A Weakened Markus–Yamabe Condition for Planar Polynomial Differential Systems of Degree

Abstract

Abstract 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 f 1 has degree one, f 2 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 g 1 and g 2 polynomials of degrees n and m , respectively.