Asymptotic behaviour of the third Painlevé transcendents in the space of initial values

Abstract

We study the asymptotic behavior of solutions of the fourth Painleve equation as the independent variable goes to infinity in its space of (complex) initial values, which is a generalization of phase space described by Okamoto. We show that the limit set of each solution is compact and connected and, moreover, that any solution that is not rational has an infinite number of poles and infinite number of zeros.