DISSECTING A RESONANCE WEDGE ON HETEROCLINIC BIFURCATIONS

The paper defines the same first return map Fμ, derives the (1,ℓ)-fixed point equations s = exp(−2ℓπ/(Kω)), y = s^δ, introduces Gℓ(ω) = s − s^δ, and proves that along the two curves ±λ = A − Gℓ(ω⋆ℓ) with ω⋆ℓ = 2ℓπ(δ − 1)/ln δ a discrete-time Bogdanov–Takens bifurcation occurs at (1,ℓ)-fixed points; at those points DFμ has a double unit eigenvalue and is not the identity, and standard 1:1-resonance nondegeneracy conditions are verified . The candidate solution reproduces these steps (solves for s and y, computes det/trace, locates ω⋆ℓ, enforces cos x0 = 0 to keep the fixed point isolated, finds a nontrivial Jordan block, and verifies a quadratic term is nonzero), matching the paper’s equations and conclusions; its transversality checks are consistent with the paper’s det/trace formulas and with the stated BT curves .