An efficient solution procedure for solving higher-codimension Hopf and Bogdanov-Takens bifurcations

The paper proves that system (8) has a Bogdanov–Takens point at E2 when (ε, k) = ((n+1)^2/(m(1−n)), n(1−n)(n+1)^2/m), with codimension 2 for n ∈ (0,1/2) ∪ (1/2,1) and codimension 3 at n = 1/2; see the explicit BT critical point and codimension split in Theorem 3.1 and the construction of the PSNF (49), where the y1y2 coefficient is −(1−2n)/n^2, vanishing exactly at n = 1/2 . The candidate solution independently verifies the double-zero eigenvalue with a single Jordan chain at E2 using the Jacobian (19), computes the BT normal-form coefficients c20 > 0 and c11 ∝ 2n^2 + n − 1 (hence c11 = 0 iff n = 1/2), and reaches the same codimension conclusions in accord with the paper’s general classification that the first non-vanishing resonant coefficient after c20 determines the codimension . Minor wording aside (the model’s remark about “codimension 3 within this two-parameter family” should be read as codimension 3 when the third parameter n is also unfolded), both are correct and consistent; the proofs differ in technique (paper: parametric simplest normal form; model: adjoint-pairing formulas for c20, c11).