We present a simpler proof of the existence of an exact number of one or morelimit cycles to the Lienard system $\dot{x}=y-F(x) $, $\dot {y}=-g(xt)$, underweaker conditions on the odd functions $F(x) $ and $g(x) $ as compared to thoseavailable in literature. We also give improved estimates of amplitudes of thelimit cycle of the Van Der Pol equation for various values of the nonlinearityparameter. Moreover, the amplitude is shown to be independent of the asymptoticnature of $F$ as $|x| \to\infty$.
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机译：我们给出了较弱条件下Lienard系统$ \ dot {x} = yF（x）$，$ \ dot {y} =-g（xt）$的较弱条件下存在一个或多个极限环的确切数目的更简单证明与文献中可用的奇数函数$ F（x）$和$ g（x）$相比。对于非线性参数的各种值，我们还给出了Van Der Pol方程极限环振幅的改进估计。此外，振幅显示为与$ F $的渐近性质无关，如$ | x |。 \ to \ infty $。
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