The Bogdanov--Takens normal-form coefficients in $\mathbb{R}^n$ as directional derivatives of the characteristic invariants

Abstract: Let $X$ be a vector field on an open set of $\mathbb{R}<sup>n$ with $X(p)=0$ and Jacobian $J=DX(p)$ of rank $n-1$ having $0$ as an eigenvalue of algebraic multiplicity two. Let $q_0$ span $\ker J$ and let $e_k(A)$ denote the sum of the principal $k\times k$ minors of $A$. Under the usual hyperbolicity assumption on the transverse block, we prove that the quadratic coefficients $a,b$ of the Bogdanov--Takens normal form on the centre manifold are $a=-\frac{1}{2}\frac{D_{q_0}e_n}{e_{n-2}}$ and $$b=\frac{D_{q_0}e_{n-1}}{e_{n-2}}-\frac{e_{n-3}D_{q_0}e_n}{e_{n-2}<sup>2}$$. The underlying spectral identity requires only invertibility of the transverse block and remains valid without hyperbolicity. Both coefficients arise as the lowest-order terms of a generating identity for the first-order spectral jet of $DX$ along $\ker J$; all higher coefficients depend on the transverse block. We prove that this dichotomy is sharp. The planar formulas $a=-\frac{1}{2}D_{q_0}\det$ and $b=D_{q_0}\operatorname{tr}$ are recovered when $n=2$. We also obtain a coordinate-free nondegeneracy test and a geometric interpretation in terms of the transversality of the kernel line to two invariant hypersurfaces. A self-contained Wolfram Language notebook accompanies the paper and verifies the results symbolically.

• The paper computes the Bogdanov–Takens normal-form coefficients $a$ and $b$ in $\mathbb{R}^n$ as directional derivatives of specific elementary symmetric functions of the Jacobian, using minimal computations and no normal-form transformation.
• The formulas happen to match existing results for $n=2$ and provide a general approach valid for $n extgreater 2$ unbiased by hyperbolicity.
• The new method reveals how certain invariants are sujet to 'contamination' through their dependence on transverse dynamics and demonstrates practical applications in parameter dependence

Overview and main result

This paper establishes that the two quadratic coefficients $a$ and $b$ of the Bogdanov–Takens (BT) normal form on the centre manifold of an equilibrium in $\mathbb{R}^n$ can be computed as first-order directional derivatives, along the kernel line, of the two lowest elementary symmetric functions of the Jacobian. Specifically, for a vector field $X$ with an equilibrium at $p$, Jacobian $J = DX(p)$ of rank $n-1$ with $0$ as an eigenvalue of algebraic multiplicity two, and transverse block invertible, the paper proves

$$a=-\frac{1}{2}\,\frac{D_{q_0}e_n}{e_{n-2}}, \qquad b=\frac{D_{q_0}e_{n-1}}{e_{n-2}}-\frac{e_{n-3}\,D_{q_0}e_n}{e_{n-2}^{2}},$$

where $q_0$ spans $b$0, $b$1 denotes the sum of principal $b$2 minors of $b$3, and $b$4 is the derivative of $b$5 at $b$6 along $b$7. For $b$8 these collapse to the planar identities $b$9 and $\mathbb{R}^n$0 established in a companion work (Castellanos et al., 14 Aug 2026), so the planar formulas are not a dimension-two accident.

The result is notable for its proof technique: it uses only Jacobi's formula for determinant differentiation and the block structure of the adjugate of a nilpotent-plus-invertible matrix. No centre manifold reduction, no normal-form transformation, no generalized eigenvectors of the adjoint, and no analytic perturbation theory appear anywhere in the proof of the main theorem. The entire argument depends only on the 2-jet of $\mathbb{R}^n$1 at $\mathbb{R}^n$2.

The generating identity and exact sharpness

The central technical device is a polynomial identity in $\mathbb{R}^n$3 describing the full first-order spectral jet of $\mathbb{R}^n$4 along $\mathbb{R}^n$5:

$\mathbb{R}^n$6

where $\mathbb{R}^n$7 is the characteristic polynomial of the transverse block $\mathbb{R}^n$8 and $\mathbb{R}^n$9 is the diagonal block of $X$0 on the transverse space. Comparing coefficients of $X$1 and $X$2 yields the main formulas; every higher coefficient is contaminated by the transverse block through the factor $X$3 multiplying the last term.

