On meromorphic complex differential equations

Abstract

In this paper, we study first-order, autonomous, complex differential equations of the form i = f (z), where f (z) is the meromorphic function of the complex variable z, defined in a simply connected domain on the Riemann sphere. We concentrate on the phase portraits of such systems, with particular attention being paid to the existence and properties of closed orbits, and orbits which reach the point at infinity (or blow-up) in finite time. Applications of the general theoy are given, as well as discussion of higher dimensional systems