Globally Asymptotically Stable Equilibrium Points in Kukles Systems - Dias, Fabio Scalco; Mello, Luis Fernando (Qualitative theory of dynamical systems, 2020)
The problem of determining the basin of attraction of equilibrium points is of great importance for applications of stability theory. In this article, we address the global asymptotic stability problem of an equilibrium point of an ordinary differential equation on the plane. More precisely, we study equilibrium points of Kukles systems from the global asymptotic stability point of view. First of all, we classify the Kukles systems satisfying the assumptions: the origin is the unique equilibrium point which is locally asymptotically stable, and the divergence is negative except possibly at the origin. Then, for each of such Kukles system, we prove that the origin is globally asymptotically stable. Poincaré compactification is used to study the systems on the complements of compact sets.