Normal forms for saddle-node bifurcations: Takens’ coefficient and applications in climate models

The paper’s Theorem 6 establishes C^k conjugacy to the extended normal form ẏ = ν − y^2 + a y^3 in three regimes (µ<0, µ=0, µ>0), with ν(µ)=p0^2 µ + O(µ^{3/2}) and a(µ)=a0 + O(√µ), where p0^2=−½ fµ fxx and a0=2 fxxx/(3 fxx^2). The proofs rely on asymptotics for the equilibria and their multipliers, an implicit-function argument to match multipliers, and a basin-by-basin extension of local conjugacies, exactly as summarized in sections 5.1–5.4 and Theorem 4 (extension to basins) and Theorem 5 (µ<0 case) of the paper. The candidate solution reproduces these ingredients with essentially the same strategy (Taylor expansions, multiplier matching via IFT, and a basin conjugacy constructed via time-of-flight coordinates), and obtains the same asymptotics for ν and a. The only point needing clarification is a quantifier subtlety in part (i): a single neighbourhood V cannot be chosen uniformly for all a∈R when µ<0; rather, V (and ν(µ)) can be chosen for each fixed a. Aside from this minor presentation issue, the arguments align. See Theorem 6 and its setup (statements and parameter asymptotics) and the proofs summarized in sections 3–5 of the paper .