ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

An imaginary eigenvalue crossing with zero Lyapunov coefficient

Example

Consider the linear system $\dot{x} = μ x - y, \dot{y} = x + μ y .$ The eigenvalues cross the imaginary axis transversely, but there is no isolated small cycle branch for nonzero $μ$ .

Facts & Assumptions

Given: The linear system in the Example and the eigenvector normalization $q^{*} q = p^{*} q = 1$ .

[F1]

Hopf spectral and multilinear normal-form data: the first Lyapunov coefficient is computed from the second and third derivatives by the displayed formula.

Verification

1.1givenalgebra

Its Jacobian is $[\begin{array}{cc} μ & - 1 \\ 1 & μ \end{array}]$ , with eigenvalues $μ \pm i$ , so $β^{'} (0) = 1$ . The vector field is linear; therefore $B = 0$ and $C = 0$ .

2.1F1step 1.1algebra

By [F1], $G_{21} = 0$ and $ℓ_{1} = 0$ . The nondegeneracy hypothesis of the generic Hopf theorem is therefore absent even though the eigenvalue crossing is transverse.

3.1step 1.1algebra∎

In polar coordinates $\dot{R} = μ R$ and $\dot{θ} = 1$ . If $μ \neq 0$ , every nonzero radius changes strictly exponentially and cannot be periodic; if $μ = 0$ , every circle $R > 0$ is periodic, so there is no isolated branch selected by crossing.

Depends on

Hopf spectral and multilinear normal-form data

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Yuri A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd ed., Springer (standard reference, not scraped)
J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, 1976 (standard reference, not scraped)
Yuri A. Kuznetsov, Andronov-Hopf Bifurcation, Scholarpedia 1(10):1858 (2006) (standard reference, not scraped)