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 $\dot H=\tfrac12(\Lambda-3H^2-u^2)+Qu$, $\dot u=-Hu$, 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 $Q\neq0$ it has two distinct rank-one nilpotent equilibria with complementary Bogdanov--Takens (BT) degeneracies: $(a,b)=(0,\neq0)$ with multiplicity $\mu=3$, and $(a,b)=(\neq0,0)$ with $\mu=2$. We show that this complementarity is forced by a single divisor-organized structure. The system is Darboux integrable, $X=\tfrac12u^4X_I$, 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 $R(p)\neq0$ has $b=0$, from $b=D_{q_0}\operatorname{tr}DX$. On the divisor, invariance of the coordinate axis forces $a=0$ because $\det DX|_{u=0}=\Phi_xG$ 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 $\mu_0=m_1\ell_y+m_2\ell_x$ gives $\mu=3,2$ without a nondegeneracy hypothesis. Both multiplicities have a mixed-covolume interpretation, with $\operatorname{Covol}=\sum_{k,i}\min(p_kq_i',p_i'q_k)$ on convenient diagrams; the corresponding general theorem for the non-convenient diagrams arising here remains open. For $\dot H=\tfrac{1-m}{2}H^2+\psi(u)$, $\dot u=-Hu$, with $\psi(0)=0$, $\psi'(0)\neq0$, $m\neq1$, the vacuum always has $\mu=3$, $a=0$, $b=-m$. Its Dumortier--Llibre--Art\'es 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 $\psi$.