An intrinsic characterization of the Bogdanov-Takens normal-form coefficients and a mixed-volume obstruction to non-isolated degeneracies

What the math gives to ML

The paper provides a coordinate-free diagnostic for Bogdanov-Takens degeneracy in a planar vector field with a nonzero nilpotent rank-one equilibrium Jacobian. The normal-form coefficients are directional derivatives of the determinant and trace of the Jacobian along its kernel direction, avoiding generalized eigenvectors and second-order tensor calculations. This mechanism transfers to two-dimensional neural ODEs, continuous-time recurrent networks, and two-dimensional center-manifold reductions as a bifurcation monitor or regularizer. The most practical use is to detect and prevent accidental transitions between ordinary BT, cusp-like, and higher-order saddle/focus/elliptic degeneracies.