Trajectories escaping to infinity in finite time

TL;DR

This paper investigates the behavior of solutions to certain complex differential equations, showing that under specific conditions, infinitely many trajectories escape to infinity in finite time.

Contribution

It establishes new conditions under which solutions to complex differential equations have infinitely many trajectories escaping to infinity in finite time.

Findings

Infinitely many trajectories tend to infinity in finite time under specified conditions.

Conditions involve transcendental meromorphic functions with finitely many poles or logarithmic singularities.

Results extend understanding of trajectory behavior in complex differential equations.

Abstract

If the function $f$ is transcendental and meromorphic in the plane, and either $f$ has finitely many poles or its inverse function has a logarithmic singularity over infinity, then the equation $\dot z = f(z)$ has infinitely many trajectories tending to infinity in finite increasing time