# Asymptotic behaviour of the fifth Painlev\'e transcendents in the space of initial values

## TL;DR

This paper investigates the asymptotic behavior of fifth Painlevé transcendents near zero and infinity, revealing properties of their limit sets, poles, and zeros, and how solutions behave with essential singularities.

## Contribution

It provides a detailed analysis of the asymptotic properties and singularity structure of fifth Painlevé transcendents in the space of initial values.

## Key findings

- Limit sets are compact and connected.
- Solutions with essential singularity at zero have infinitely many poles and zeros.
- Solutions with essential singularity at infinity have infinitely many poles and take the value 1 infinitely often.

## Abstract

We study the asymptotic behaviour of the solutions of the fifth Painlev\'e equation as the independent variable approaches zero and infinity in the space of initial values. We show that the limit set of each solution is compact and connected and, moreover, that any solution with the essential singularity at zero has an infinite number of poles and zeroes, and any solution with the essential singularity at infinity has infinite number of poles and takes value $1$ infinitely many times.