Basins of Equilibria and geometry of Global Sectors in Holomorphic Flows

In this follow-up paper to [Local geometry of Equilibria and a Poincare-Bendixson-type Theorem for Holomorphic Flows, Nicolas Kainz, Dirk Lebiedz (2024)] we investigate the global topology and geometry of dynamical systems $\dot{x} = F(x)$ with entire vector field $F$ by exploiting and extending the local structure of finite elliptic decompositions. We prove topological properties of the basins of centers, nodes, and foci, while excluding isolated equilibria at the boundaries of the latter two. We propose a definition of global elliptic sectors and introduce the notion of a sector-forming orbit allowing global topological analysis. The structure of heteroclinic regions between two equilibria is characterized.