# Characterization of Separatrices in Holomorphic Dynamical Systems

## TL;DR

This paper explores the topological characterization of separatrices in holomorphic dynamical systems, revealing how complex time offers new insights into the bundling of orbits and the structure of invariant manifolds.

## Contribution

It introduces numerical methods to approximate slow invariant manifolds in holomorphic flows and demonstrates how separatrices can be topologically characterized.

## Key findings

- Separatrices can be characterized topologically in holomorphic systems.
- Complex time provides new insights into the structure of holomorphic dynamical systems.
- Numerical methods effectively approximate invariant manifolds in these systems.

## Abstract

Multiple time scales in dynamical systems lead to a bundling of trajectories onto slow invariant manifolds (SIMs). Although they are absent in two-dimensional holomorphic dynamical systems, a bundling of orbits is often observed as well. They bundle onto special trajectories called separatrices. We apply numerical methods for the approximation of SIMs to holomorphic flows and show how a separatrix between two regions of periodic orbits can be characterized topologically. Complex time reveals a new perspective on holomorphic dynamical systems.