Le Canard de Painlevé

Abstract

We consider the problem of a slender rod slipping along a rough surface. Painlevé [44,45,46] 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 µ P . Subsequently Génot and Brogliato [19] proved that, from a consistent state, the rod cannot reach an inconsistent state through slipping. Instead there is a special solution for µ > µ C > µ P , with µ 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 canard [1]. 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. * K.Remark 3 For µ P < µ < µ C so that ξ > 1, then the direction θ = θ 1 is strong while (18) is weak. However, by evaluating the slope of the curve b(θ, φ) = 0 at the point P and comparing the result with s/(1 − ξ), it is straightforward to show that the weak eigendirection (18) is not contained within F ∪ P ∪ T . The reduced problem is only defined within F ∪ T and hence the classical Painlevé problem does therefore not support weak singular canards. 2 7 The result µ C (3) = 8 3 √ 3 does appear in [19].

Blowup for flat slow manifolds

“…However, in the case of dynamic Hopf, it is shown in [15] that blowup can be successfully applied when combined with the technique, popularized by Neishtadt in [36,37], of complex time. The reference [25] deals with fast oscillations using a different approach. Similarly, the blowup approach does not appear to help when applied directly to the problem of bifurcation delay [39].…”

“…A frequently used approach is therefore to apply regularization. This has for example been done in the references [27,28,29,16,25], to deal with problems associated with lack of uniqueness in PWS systems, and in [1]. These references, however, exclude regularization functions such as tanh due its special asymptotic properties that leads to loss of hyperbolicity at an exponential rate.…”

On the regularization of impact without collision: the Painlevé paradox and compliance

“…Note that we are not attempting to describe all the dynamics around P . There is a canard connecting the third quadrant with the first and the analysis is exceedingly complicated [18] due to fast oscillatory terms. Instead, we follow [37] and consider that the rod dynamics starts in a configuration with p + (θ 0 ) < 0.…”

“…We give exact and asymptotic expressions for key quantities as well as providing a geometric interpretation of our results, for a large class of rigid bodies, in the presence of a large class of normal forces, as well as giving a precise estimate for the time of (regularized) IWC, all in the presence of both stiffness and damping. Note that we are not attempting to describe all the dynamics around P. There is a canard connecting the third quadrant with the first, and the analysis of it is exceedingly complicated [29] due to fast oscillatory terms. Instead, we follow [24] and consider that the rod dynamics starts in a configuration with p + (θ 0 ) < 0.…”

On generalized Coulomb–Amontons’ law in the context of rigid body dynamics

“…The indeterminacy aspect in recent works is actively studied by means of regularization in the neighbour-hood of the point of contact (see, e.g. [10,11]). However, in this direction, our knowledge is yet limited mostly to particular scenarios in certain planar problems with a single point of contact.…”