On the global attractivity and asymptotic stability for autonomous systems of differential equations on the plane

Abstract:

The autonomous system of differential equations \[ x’=f(x),\qquad (x=(x_1,x_2)^T\in \mathbb {R}^2,\ f(x)=(f_1(x),f_2(x))^T),\] is considered, and sufficient conditions are given for the global attractivity of the unique equilibrium $x=0$. This property means that all solutions tend to the origin as $t\to \infty$. The two cases (a) $\operatorname {div} f(x)\le 0$ ($x\in \mathbb {R}^2$) and (b) $\operatorname {div} f(x)\ge 0$ ($x\in \mathbb {R}^2$) are treated, where $\operatorname {div} f(x):=\partial f_1(x)/\partial x_1+\partial f_2(x)/\partial x_2$. Earlier results of N. N. Krasovskiĭ and C. Olech about case (a) are improved and generalized to case (b). Three types of assumptions are required: certain stability properties of the origin (local attractivity, stability), boundedness above in some sense for $\operatorname {div} f(x)$, and assumptions that $|f(x)|$ is not as small as $|x|\to \infty$. The conditions of the second and third types are connected with each other.