Separatrix configurations in holomorphic flows

TL;DR

This paper studies boundary orbits in holomorphic flows, classifies their configurations, and explores blow-up scenarios, providing new insights into the structure of separatrices in various flow regions.

Contribution

It offers a detailed classification of boundary orbits and separatrix configurations in holomorphic flows, including new results on blow-up behavior and a counterexample to previous claims.

Findings

Separatrices of basin centers are composed of double-sided separatrices with finite-time blow-up.

Separatrices of node and focus basins blow up in the same time direction as the flow towards the equilibrium.

The boundary of elliptic sectors includes multiple equilibria and countably many double-sided separatrices.

Abstract

We investigate properties of boundary orbits (separatrices) of canonical regions (basins/neighbourhoods of equilibria) in holomorphic flows with real-valued time. We establish the continuity of transit times along these boundary orbits and classify possible path components of the boundary of flow-invariant domains. Thus, we provide central tools for topological and geometric constructions aimed at examining the role of blow-up scenarios in separatrix configurations of basins of simple equilibria and global elliptic sectors: First, we prove that the separatrices of basins of centers is entirely composed of double-sided separatrices with a blow-up in finite positive and finite negative time. Second, we show that the separatrices of node and focus basins (sinks and sources) exhibit a finite-time blow-up in the same time direction in which the orbits within the basin tend towards the equilibrium. Additionally, we propose a counterexample to the claim in Theorem 4.3 (3) in ["The structure of sectors of zeros of entire flows", K. Broughan (2003)], demonstrating that a blow-up does not necessarily have to occur in both time directions. Third, we describe the boundary structure of global elliptic sectors. It consists of the multiple equilibrium, one incoming and one outgoing separatrix attached to it, and at most countably many double-sided separatrices.