Newton geometry of Bogdanov-Takens degeneracies in integrable dilatonic models: invariant divisors and boundary multiplicity
Abstract
The Kantowski--Sachs interior of Grumiller's two-dimensional dilaton gravity model reduces to Ḣ=1/2(Λ−3H2−u2)+Qu, u̇=−Hu, where H is the expansion rate of the orbit two-spheres, u their inverse areal radius, and Q is minus twice the Rindler acceleration. For Q≠0 it has two distinct rank-one nilpotent equilibria with complementary Bogdanov--Takens (BT) degeneracies: (a,b)=(0,≠0) with multiplicity μ=3, and (a,b)=(≠0,0) with μ=2. We show that this complementarity is forced by a single divisor-organized structure. The system is Darboux integrable, X=1/2u4XI, where I is the mass function and u4 is an inverse integrating factor whose zero divisor is the invariant axis {u=0}. Off the divisor, every nilpotent equilibrium of RXI with R(p)≠0 has b=0, from b=Dq0trDX. On the divisor, invariance of the coordinate axis forces a=0 because det DX|u=0=ΦxG and both factors vanish at the corner. Hence ab=0 throughout the class, so a versal two-parameter BT unfolding is impossible. The Bernstein--Kushnirenko bound fails at both points; instead, the exact local identity μ0=m1ℓy+m2ℓx gives μ=3,2 without a nondegeneracy hypothesis. Both multiplicities have a mixed-covolume interpretation, with Covol=∑k,imin(pkqi',pi'qk) on convenient diagrams; the corresponding general theorem for the non-convenient diagrams arising here remains open. For Ḣ=(1−m)/2H2+ψ(u), u̇=−Hu, with ψ(0)=0, ψ'(0)≠0, m≠1, the vacuum always has μ=3, a=0, b=−m. Its Dumortier--Llibre--Artés discriminant is b12+8a3=(m−2)2, so for every m>1 it has one hyperbolic and one elliptic sector, independently of ψ.
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BibTeX
@misc{chanlopez2026newton,
title = {{Newton geometry of Bogdanov-Takens degeneracies in integrable dilatonic models: invariant divisors and boundary multiplicity}},
author = {E. Chan-López and A. Martín-Ruiz and J. M. Paulin Fuentes},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16677},
eprint = {2609.16677},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16677},
}
RIS
TY - UNPB
TI - Newton geometry of Bogdanov-Takens degeneracies in integrable dilatonic models: invariant divisors and boundary multiplicity
AU - E. Chan-López
AU - A. Martín-Ruiz
AU - J. M. Paulin Fuentes
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16677
UR - https://arxiv.org/abs/2609.16677
AB - The Kantowski--Sachs interior of Grumiller's two-dimensional dilaton gravity model reduces to Ḣ=1/2(Λ−3H^2−u^2)+Qu, u̇=−Hu, where H is the expansion rate of the orbit two-spheres, u their inverse areal radius, and Q is minus twice the Rindler acceleration. For Q≠0 it has two distinct rank-one nilpotent equilibria with complementary Bogdanov--Takens (BT) degeneracies: (a,b)=(0,≠0) with multiplicity μ=3, and (a,b)=(≠0,0) with μ=2. We show that this complementarity is forced by a single divisor-organized structure. The system is Darboux integrable, X=1/2u^4X_I, where I is the mass function and u^4 is an inverse integrating factor whose zero divisor is the invariant axis {u=0}. Off the divisor, every nilpotent equilibrium of RX_I with R(p)≠0 has b=0, from b=D_q_0trDX. On the divisor, invariance of the coordinate axis forces a=0 because det DX|_u=0=Φ_xG and both factors vanish at the corner. Hence ab=0 throughout the class, so a versal two-parameter BT unfolding is impossible. The Bernstein--Kushnirenko bound fails at both points; instead, the exact local identity μ_0=m_1ℓ_y+m_2ℓ_x gives μ=3,2 without a nondegeneracy hypothesis. Both multiplicities have a mixed-covolume interpretation, with Covol=∑_k,imin(p_kq_i',p_i'q_k) on convenient diagrams; the corresponding general theorem for the non-convenient diagrams arising here remains open. For Ḣ=(1−m)/2H^2+ψ(u), u̇=−Hu, with ψ(0)=0, ψ'(0)≠0, m≠1, the vacuum always has μ=3, a=0, b=−m. Its Dumortier--Llibre--Artés discriminant is b_1^2+8a_3=(m−2)^2, so for every m>1 it has one hyperbolic and one elliptic sector, independently of ψ.
ER -