The Poincar\'e-Bendixson Theorem and the Non-linear Cauchy-Riemann Equations

TL;DR

This paper extends the Poincaré-Bendixson theorem to bounded solutions of non-linear Cauchy-Riemann equations, using an abstract flow theorem with a discrete Lyapunov function.

Contribution

It demonstrates a Poincaré-Bendixson type result for non-linear Cauchy-Riemann equations, broadening the theorem's applicability.

Findings

Poincaré-Bendixson theorem applies to non-linear Cauchy-Riemann equations

Use of an abstract flow theorem with a discrete Lyapunov function

Establishment of bounded solution behavior in complex PDEs

Abstract

Fiedler and Mallet-Paret prove a version of the classical Poincar\'e-Bendixson Theorem for scalar parabolic equations. We prove that a similar result holds for bounded solutions of the non-linear Cauchy-Riemann equations. The latter is an application of an abstract theorem for flows with a(n) (unbounded) discrete Lyapunov function.