Complex potentials and holomorphic differential equations

TL;DR

This paper extends the concept of complex potentials to general holomorphic dynamical systems, providing a framework for classifying phase portraits and addressing fundamental problems in planar vector fields.

Contribution

It generalizes complex potentials beyond classical harmonic functions to broader holomorphic systems, introducing a rectification mapping for topological classification.

Findings

Provides a framework for classifying phase portraits of polynomial vector fields

Addresses bounds on the number of limit cycles in piecewise systems

Analyzes local curvature line configurations around umbilic points

Abstract

A complex potential is a holomorphic function $\Omega:\mathbb{C} \to \mathbb{C}$ whose real and imaginary parts generate a pair of orthogonal foliations, representing the equipotential lines and the streamlines of $\dot{z} = \overline{\Omega'(z)}$. In this work, we generalize the concept of potential to the broader class of dynamical systems of the form $\dot{z} = f(z)$, with $f:\mathbb{C} \to \mathbb{C}$ holomorphic. The resulting potential induces a rectification mapping providing a natural framework for the topological classification of phase portraits of planar polynomial vector fields. The existence of complex potentials serves as a powerful tool in addressing fundamental problems, such as the establishment of bounds for the number of limit cycles in piecewise-smooth systems, and the local configuration of curvature lines around umbilic points, among others.