The paper proves this dichotomy is exact. Using a gap lemma for polynomials — proved here for want of a reference, and shown to be sharp via $X$4 — the author establishes that all adjugate coefficient matrices $X$5 are nonzero when $X$6 is invertible. Consequently, for $X$7, no directional derivative $X$8 with $X$9 is determined by $p$0 and $p$1 alone: the invariants $p$2 and $p$3 are the only transversally clean ones. This gives precise content to what is often described informally as "hyperbolic contamination": the trace identity $p$4 shows explicitly how the naive candidate $p$5 exceeds $p$6 by the variation of the transverse trace, with the second term of the formula for $p$7 being exactly the correction removing this contribution.

Hyperbolicity is used only to name the answer

A structural point worth emphasizing: the algebraic identities require only invertibility of the transverse block ($p$8), not hyperbolicity. They remain valid when the transverse spectrum contains a purely imaginary pair or a resonance $p$9, situations where the perturbative route via centre manifolds breaks down entirely. Hyperbolicity enters exactly once, in the identification lemma connecting the algebraically defined pair $J = DX(p)$0 to the coefficients of the BT normal form on the centre manifold; the paper keeps these two statements strictly apart. In the non-hyperbolic case the formulas still hold as identities between 2-jet quantities, though the reading of $J = DX(p)$1 as coefficients of a two-dimensional germ is lost.

Geometric reading and nondegeneracy

Defining the central determinant $J = DX(p)$2 and central trace $J = DX(p)$3 — both rational functions of the principal minors of $J = DX(p)$4 requiring no eigenvector computation — the paper shows $J = DX(p)$5 and

$J = DX(p)$6

Thus the planar formulas hold verbatim in every dimension once determinant and trace are replaced by these central analogues. The nondegeneracy conditions become transversality statements: $J = DX(p)$7 if and only if the kernel line $J = DX(p)$8 meets the hypersurface $J = DX(p)$9 transversally at $n-1$0, and similarly $n-1$1 corresponds to transversality to $n-1$2. The sign of $n-1$3, which determines the topological type of the versal unfolding, becomes the sign of an explicit polynomial in the entries of $n-1$4 and $n-1$5:

$n-1$6

The paper is careful about scope: neither $n-1$7, nor $n-1$8, nor $n-1$9 is coordinate-free — rescaling $0$0 sends $0$1 — but the vanishing conditions and the sign of $0$2 are invariant under both normalization changes and $0$3 changes of phase-space coordinates, the latter established via an invariance lemma whose proof requires both $0$4 and $0$5; a counterexample shows invariance fails away from the kernel line.

