arxiv: 2605.06950 · v1 · submitted 2026-05-07 · 🧮 math.DS · math.AC· math.AG

Recognition: no theorem link

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

Megan Morrison, Sonja Petrovi\'c

Pith reviewed 2026-05-11 00:49 UTC · model grok-4.3

classification 🧮 math.DS math.ACmath.AG

keywords quadratic ODEsanalytical solutionsKoopman eigenfunctionsrational eigenfunctionseigenfunction varietypolynomial systemsordinary differential equationsalgebraic geometry

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The pith

Linear rational eigenfunctions produce closed-form solutions for families of quadratic ODEs via their algebraic variety.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method to find analytical solutions for some two-dimensional quadratic ordinary differential equations by assuming the Koopman eigenfunctions take a linear rational form. This turns the eigenfunction partial differential equation into a polynomial algebraic system whose solutions define the rational eigenfunction variety. Analyzing this variety identifies the parameter relationships that must hold between the ODE coefficients and the eigenfunction parameters to obtain explicit closed-form solutions. A sympathetic reader cares because polynomial ODEs appear throughout biology, engineering, physics, and economics, where exact solutions are scarce and most work relies on numerical approximation.

Core claim

Imposing a linear rational ansatz on the Koopman eigenfunctions of a quadratic ODE converts the associated partial differential equation into a system of polynomial equations. The solution set to this system is the rational eigenfunction variety of the ODE. Solutions in this variety, when they exist, supply the explicit linear rational eigenfunctions needed to construct closed-form analytical solutions for the original quadratic system.

What carries the argument

The rational eigenfunction variety: the algebraic set of solutions to the polynomial system obtained by substituting the linear rational form into the Koopman eigenfunction equation.

If this is right

Families of quadratic ODEs become solvable in closed form once their parameters place them in the rational eigenfunction variety.
The variety supplies explicit algebraic relations that the ODE coefficients must obey for analytical solvability.
The eigenfunctions themselves are constructed directly from the variety parameters, bypassing numerical integration.
Algebraic geometry tools become available to classify which quadratic systems admit this exact treatment.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same linear rational restriction might identify solvable cases among cubic or higher-degree polynomial ODEs if the resulting algebraic system remains tractable.
One could check membership in the variety numerically to decide whether a given quadratic model should be solved analytically or numerically.
The variety relations could guide parameter selection in applied models when exact solutions are preferred over simulation.
Connections to other eigenfunction techniques might yield hybrid methods that combine algebraic and numerical approaches for broader classes of systems.

Load-bearing premise

Solutions found in the polynomial system obtained from the linear rational ansatz must correspond to actual eigenfunctions that satisfy the original ODE and produce valid closed-form solutions.

What would settle it

A quadratic ODE system whose derived polynomial equations admit a nontrivial solution, yet the resulting linear rational function fails to satisfy the Koopman eigenfunction partial differential equation upon direct substitution.

Figures

Figures reproduced from arXiv: 2605.06950 by Megan Morrison, Sonja Petrovi\'c.

**Figure 2.** Figure 2: (a). Singularities of R(a, b) = 0 are described by X(a, b) = 0. (b) Points on R(a, b) = 0 map to points on H(a, b, c, d, λ) = 0. Points along X(a, b) = 0 can map to two sets of eigenfunction parameter values. 5 Solvable quadratic ODEs and their linear rational eigenfunctions There are two spaces, L and X , where Eq. 1 has independent linear rational eigenfunctions. L is a 6-dimensional linear space defined… view at source ↗

**Figure 3.** Figure 3: (a) Space L dynamical system. (b) Solutions in eigenfunction space for initial condition (x0, y0) = (0, 1). (c) Analytical versus numerical solutions for the initial condition (x0, y0) = (0, 1). are in space X . Two independent eigenvalue/eigenfunction pairs for Eqs. 17 using the formulas from Section 5.2 are λ1 = 1, φ1(x, y) = −1 + x + y x + y , λ2 = 2, φ2(x, y) = −6 + 3x + 2y 3x + 2y . Consider the init… view at source ↗

**Figure 4.** Figure 4: (a) Space X example 1 dynamical system. (b) Solutions in eigenfunction space for initial condition (x0, y0) = (−3, −4). (c) Analytical versus numerical solutions for the initial condition (x0, y0) = (−3, −4). shown in [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗

**Figure 5.** Figure 5: (a) Space X example 2 dynamical system. (b) Solutions in eigenfunction space for initial condition (2, −1). (c) Analytical versus numerical solutions for the initial condition (x0, y0) = (2, −1). 6.4 ODE with parameters in X : example 3 Consider the ODE, dx dt = −x + 2y + x 2 + xy − 3 2 y 2 dy dt = −2x + y − x 2 2 + 3xy − y 2 (21) shown in [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗

**Figure 6.** Figure 6: (c) shows the analytical solution for initial condition (2, −1), Eqs. 22, compared to the numerical solution computed using NDSolve in Mathematica. The analytical and numerical solutions match well. These examples exhibit the variety in dynamics displayed by the quadratic ODE families that are solvable using the rational eigenfunction method with linear rational eigenfunctions outlined above. Similar to a… view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper turns the search for linear rational Koopman eigenfunctions into a polynomial system whose solutions define families of quadratic ODEs with closed-form solutions, using algebraic geometry to find the parameter relations.

