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 $\dot{H}=\frac{1}{2}(\Lambda -3H^2-u^2)+Qu$, $\dot{u}=-Hu$, 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 $Q\ne 0$ it has two distinct rank-one nilpotent equilibria with complementary Bogdanov--Takens (BT) degeneracies: $(a,b)=(0,\ne 0)$ with multiplicity $\mu =3$, and $(a,b)=(\ne 0,0)$ with $\mu =2$. We show that this complementarity is forced by a single divisor-organized structure. The system is Darboux integrable, $X=\frac{1}{2}{u}^4{X}_I$, 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 $R{X}_I$ with $R(p)\ne 0$ has $b=0$, from $b=D_{q_0}\mathrm{tr}DX$. On the divisor, invariance of the coordinate axis forces $a=0$ because $\det DX{|}_{u=0}={\Phi }_{x}G$ 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 $\mathrm{Covol}={\sum }_{k,i}\min (p_k{q}_i^{\prime },p_i^{\prime }{q}_k)$ on convenient diagrams; the corresponding general theorem for the non-convenient diagrams arising here remains open. For $\dot{H}=\frac{1-m}{2}{H}^2+\psi (u)$, $\dot{u}=-Hu$, with $\psi (0)=0$, ${\psi }^{\prime }(0)\ne 0$, $m\ne 1$, the vacuum always has $\mu =3$, $a=0$, $b=-m$. 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 $\psi$.

22 pages, 1 figure, 1 supplemental Wolfram Language notebook