We consider the problem of a slender rod slipping along a rough surface. Painlevé [C. R. Séances Acad. Sci., 121 (1895), pp. 112–115; C. R. Séances Acad. Sci., 141 (1905), pp. 401–405; C. R. Séances Acad. Sci., 141 (1905), pp. 546–552] showed that the governing rigid body equations for this problem can exhibit multiple solutions (the indeterminate case) or no solutions at all (the inconsistent case), provided the coefficient of friction µ exceeds a certain critical value µP. Subsequently Génot and Brogliato [Eur. J. Mech. A Solids, 18 (1999), pp. 653–677] proved that, from a consistent state, the rod cannot reach an inconsistent state through slipping. Instead the rod will either stop slipping and stick or it will lift off from the surface. Between these two cases is a special solution for µ > µC > µP, where µC is a new critical value of the coefficient of friction. Physically, the special solution corresponds to the rod slipping until it reaches a singular “0/0” point P. Even though the rigid body equations cannot 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 canard [E. Benoît et al., Collect. Math., 31-32 (1981), pp. 37–119]. To overcome the inadequacy of the rigid body equations beyond P, the rigid body assumption is relaxed in the neighborhood 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 four-dimensional slow-fast phase space, connecting a two-dimensional focus-type slow manifold with the stable manifold of a two-dimensional 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.
- Impact without collision
- Painlevé paradox