We consider the gradient flow Γ̇=−∇P(Γ;μ) on the real 4×4 matrices, with potential P(Γ)=‖Γ‖²+μ·detΓ+β‖Γ‖⁴ (plus an optional sextic regularizer), and its damped second-order counterpart Γ̈+γΓ̇+∇P=0. Near a degeneration of the Hessian —a simple soft mode— center-manifold reduction yields the local normal forms of bifurcation theory. The cubic coefficient of the reduced flow comes from the determinant through its cofactor matrix, the only anisotropic nonlinearity of the field; it is the sole source of the cubic when the soft mode is orthogonal to Γ*. We classify the accessible organizing centers: fold and pitchfork in codimension 1,the cusp in the gradient sector, and Bogdanov–Takens in codimension 2, whose existence is topologically obstructed in the gradient limit and requires lifting the system to the second-order (inertial) flow. The homoclinic orbit closes the Bogdanov–Takens portrait, and the non-gradient part of the field, which carries the rotation, numerically sustains a Shilnikov-type chaotic regime. Analytical results are accompanied by reproducible simulations verifying the critical scalings: the Kramers law, a pseudo-arclength continuation through the fold, the 3/2 law of the cusp, and the Lyapunov exponent of the chaotic regime.
Publication Date: 2026-06-20