Global Phase Portrait and Local Integrability of Holomorphic Systems

Planar holomorphic systems x˙ = u(x, y) , y˙ = v(x, y) are those that u= Re (f) and v= Im (f) for some holomorphic function f(z). They have important dynamical properties, highlighting, for example, the fact that they do not have limit cycles and that center-focus problem is trivial. In particular, the hypothesis that a polynomial system is holomorphic reduces the number of parameters of the system. Although a polynomial system of degree n depends on n2+ 3 n+ 2 parameters, a polynomial holomorphic depends only on 2 n+ 2 parameters. In this work, in addition to prove that holomorphic systems are locally integrable, we classify all the possible global phase portraits, on the Poincaré disk, of systems z˙ = f(z) and z˙ = 1 / f(z) , where f(z) is a polynomial of degree 2, 3 and 4 in the variable z∈ C. We also classify all the possible global phase portraits of Moebius systems z˙=Az+BCz+D, where A, B, C, D∈ C, AD- BC≠ 0.