Parameter Inference with Bifurcation Diagrams

The paper’s Appendix C derives a generalized Leibniz rule for line integrals over curves defined implicitly by Fθ(z)=0, yielding d/dθ ∫_{Fθ=0} Lθ(z) dz = ∫_{Fθ=0} [∂θL + ∂zL · ϕθ + L T̂θ · (∂ϕθ/∂z) · T̂θ] dz, where ϕθ = −(∂zF)ᵀ(∂zF(∂zF)ᵀ)^{-1}∂θF and T̂θ is the unit tangent along the level set. This is shown by reparameterizing the integral over a fixed domain, differentiating under the integral sign, computing the rate of change of the arclength element, and using the normal deformation field for the implicit curve (eqs. (24)–(34) in the paper) . The candidate solution reproduces the same formula via a slightly more rigorous geometric setup: it constructs the 2D manifold M = {(z,θ): F(z,θ)=0}, generates a θ-flow using the canonical normal velocity ϕθ, differentiates the pulled-back integral, computes ∂θ|∂sγ| = |∂sγ| T̂·(∂ϕ/∂z)·T̂, and assembles terms, also noting invariance under added tangential velocity on closed components. This closely matches the paper’s derivation, including the same canonical expression for ϕθ and the tangent-field construction (via the determinant/wedge) used to define T̂θ . Minor gaps in the paper (e.g., conditions for differentiating under the integral sign and boundary terms for open curves) are acknowledged briefly via a citation to Flanders (1973) and by choosing a normal deformation field, while the model fills these with standard regularity and compactness assumptions. Overall, the two arguments are substantively the same and correct, with the model providing a slightly more explicit justification of hypotheses.