Modeling Nonlinear Dynamics from Videos

The paper’s claims about (i) reconstructing an SSM in a delay-embedded observable space for p ≥ 2d+1 and generic (μ, Δt), (ii) using the extended normal form with ρ̇ = γ(ρ)ρ and θ̇ = ω(ρ) under nonresonance to define damping/frequency backbone curves, and (iii) revealing unstable fixed points and unstable limit cycles not directly visited by training trajectories, are consistent with standard results it cites and with its demonstrations (Takens embedding, Sec. 3.3; normal-form/“backbone” definitions, Sec. 3.4; examples using ρ̇/ρ polynomials and zeros of γ) . The candidate solution gives a more formal justification: it adds compact-set/locality qualifiers for Takens embeddings on W^E, spells out the push-forward vector field Y = DΦ·X on the embedded manifold, derives the żj = zj Φj(|z|^2) form and the stability test sign γ′(ρ*) for limit cycles, and argues generic monotonicity of an amplitude map A(ρ), which the paper uses implicitly. A minor caveat is that the paper’s empirical claim about revealing unstable cycles is not proved (only demonstrated), whereas the model solution frames this locally (on K) but does not provide data-driven error guarantees. Net: the two are consistent and correct; the model provides a more explicit (and slightly stronger) mathematical framing of what the paper informally states and demonstrates.