For a general autonomous planar polynomial differential system, it is difficult to find conditions that are easy to verify and which guarantee global asymptotic stability, weakening the Markus–Yamabe condition. In this paper, we provide three conditions that guarantee the global asymptotic stability for polynomial differential systems of the form $x^{\prime}=f_1(x,y)$, $y^{\prime}=f_2(x,y)$, where f1 has degree one, f2 has degree $n\ge 1$ and has degree one in the variable y. As a consequence, we provide sufficient conditions, weaker than the Markus–Yamabe conditions that guarantee the global asymptotic stability for any generalized Liénard polynomial differential system of the form $x^{\prime}=y$, $y^{\prime}=g_1(x) +y g_2(x)$ with g1 and g2 polynomials of degrees n and m, respectively.

对于一般的自治平面多项式微分系统，很难找到易于验证且保证全局渐近稳定的条件，削弱了Markus-Yamabe条件。在本文中，我们提供了三个条件来保证形式为 $x^{\prime}=f_1(x,y)$< 的多项式微分系统的全局渐近稳定性。 a i=2>, $y^{\prime}=f_2(x,y)$，其中 f 1 具有一级，f2 的度数为 $n\ge 1$ 并且变量的度数为 1 y< i=14>.因此，我们提供了比 Markus-Yamabe 条件更弱的充分条件，保证了任何广义 Liénard 多项式微分系统的全局渐近稳定性，其形式为 $x^{\prime}=y $, $y^{\prime}=g_1(x) +y g_2(x)$ 与 < /span>2< /span> ，分别。m 和 n 次数多项式 g 和 1g