2022/01/11 by L. F. S. Gouveia, Gouveia, L. F. S., G. Rondón +3
Mathematics · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2201.04159
openalex publication_date 2022/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 making a general overview of the theory of holomorphic systems, 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. 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.