Analytical solutions for some quadratic ODEs found via linear rational eigenfunctions and the rational eigenfunction variety

TL;DR

This paper introduces a novel algebraic framework leveraging linear rational Koopman eigenfunctions to analytically solve a broad class of two-dimensional quadratic ODEs, which are common in various scientific fields.

Contribution

It develops a systematic algebraic method to identify and solve quadratic ODEs using the rational eigenfunction variety, expanding the set of analytically solvable systems.

Findings

Identified families of quadratic ODEs with analytical solutions.

Characterized eigenfunction parameters for these ODEs.

Produced closed-form solutions for specific quadratic systems.

Abstract

Many important systems across biology, engineering, physics, and economics are characterized by polynomial ordinary differential equations (ODEs), yet analytical solutions are rare. We develop a framework for identifying and solving a broad class of two-dimensional quadratic ODEs using linear rational Koopman eigenfunctions. By imposing a linear rational form on the eigenfunctions, we convert the Koopman eigenfunction PDE into a large algebraic system of polynomials. We then study the solutions of this polynomial system that satisfy the ODE restrictions; we call the solution set the rational eigenfunction variety of an ODE system. The nonlinear algebra method uses formal algebraic geometry theory to analyze and solve systems otherwise intractable and to discover relationships between ODE and eigenfunction parameters that must hold to extract eigenfunctions. We identify families of quadratic ODEs that can be solved analytically, characterize their eigenfunction parameters, and use the resulting eigenfunctions to produce closed-form analytical solutions.