Computations for the first Lyapunov coefficient

Isabella Cravero; Marino Badiale

arxiv: 2511.08428 · v2 · submitted 2025-11-11 · 🧮 math.DS

Computations for the first Lyapunov coefficient

Marino Badiale , Isabella Cravero This is my paper

Pith reviewed 2026-05-17 23:28 UTC · model grok-4.3

classification 🧮 math.DS

keywords Hopf bifurcationLyapunov coefficientasymptotic expansionHANDY modelbifurcation theorydynamical systemsmultilinear formsresolvents

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The pith

Asymptotic expansions yield explicit formulas for the first Lyapunov coefficient in a Hopf bifurcation of HANDY-type models.

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

This supplementary note works out the explicit value of the first Lyapunov coefficient for a Hopf bifurcation in HANDY-type models. It follows the standard Kuznetsov procedure by calculating the multilinear forms B and C, the normalized left and right eigenvectors, and the two required resolvents of the linear operator A. With those quantities in hand the authors insert asymptotic expansions in the small parameter ε to obtain closed-form expressions for the critical value μ(ε), the frequency ω₀, and the coefficient a itself. The sign of a then tells whether the bifurcation is supercritical or subcritical and therefore whether the periodic orbits that appear are stable.

Core claim

By computing the multilinear forms B and C, normalizing the right and left eigenvectors, and evaluating the resolvents A^{-1} and (2iω₀ I - A)^{-1}, the authors obtain, via asymptotic expansions in the small parameter ε, explicit formulas for μ(ε), ω₀, and the Lyapunov coefficient a(μ(ε), ε) that fix the criticality of the Hopf bifurcation occurring at a simple pair of complex eigenvalues in the main HANDY-type system.

What carries the argument

The first Lyapunov coefficient a, obtained from the multilinear forms B and C together with the resolvents of the linear operator A at the critical point, which determines the stability of the bifurcating periodic orbits.

If this is right

The sign of a determines whether the Hopf bifurcation is supercritical, producing stable periodic orbits, or subcritical, producing unstable ones.
The explicit expressions in ε allow the criticality to be read off analytically for small values of the parameter.
These formulas apply directly to the HANDY-type model examined in the companion paper and fix its bifurcation type.

Where Pith is reading between the lines

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

The same expansion technique could be applied to other ecological models that contain a comparable small parameter and undergo a Hopf bifurcation.
Higher-order terms in the expansion might supply an estimate for the amplitude of the emerging oscillations.
Direct numerical continuation of periodic orbits in the full system would provide an independent check on the accuracy of the asymptotic coefficient a.

Load-bearing premise

The derivation assumes that the Hopf bifurcation occurs at a simple pair of complex eigenvalues crossing the imaginary axis and that the standard normalization and resolvent formulas apply without extra degeneracies.

What would settle it

A numerical integration of the HANDY-type system near the predicted μ(ε) that produces periodic orbits whose stability contradicts the sign of the computed a would falsify the explicit formulas.

read the original abstract

These notes are a supplementary file to the paper Hopf bifurcations for HANDY-type models (M. Badiale and I. Cravero, under submission), providing full details of the computations developed in Section 4.2. The purpose of this supplement is to derive explicitly the first Lyapunov coefficient associated with a Hopf bifurcation, following the framework of Yu. A. Kuznetsov (Elements of Applied Bifurcation Theory, Springer, 4th ed., 2023). We compute the multilinear forms $B$ and $C$, the right and left eigenvectors and their normalization, and the resolvents $A^{-1}$ and $(2i\omega_0 I - A)^{-1}$. Using asymptotic expansions with respect to the small parameter $\varepsilon$, we derive explicit formulas for $\mu(\varepsilon)$, $\omega_0$, and the Lyapunov coefficient $a(\mu(\varepsilon),\varepsilon)$, which characterize the criticality of the Hopf bifurcation in the main model.

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

This supplement gives explicit asymptotic formulas for the Lyapunov coefficient in the HANDY model but stays inside the standard Kuznetsov calculation.