What does not transfer to Hopf points

The paper examines whether the programme extends to Hopf bifurcations and answers negatively at the relevant order. The first Lyapunov coefficient $0$6 depends on $0$7, whereas $0$8 depends only on the 2-jet; the planar family $0$9, $$a=-\frac{1}{2}\,\frac{D_{q_0}e_n}{e_{n-2}}, \qquad b=\frac{D_{q_0}e_{n-1}}{e_{n-2}}-\frac{e_{n-3}\,D_{q_0}e_n}{e_{n-2}^{2}},$$0 has identical 2-jets for all $$a=-\frac{1}{2}\,\frac{D_{q_0}e_n}{e_{n-2}}, \qquad b=\frac{D_{q_0}e_{n-1}}{e_{n-2}}-\frac{e_{n-3}\,D_{q_0}e_n}{e_{n-2}^{2}},$$1 while $$a=-\frac{1}{2}\,\frac{D_{q_0}e_n}{e_{n-2}}, \qquad b=\frac{D_{q_0}e_{n-1}}{e_{n-2}}-\frac{e_{n-3}\,D_{q_0}e_n}{e_{n-2}^{2}},$$2. Hence $$a=-\frac{1}{2}\,\frac{D_{q_0}e_n}{e_{n-2}}, \qquad b=\frac{D_{q_0}e_{n-1}}{e_{n-2}}-\frac{e_{n-3}\,D_{q_0}e_n}{e_{n-2}^{2}},$$3 is provably not a function of $$a=-\frac{1}{2}\,\frac{D_{q_0}e_n}{e_{n-2}}, \qquad b=\frac{D_{q_0}e_{n-1}}{e_{n-2}}-\frac{e_{n-3}\,D_{q_0}e_n}{e_{n-2}^{2}},$$4. The author characterizes this as an obstruction of order rather than structure: second-order jets of the characteristic polynomial do encode ordered compositions of two copies of $$a=-\frac{1}{2}\,\frac{D_{q_0}e_n}{e_{n-2}}, \qquad b=\frac{D_{q_0}e_{n-1}}{e_{n-2}}-\frac{e_{n-3}\,D_{q_0}e_n}{e_{n-2}^{2}},$$5 mediated by a resolvent, and whether some higher-order jet determines $$a=-\frac{1}{2}\,\frac{D_{q_0}e_n}{e_{n-2}}, \qquad b=\frac{D_{q_0}e_{n-1}}{e_{n-2}}-\frac{e_{n-3}\,D_{q_0}e_n}{e_{n-2}^{2}},$$6 remains open. The structural reason for the cleanliness of the BT case is identified precisely: the constant term of $$a=-\frac{1}{2}\,\frac{D_{q_0}e_n}{e_{n-2}}, \qquad b=\frac{D_{q_0}e_{n-1}}{e_{n-2}}-\frac{e_{n-3}\,D_{q_0}e_n}{e_{n-2}^{2}},$$7 is the rank-one matrix $$a=-\frac{1}{2}\,\frac{D_{q_0}e_n}{e_{n-2}}, \qquad b=\frac{D_{q_0}e_{n-1}}{e_{n-2}}-\frac{e_{n-3}\,D_{q_0}e_n}{e_{n-2}^{2}},$$8, which isolates single entries of the central block visible to a first derivative of a scalar invariant.

Algorithmic consequences and validation

The formulas yield a four-step algorithm requiring only one null vector computation, one univariate characteristic polynomial evaluation along the kernel line truncated at first order, and no linear solves over function fields — avoiding the null vector of $$a=-\frac{1}{2}\,\frac{D_{q_0}e_n}{e_{n-2}}, \qquad b=\frac{D_{q_0}e_{n-1}}{e_{n-2}}-\frac{e_{n-3}\,D_{q_0}e_n}{e_{n-2}^{2}},$$9 and generalized eigenvector $q_0$0 demanded by the classical Kuznetsov formula. A self-contained Wolfram Language script verifies all 101 assertions symbolically, including cross-validation against the classical bilinear formula over dimensions $q_0$1 with real, complex, purely imaginary, and resonant transverse spectra.

The worked examples are instructive. In the Dias–Mello quadratic family, the new method reproduces published values of $q_0$2 and $q_0$3 "on the nose" once the generator normalization is accounted for via the covariance law $q_0$4 — resolving an apparent discrepancy that would otherwise be an unexplained constant factor. The generic three-dimensional germ demonstrates that none of the twelve transverse coupling coefficients enters $q_0$5 or $q_0$6, consistent with the theory.

Limitations and open questions

Several boundaries of the results deserve plain statement. First, the nondegeneracy test concerns the germ of a single vector field; genericity of a two-parameter family additionally requires transversality of the unfolding, a condition on parameter dependence not addressed here. Second, the degenerate cases $q_0$7 or $q_0$8 are governed by higher-order normal-form data; whether the second-order jet of $q_0$9 and $b$00 along $b$01 carries the cusp coefficient is posed but unresolved, with the second-order identity indicating the shape such a formula would take. Third, the claim about $b$02 leaves open whether some higher-order jet of the characteristic invariants suffices. Fourth, the geometric interpretation of $b$03 and $b$04 is deliberately modest: they reproduce the determinant and trace of the reduced Jacobian only to first order along the kernel line, and are not determinants or traces of any invariantly defined $b$05 block beyond that.

Conclusion

The paper shows that the BT normal-form data in $b$06 are first-order variations, along the kernel of the linearization, of the two lowest characteristic coefficients normalized by the transverse determinant; that this characterization is embedded in a generating identity describing the full first-order spectral jet; and that the dichotomy between clean and contaminated invariants is provably exact. The elimination-theoretic reformulation — the BT locus as a saturation of an ideal generated by polynomial conditions on the jet of $b$07 — reduces the systematic search for BT points in parametrized models to routine Gröbner computation, replacing the adjoint-eigenvector formulation of the classical theory.