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arXiv · 2410.04895

Basins of Equilibria and geometry of Global Sectors in Holomorphic Flows

Nicolas Kainz·Dirk Lebiedz
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Abstract

In this follow-up paper, we investigate the global geometry and topology of dynamical systems $\dot{x} = F(x)$ with entire vector field $F$, building on and constructively extending the local structure of simple and higher-order equilibria. We provide a step-by-step analysis to reveal 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 concept of sector-forming orbits based on the geometry within a finite elliptic decomposition of multiple equilibria. Finally, we characterize the structure of heteroclinic regions connecting two equilibria.

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Nicolas Kainz, Dirk Lebiedz. 2025-09-04. Basins of Equilibria and geometry of Global Sectors in Holomorphic Flows. https://arxiv.org/abs/2410.04895

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