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| The Non-Existence of Limit Cycle of Some Quadratic Differential System |
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Citation： |
Wang Mingshu,Lin Yingju.The Non-Existence of Limit Cycle of Some Quadratic Differential System[J].Chinese Annals of Mathematics B,1982,3(6):721~724 |
| Page view： 813
Net amount： 956 |
Authors： |
Wang Mingshu; Lin Yingju |
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| Abstract： |
In this paper, we consider the system
$\frac {dx}{dt}=-y+lx^2+5axy,\frac {dy}{dt}=x(1+ax+3ly)$ (1)
which has a fine focus of -the third order O(0,0) .
We prove the following
Theorem. No limit cycle of system (1) can exist in the neighborhood of O(0,0).
In fact, using a few transformations, we can change system (1) into the Van Der Pol's system
$\frac{dx}{dt}=-y,\frac {dy}{dt}=-g(x)-f(x)y$(2)
where $g(x)=x^-3(x-1)[3x(x+4)+\delta(6x-1)]$
$f(x)=-x^-2(x-1)(x+4),\delta=(5a^2-3l^2)/l^2$
Hence the problem of the existence or non-existence of limit oyole for system (1) around O changes into that for system (2) around (1, 0) .
Again through Филиппов transformation, we arrive at the following system
$\frac {dy}{dz}=\frac {1}{F_1(z)-y},\frac {dy}{dz}=\frac {1}{F_2(z)-y}$
where
$\int_1^x {f(\xi)d \xi=F(x)=F_1(z),(x>1)}$
$\int_1^x {f(\xi)d \xi=F(x)=F_2(z),(0 |
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