Global behavior of solutions of nonlinear ODEs in $\CC$: first order equations

TL;DR

This paper demonstrates that solutions to first order nonlinear ODEs in the complex plane can be globally controlled using constants of motion, allowing for analysis of their behavior and singularities.

Contribution

It introduces a method to control and analyze solutions of nonlinear ODEs in the complex domain via finite constants of motion.

Findings

Solutions can be controlled globally in the complex plane.

Constants of motion determine solution behavior and singularity locations.

Quantitative analysis of solutions far from the origin is possible.

Abstract

We show that the solutions of first order nonlinear ODEs can be controlled globally in the complex domain, using a finite set of constants of motion defined in regions of $\CC$. These constants of motion enable us to obtain quantitative behaviors of the solutions far away from the origin, as well as to determine the position of singularities of the solution.