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

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.