Singularities that induce a breakdown of determinism in mechanics

We will show that determinism can break down at certain singularities that arise in the mechan- ics of semi-rigid bodies. An example system consists of a wheel slipping and rolling along a surface, such that dry friction applies a torque to the wheel via a standard nonlinear tyre-like interaction. In a certain configuration the force on the wheel and its ensuing motion acquire many possible values. The subsequent motion is there- fore infinitely sensitive to perturbation, constituting a point of non-determinism in an otherwise deterministic system. Moreover the device returns to the singularity repeatedly, so that it suffers repeated bursts of unpredictability, constituting an extreme non-deterministic form of chaos. A non-deterministic chaotic attractor was found for a range of parameters and tyre/friction models in (2013). Here, we explore the building blocks of this dynamics by stripping the model down to its key elements. We investigate the non-deterministic phenomenon when subjected to relaxations of the model that allow for smoothing, noise, and hysteresis. This is part of a wider theory currently emerging in the study of dynamical systems that undergo sharp switches in behaviour, whereby singularities can violate traditional rules of smooth dynamical systems.