Bifurcation analysis of Bogdanov–Takens bifurcations in delay differential equations

The paper derives orbital BT normal forms on a parameter-dependent center manifold for DDEs, applies the singular blow-up to obtain the perturbed second-order oscillator u¨ = u^2 − 4 + ε u̇(u + τ) + O(ε^4), and reuses known third-order homoclinic templates (including τ = 10/7 + 288/2401 ε^2). The candidate solution follows the same steps and scalings (β1, β2, w0, w1, s), uses the same homoclinic and Melnikov calculation for τ0 = 10/7, and maps back via the same H, K, ϑ expansions—precisely matching the paper’s formulas and order bookkeeping.