Scalar polynomial vector fields in real and complex time

TL;DR

This paper analyzes scalar polynomial ODEs with complex zeros, describing their solutions on Riemann surfaces, classifying real phase portraits, and exploring the structure of their compactifications and associated combinatorial objects.

Contribution

It provides a detailed classification of polynomial scalar ODE solutions on Riemann surfaces and real phase portraits, including novel residue conditions and combinatorial structures.

Findings

Explicit solutions via separation of variables

Classification of real-time phase portraits

Connection to planar trees and chord diagrams

Abstract

In the present paper, the simplest scalar ODE case is studied for polynomials $$ \dot{w}=f(w)=(w-e_0)\cdot\ldots\cdot(w-e_{d-1}) $$ of degree $d$ with $d$ simple complex zeros. The explicit solution by separation of variables and explicit integration is an almost trivial matter. In a classical spirit, indeed, we describe the complex Riemann surface $\mathcal{R}$ of the global nontrivial solution $(t,w(t))$ in complex time, as an unbranched cover of the punctured Riemann sphere. The flow property, however, fails at $w=\infty$. The global consequences depend on the period map of the residues $2\pi\mathrm{i}/f'(e_j)$ of $1/f$ at the punctures, in detail. We therefore show that polynomials $f$ exist for arbitrarily prescribed residues with zero sum. This result is not covered by standard interpolation theory. Motivated by the PDE case, we also classify the planar real-time phase portraits of the above ODE. Poincar\'e compactification of $w$ regularizes $w=\infty$ by $2(d-1)$ equilibria, alternatingly stable and unstable within the invariant circle boundary at infinity. In structurally stable cases, we classify all compactified phase portraits, up to orientation preserving orbit equivalence and time reversal. Combinatorially, their connection graphs are equivalent to certain unrooted, unlabeled, undirected planar trees of $d$ vertices or, dually, to certain chord diagrams with $d-1$ nonintersecting chords. We sketch a proof that all planar trees are actually realized by Poincar\'e compactifications. Not least, we offer a 1,000 Euro reward for the discovery, or refutation, of complex entire homoclinic orbits.