On Planar Holomorphic Systems

• Published in: arXiv.org 2022-01
• Main Authors: Gouveia, L F S, Rondón, G, da Silva, P R
• Format: Article
• Language: English
• Subjects: Analytic functions; Classification; Fluid dynamics; Mathematical analysis; Parameters; Polynomials
• ISSN: 2331-8422

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.