x+yF(x,y) (2)for which we have the following: Theorem. If F(x,y)=0 is a real ellipse, which contains the origin in tis interior, then the system(2)has a unique limit-cycle, which is stable when F(0,0)>0, and unstable when F(0,0)<0; if F(x,y)=0 is an imaginary ellipse or a point, then the system (2) has no periodic solution. This Is a generalization of two theorems in [2], Part Ⅳ, chapter 3, §1. If g=h=r=s=0 and F_2(x,y)≡0, then system(1)takes the form dx/dt=-y+xF(x,y),dx/dt=x(3) for which we have the foolowing: Theorem. If F(x,y)=0 is a real ellipse which contains the origin in its interior, then the system (3) has a stable limit-cycle when F(0,0)>0, and an unstable limit-cycle when F(0,0)<0. If the distance between the center of the ellipse and the origin is not greater than the distance between the ellipse and origin, then the limit-cycle is unique. Besides, we have also inverstaged the system (1) and some special cases of system (1) in §1, the system (3) as well as the more general system dx/dt=-y+xF(x,y)+gy~2+hy~3, dy/dt=xin §2, and established certain sufficient conditions for the existance of limit-cycles or non-existance of period solutions. 討論题目中P_3(x,y)与Q_3(y,x)是具实系数的三次多項式时的方程組的週期解与极限环問题。設方程組至少有一指标为+1的初等奇点且可写成下列形式 dx/dt=-y+xF_1(x,y)+gy~2+hy~3,dy/dt=x+yF_2(x,y)+rx~2+sx~3其中,F_1(x,y)与F_2(x,y)均为二次多項式。当g=h=r=s=0时,在F_1(x,y)≡F_2(x,y)或F_2(x,y)≡0的假設下,就上列方程組建立了极限环的存在唯一性定理。此外,对上列方程組本身以及其他一些特殊情况分別給出了存在极限环或不存在週期解的充分性条件。
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