Global Phase Portrait and Local Integrability of Holomorphic Systems

Abstract

Planar holomorphic systems ${\dot{x}}=u(x,y)$, ${\dot{y}}=v(x,y)$ are those that $u={\text {Re}}(f)$ and $v={\text {Im}}(f)$ 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 $n^2 +3n+2$ 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 ${\dot{z}}=f(z)$ and ${\dot{z}}=1/f(z)$, where f(z) is a polynomial of degree 2, 3 and 4 in the variable $z\in {\mathbb {C}}$. We also classify all the possible global phase portraits of Moebius systems ${\dot{z}}=\frac{Az+B}{Cz+D}$, where $A,B,C,D\in {\mathbb {C}}, AD-BC\ne 0$.