Higher-order characteristic-invariant jets and the first Lyapunov coefficient

Determine whether some higher-order jet of the characteristic invariants of the Jacobian along a direction can determine the first Lyapunov coefficient at a Hopf point, extending beyond the first-order directional invariant data considered in the paper.

Background

The paper investigates whether bifurcation coefficients can be expressed using directional derivatives of scalar characteristic invariants of the Jacobian, rather than eigenvectors and multilinear normal-form formulas. For the Bogdanov–Takens case, first-order directional derivatives of the elementary symmetric invariants suffice to recover the quadratic normal-form coefficients.

At a Hopf point, the first Lyapunov coefficient depends on third-order derivatives of the vector field, whereas the first directional derivatives of the characteristic invariants depend only on the 2-jet. The paper derives a second-order characteristic-polynomial identity that can encode certain ordered compositions involving the quadratic derivative, but it does not establish whether any higher-order jet of characteristic invariants is sufficient to recover the first Lyapunov coefficient.

References

Whether some higher-order jet of the characteristic invariants determines \ell_{1} is an open question that we do not address.

— The Bogdanov--Takens normal-form coefficients in $\mathbb{R}^n$ as directional derivatives of the characteristic invariants  (2608.19018 - Chan-López, 19 Aug 2026) in Section “What transfers to a Hopf point, and what does not,” subsection “The nonlinear level: an obstruction of order, not of structure”