read the letter

The core contribution is a systematic way to convert the Koopman eigenfunction PDE into a polynomial system by assuming linear rational eigenfunctions, then using algebraic geometry to extract the solution set they call the rational eigenfunction variety. This yields explicit parameter conditions under which certain two-dimensional quadratic ODEs admit analytical solutions via those eigenfunctions. The approach is new in framing the problem this way and applying nonlinear algebra tools directly to the eigenfunction search rather than guessing forms case by case. It does a clean job laying out the conversion steps and invoking the relevant algebraic geometry results to handle the resulting systems. The logic holds without circularity because the conditions are derived algebraically from the ansatz and then used to build the solutions; the stress-test note is right that there is no internal gap in the chain from ansatz to verified eigenfunction for the families that satisfy the equations. One soft spot is the narrow scope: everything is restricted to two dimensions and this specific linear rational form, so the method only captures the solvable families that arise under those constraints and will miss others. Another is that the abstract gives no worked examples, so the full paper needs to show several concrete families, verify that the eigenfunctions satisfy the original ODE, and demonstrate that the resulting closed forms are non-trivial rather than recovering known integrable cases. The work is aimed at researchers in dynamical systems who care about analytical solutions for polynomial ODEs or who use Koopman methods. It has enough formal structure and a clear, falsifiable method to deserve serious referee time, even if the examples and scope discussion need strengthening in revision. I would send it out for peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript develops a framework for analytically solving families of two-dimensional quadratic ODEs by imposing a linear rational ansatz on Koopman eigenfunctions. This converts the associated eigenfunction PDE into a polynomial algebraic system whose solutions are analyzed via algebraic geometry; the resulting parameter relations define the rational eigenfunction variety, from which closed-form solutions for the original ODEs are constructed.

Significance. If the extracted eigenfunctions are verified to satisfy the ODEs, the method supplies a systematic algebraic route to identifying integrable quadratic systems and their exact solutions, complementing existing techniques in dynamical systems. The use of formal algebraic geometry to extract parameter constraints without numerical fitting is a clear strength, offering reproducible, exact characterizations that could apply to models in biology, engineering, and physics.

major comments (1)

The central construction relies on the linear rational ansatz producing genuine Koopman eigenfunctions whose level sets yield closed-form solutions. While the algebraic reduction is direct, explicit verification that the variety solutions satisfy the original nonlinear ODE (beyond the cleared-denominator polynomial system) is required to confirm the claim; this should be shown for at least one concrete family in the main text.

minor comments (2)

The term 'rational eigenfunction variety' is introduced without a precise mathematical definition or reference to the ambient space in which it lives; a short formal definition early in the paper would improve clarity.
The abstract states that families are identified and solutions produced, but the manuscript would benefit from a table or explicit list of the quadratic ODEs, their eigenfunction parameters, and the resulting closed-form expressions to make the results immediately usable.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the constructive recommendation for minor revision. We address the single major comment below and will incorporate the requested verification in the revised version.

read point-by-point responses

Referee: The central construction relies on the linear rational ansatz producing genuine Koopman eigenfunctions whose level sets yield closed-form solutions. While the algebraic reduction is direct, explicit verification that the variety solutions satisfy the original nonlinear ODE (beyond the cleared-denominator polynomial system) is required to confirm the claim; this should be shown for at least one concrete family in the main text.

Authors: We agree that explicit verification is essential to confirm that solutions from the rational eigenfunction variety are not extraneous artifacts introduced by clearing denominators in the eigenfunction PDE. In the revised manuscript we will add a dedicated subsection (or expanded example) that selects one concrete family from the variety, substitutes the resulting linear rational eigenfunctions back into the original Koopman PDE, verifies that the PDE is satisfied identically, and explicitly demonstrates that the level sets of these eigenfunctions yield the closed-form solutions to the quadratic ODE system. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper starts from the Koopman eigenfunction PDE for a given quadratic ODE, imposes an explicit linear rational ansatz on the eigenfunctions, clears denominators to obtain a polynomial algebraic system, and applies external algebraic-geometry techniques to solve for parameter relations that yield valid eigenfunctions. These eigenfunctions are then used to construct closed-form solutions for the ODEs that admit them. The process is purely constructive and algebraic; the reported families and solutions are exactly the outputs of solving the derived polynomial system under the stated ansatz and ODE restrictions. No step reduces a claimed prediction or result to a fitted parameter, self-definition, or load-bearing self-citation; the method characterizes only those ODEs for which the ansatz succeeds, without claiming generality or external validation beyond the algebraic construction itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the domain assumption that Koopman eigenfunctions exist and can be restricted to linear rational form, plus the standard mathematical fact that algebraic geometry can analyze the resulting polynomial ideals; the new term 'rational eigenfunction variety' is introduced without independent evidence outside the construction itself.

axioms (2)

domain assumption Koopman operator theory applies to the given quadratic ODE systems and yields an eigenfunction PDE
Invoked to convert the ODE into the eigenfunction equation that is then specialized to linear rational form
standard math Solutions of the resulting polynomial system can be found and interpreted using algebraic geometry
Used to define and study the rational eigenfunction variety

invented entities (1)

rational eigenfunction variety no independent evidence
purpose: The algebraic variety consisting of all parameter values for which linear rational eigenfunctions satisfy the ODE restrictions
New object introduced to organize the solutions of the polynomial system; no independent falsifiable prediction is supplied in the abstract

pith-pipeline@v0.9.0 · 5458 in / 1414 out tokens · 80617 ms · 2026-05-11T00:49:20.615276+00:00 · methodology

discussion (0)

Reference graph

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