Abstract:

We are interested in the study (local and global) of the complex differential equation $dz/dt\; =\; f(z)$, where f(z) is an analytic o meromorphic map. In the local approach we study the phase portrait of $dz/dt\; =\; f(z)$ near an analytic point or a pole or a essential singularity obtaining, in the first two cases, the corresponding normal forms. In the global approach we recall some known properties (non existence of limit cycles) of these systems. Finally, we use both parts to classify all the systems of the form $dz/dt\; =\; P(z)/Q(z)$, where $P(z)$ and $Q(z)$ are polynomial with degree at most 2.