Resumen de Global Phase Portrait and Local Integrability of Holomorphic Systems

Luiz F.S. Gouveia, Paulo Ricardo da Silva, Gabriel Rondón

• 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 + 3n + 2 parameters, a polynomial holomorphic depends only on 2n+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+B Cz+D , where A, B,C, D ∈ C, AD − BC = 0.