read the letter

This note supplies the detailed algebra for the first Lyapunov coefficient in the HANDY-type models from the companion paper. The central result is a set of explicit leading-order expressions in the small parameter ε for the critical value μ(ε), the imaginary part ω0, and the coefficient a that tells you if the Hopf bifurcation is super- or subcritical. The authors apply the Kuznetsov procedure in full: they calculate the bilinear and trilinear forms B and C, normalize the left and right eigenvectors for the critical eigenvalues, find the resolvents for the linear operator A and for 2iω0 I - A, and then expand all quantities asymptotically. For this particular family the explicit formulas are new, and they turn the abstract bifurcation analysis into something one can evaluate directly. The work follows the standard framework without deviation, and the assumptions about a simple pair of eigenvalues crossing the axis are stated up front. That keeps the derivation on solid ground. The citation to Kuznetsov is appropriate, and there is no sign of circular reasoning or fitted parameters. The obvious limitation is that these are lengthy hand calculations. A reader cannot easily spot an arithmetic error in the expansions without redoing the steps, though the overall structure matches what one expects. If the sign of a depends on higher-order terms that were dropped, that could matter for the conclusion, but the paper appears to stop at the leading order as usual. This is for people already engaged with the HANDY model or similar low-dimensional systems where they need the explicit criticality condition for further analysis or simulation. It is not aimed at readers looking for new theoretical tools in bifurcation theory. I would recommend sending the supplement to peer review alongside the main paper. The computation is load-bearing for the bifurcation claims, so a careful check of the algebra is worth the referee's time.

Referee Report

0 major / 3 minor

Summary. This supplementary note details the computations for the first Lyapunov coefficient of a Hopf bifurcation in HANDY-type models. Following Kuznetsov's framework, it calculates the multilinear forms B and C, the normalized right and left eigenvectors, the resolvents A^{-1} and (2iω₀I - A)^{-1}, and performs asymptotic expansions in the small parameter ε to obtain explicit leading-order formulas for μ(ε), ω₀, and the Lyapunov coefficient a(μ(ε), ε) that determine the criticality of the bifurcation.

Significance. If the derived expressions are correct, the explicit formulas provide an analytical means to classify the Hopf bifurcation as super- or subcritical, offering insight into the local dynamics near the bifurcation point in the HANDY model. The decision to publish the full algebra as a supplement is a strength, as it supports reproducibility and allows independent verification of the arithmetic steps without requiring the reader to reconstruct the multilinear forms from the companion paper.

minor comments (3)

In the section computing the multilinear forms B and C, explicitly list the relevant partial derivatives of the vector field evaluated at the equilibrium to allow direct checking against the model equations from the main paper.
For the asymptotic expansions of μ(ε), ω₀, and a, state the truncation order (e.g., O(ε²)) and briefly indicate how the neglected terms affect the leading-order sign of the Lyapunov coefficient.
The normalization condition ⟨p, q⟩ = 1 for the left and right eigenvectors should be written explicitly at the point where the eigenvectors are introduced.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our supplementary note, including the recognition that publishing the full algebra supports reproducibility. The recommendation for minor revision is noted. As the report contains no specific major comments to address, we have no point-by-point responses at this stage and will proceed with any minor clarifications required in the revised version.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The supplement applies the standard Kuznetsov procedure to compute multilinear forms B and C, normalized eigenvectors, and resolvents, followed by asymptotic expansions in the small parameter ε to obtain explicit leading-order expressions for μ(ε), ω₀, and the first Lyapunov coefficient a. These steps rest on the model equations supplied by the companion paper and on external references to Kuznetsov (2023); no quantity is defined in terms of a fitted parameter that is then renamed as a prediction, no self-citation chain carries the central claim, and the derivation does not reduce to its own inputs by construction. The paper is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The computations rely on the standard Hopf bifurcation formulas from Kuznetsov and on the specific vector field of the HANDY-type model; no new free parameters, ad-hoc axioms, or invented entities are introduced in the supplement itself.

axioms (1)

domain assumption The system possesses a Hopf bifurcation at a parameter value where a simple pair of complex-conjugate eigenvalues crosses the imaginary axis with nonzero speed.
Invoked when the authors apply the Kuznetsov procedure to the HANDY-type model.

pith-pipeline@v0.9.0 · 5457 in / 1416 out tokens · 23194 ms · 2026-05-17T23:28:08.112564+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

a(μ(ε),ε)=1/(2ω0) Re[⟨p,C(q,q,q̄)⟩−2⟨p,B(q,A⁻¹B(q,q̄))⟩+⟨p,B(q̄,(2iω0I−A)⁻¹B(q,q))⟩]
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

asymptotic expansions … μ(ε)=μ0+μ1ε+O(ε²), ω0=√(7μ0+μ3ε)+O(ε²)

What do these tags mean?

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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
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Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

Badiale and I

M. Badiale and I. Cravero. Hopf bifurcations for handy-type models. To be submitted

work page
[2]

Y. A. Kuznetsov.Elements of Applied Bifurcation Theory. Springer, 4th edition edition, 2022. 12

work page 2022

[1] [1]

Badiale and I

M. Badiale and I. Cravero. Hopf bifurcations for handy-type models. To be submitted

work page

[2] [2]

Y. A. Kuznetsov.Elements of Applied Bifurcation Theory. Springer, 4th edition edition, 2022. 12

work page 2022