On Planar Holomorphic Systems

TL;DR

This paper classifies all global phase portraits of planar holomorphic systems, especially polynomial and Möbius types, on the Poincaré disk, and provides explicit first integrals relevant to fluid dynamics.

Contribution

It offers a comprehensive classification of global phase portraits for holomorphic systems of degrees 2, 3, and 4, and Möbius systems, with explicit first integrals and applications.

Findings

Classified all global phase portraits for polynomial holomorphic systems of degrees 2, 3, and 4.

Classified all global phase portraits of Möbius systems.

Derived explicit first integrals for holomorphic and conjugated systems.

Abstract

Planar holomorphic systems $\dot{x}=u(x,y)$, $\dot{y}=v(x,y)$ are those that $u=\operatorname{Re}(f)$ and $v=\operatorname{Im}(f)$ 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 making a general overview of the theory of holomorphic systems, we classify all the possible global phase portraits, on the Poincar\'{e} 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 $A,B,C,D\in\mathbb{C}, AD-BC\neq0$. Finally, we obtain explicit expressions of first integrals of holomorphic systems and of conjugated holomorphic systems, which have important applications in the study of fluid dynamics.