Averaging Theory and Catastrophes: The Persistence of Bifurcations Under Time-Varying Perturbations

The paper’s Theorem 1 states that, under K-universal (versal) hypotheses on the guiding field g_ℓ, the catastrophe surface M_Π of the stroboscopic Poincaré map equals the image of Z_{g_ℓ}×R by a strongly fibred diffeomorphism, up to the trivial branch ε=0. The proof in the paper proceeds via the displacement function of order ℓ, Δ_ℓ, using the identity Π = Id + ε^ℓ Δ_ℓ, and K-equivalence to transfer zero sets; it constructs a strongly fibred diffeomorphism whose third component is ε and shows invariance of Z_{g_ℓ}×{0} (Theorem 1 and its proof) . The candidate’s argument mirrors this exactly: (i) adopt the stroboscopic near-identity change of variables normalized at t=0 so the stroboscopic map aligns with the averaged coordinates ; (ii) define F = (Π−Id)/ε^ℓ and extend smoothly at ε=0 so F(x,μ,0)=T g_ℓ(x,μ), i.e., F=Δ_ℓ by the paper’s relation Π=Id+ε^ℓ Δ_ℓ ; (iii) invoke K-universality so F is K-induced by g_ℓ, producing Q, α, h and a strongly fibred diffeomorphism Θ=(α,h,ε) with hex=(h,ε) a local diffeomorphism, exactly as in the paper’s Lemma 2 and proof of Theorem 1 ; (iv) conclude M_Π = Φ((Z_{g_ℓ}×R)∩U) ∪ V_{ε=0} and invariance of Z_{g_ℓ}×{0} . No substantive logical gaps remain; both proofs are materially the same.