Le canard de Painlev\'e

Abstract

We consider the problem of a slender rod slipping along a rough surface. Painlev\'e \cite{Painleve1895, Painleve1905a,Painleve1905b} showed that the governing rigid body equations for this problem can exhibit multiple solutions (the {\it indeterminate} case) or no solutions at all (the {\it inconsistent} case), provided the coefficient of friction $\mu$ exceeds a certain critical value $\mu_P$. Subsequently G\'enot and Brogliato \cite{GenotBrogliato1999} proved that, from a consistent state, the rod cannot reach an inconsistent state through slipping. Instead there is a special solution for $\mu>\mu_C>\mu_P$, with $\mu_C$ a new critical value of the coefficient of friction, where the rod continues to slip until it reaches a singular `$0/0$' point $P$. Even though the rigid body equations can not describe what happens to the rod beyond the singular point $P$, it is possible to extend the special solution into the region of indeterminacy. This extended solution is very reminiscent of a {\it canard} \cite{Benoit81}. To overcome the inadequacy of the rigid body equations beyond $P$, the rigid body assumption is relaxed in the neighbourhood of the point of contact of the rod with the rough surface. Physically this corresponds to assuming a small compliance there. It is natural to ask what happens to both the point $P$ and the special solution under this regularization, in the limit of vanishing compliance. In this paper, we prove the existence of a canard orbit in a reduced $4D$ slow-fast phase space, connecting a $2D$ focus-type slow manifold with the stable manifold of a $2D$ saddle-type slow manifold. The proof combines several methods from local dynamical system theory, including blowup. The analysis is not standard, since we only gain ellipticity rather than hyperbolicity with our initial blowup.