Local and global phase portrait of equation \(\dot z=f(z)\) (Q2372729)

The authors study the complex first order differential equation \[ \dot{z}(t)=f(z);\quad z\in\mathbb{C},\;t\in\mathbb{R} \] where \(f\) is an analytic function in \(\mathbb{C}\) except, possibly, in a punctured neighborhood of the origin \(z=0\) (isolated singularities). The phase portrait of the local normal form in the vicinity of the singular origin \(f(0)=0\) -- or a singularity given by a pole- is determined and global properties of the solutions are given. The homotopy method using Lie symmetries is employed. As a consequence, nonexistence of limit cycles is proved. In case \(f\) is a complex rational function \(f(z)=\frac{P(z)}{Q(z)}\), where \(P\) and \(Q\) are polynomials in the \(z\) variable, the authors describe and give a sketch of the topological classification of the phase portraits in the Poincaré disc. The notion of conformally conjugated systems near the origin -- which is stronger than the classical topological equivalency of differential systems- is also presented.