Complex potentials and holomorphic differential equations

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.