A Stiction Oscillator with Canards: On Piecewise Smooth Nonuniqueness and Its Resolution by Regularization Using Geometric Singular Perturbation Theory
In mechanics, one often describes microscopic processes such as those leading to friction between relative interfaces using macroscopic variables (relative velocity, temperature, etc.) in order to avoid models of intangible complexity. As a consequence, such macroscopic models are frequently nonsmooth, a prominent example being the Coulomb law of friction. In many cases, these models are perfectly adequate for engineering purposes. Formally, however, since the Fundamental Theorem of Existence and Uniqueness does not apply to these situations, one generally expects that these models possess forward nonuniqueness of solutions. Consequently, numerical computations of such systems might possibly unknowingly discard certain solutions. In this paper, we try to shed further light on this issue by studying solutions of a simple friction oscillator subject to stiction friction. The stiction law is a simple nonsmooth model of friction that is a modification of Coulomb based on the fundamental observation that the dynamic friction force, when the mass is in motion, is smaller than the static friction force during stick. The resulting piecewise smooth vector field of this discontinuous model does not follow the classical Filippov convention, and the concept of a Filippov solution cannot be used. Furthermore, some Carathéodory solutions, i.e., absolutely continuous solutions satisfying the differential equation in a weaker sense, are nonphysical. Therefore, we introduce the concept of stiction solutions. These are the Carathéodory solutions that are physically relevant, i.e., the ones that follow the stiction law. However, we find that some of the stiction solutions are forward nonunique in subregions of the slip onset. We call these solutions singular, in contrast to the regular stiction solutions that are forward unique. In order to further understanding of the nonunique dynamics, we then introduce a general regularization of the model. This gives a singularly perturbed problem that captures the main features of the original discontinuous problem. Using geometric singular perturbation theory, we identify a repelling slow manifold that separates the forward slipping from the forward sticking solutions, leading to high sensitivity to the initial conditions. On this slow manifold we find canard trajectories that have the physical interpretation of delaying the slip onset. Most interestingly, we find that these new solutions do not correspond to stiction solutions in the piecewise-smooth limit, and are therefore seemingly nonphysical, yet they are robust and appear generically in the class of regularizations we consider. Finally, we show that the regularized problem has a family of periodic orbits interacting with the canards. We observe that this family has a saddle stability and that it connects, in the rigid body limit, the two regular, slip-stick branches of the discontinuous problem, which are otherwise disconnected.
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