Plane autonomous systems with rational vector fields

Abstract

The differential equation z ˙ = R ( z ) \dot z= R(z) is studied, where R R is an arbitrary rational function. It is shown that the Riemann sphere is decomposed into finitely many open sets, on each of which the flow is analytic and, in each time direction, there is common long-term behavior. The boundaries of the open sets consist of those points for which the flow fails to be analytic in at least one time direction. The main idea is to express the differential equation as a continuous Newton method z ˙ = − f ( z ) / f ′ ( z ) \dot z = - f(z)/f’\;(z) , where f f is an analytic function which can have branch points and essential singularities. A method is also given for the computer generation of phase plane portraits which shows the correct time parametrization and which is noniterative, thereby avoiding the problems associated with the iteration of rational functions.