Local geometry of Equilibria and a Poincar\'e-Bendixson-type Theorem for Holomorphic Flows

TL;DR

This paper investigates the local geometry of holomorphic dynamical systems in the plane, establishing a Poincaré-Bendixson-type theorem that characterizes the nature of bounded orbits.

Contribution

It introduces new geometric conditions for equilibria of holomorphic flows and proves a Poincaré-Bendixson-type theorem specific to these systems.

Findings

Bounded non-periodic orbits are homoclinic or heteroclinic.

Established geometric criteria for higher-order equilibria.

Constructed a finite elliptic decomposition for holomorphic flows.

Abstract

In this paper, we explore the local geometry of dynamical systems $\dot{x}=F(x)$ with real time parameterization, where $F$ is holomorphic on connected open subsets of $\mathbb{C}\stackrel{\sim}{=}\mathbb{R}^2$. We describe the geometry of first-order equilibria. For equilibria of higher orders, we establish an equivalent condition for "definite directions", allowing us to reverse the implication in Theorem 2 of Chapter 2.10 in [Differential equations and dynamical systems, Lawrence Perko (1990)] under the additional condition of holomorphy. This enables the geometric construction of a finite elliptic decomposition. We derive a holomorphic Poincar\'e-Bendixson-type theorem, leading to the conclusion that bounded non-periodic orbits are always homoclinic or heteroclinic.