The Bogdanov--Takens normal-form coefficients in $\mathbb{R}^n$ as directional derivatives of the characteristic invariants
arXiv:2608.19018
2026
Dynamics
2 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper provides a coordinate-free method for computing the two quadratic Bogdanov–Takens coefficients from directional derivatives of characteristic-polynomial invariants, avoiding center-manifold construction, Jordan chains, adjoint eigenvectors, and explicit spectral factorization. Its transferable asset is a cheap local bifurcation diagnostic for neural ODEs, continuous-time RNNs, and learned state-space models. The most useful application is to monitor or constrain the central determinant and central trace near an equilibrium, predicting when a recurrent system approaches a codimension-two instability. A second application is deliberate BT-critical initialization to create controllable long-memory and oscillatory regimes.
Ideas from this paper
✗ Failed on benchmark
2026
Add a local bifurcation monitor to a neural ODE, continuous-time RNN, or state-space model by computing the central determinant and central trace from characteristic invariants of the state Jacobian. Their directional derivatives along the zero-eigenvalue direction estimate the BT coefficients and predict whether the model is approaching a codimension-two transition, allowing training to avoid destructive criticality or intentionally preserve a useful long-memory regime.
Useful7/10
Difficulty5/10
Novelty8/10
Unverified
2026
Construct a recurrent or state-space layer whose equilibrium Jacobian is placed near a nondegenerate Bogdanov–Takens point, then use a small unfolding parameter to move between damped, oscillatory, and slowly relaxing regimes. Unlike eigenvalue-only initialization near one, this controls both the double-zero center structure and the quadratic nonlinear coefficients that determine the local phase portrait.
Useful6/10
Difficulty6/10
Novelty8/10