A NONSMOOTH COULOMB-FRICTION OSCILLATOR

A forced Coulomb friction oscillator, whose frictional force is allowed to vary with displacement, is analyzed geometrically. The equation of motion for the oscillator is piecewise linear. We geometrically observe the nature of the flow in each region of solvability, and then see how these solutions interact at the boundary of the regions. The dynamics of the flow is viewed in terms of a map on the boundary between the regions. For chaotic motion, we geometrically construct the strange attractor, and show that its exact behavior is that of a one-dimensional map. The following dynamical properties arise from the nonsmooth nature of the Coulomb friction law: the flow may not be invertible; the flow may reach its attractor in finite time; the dimension of the attractor may be less than or equal to two; embeddings of an observable may not be diffeomorphic to the full phase flow.