Self-sustained oscillations of nonlinearly viscoelastic layers

We study the motion of an incompressible nonlinearly viscoelastic layer sub ject to a slip-stick frictional force applied to one of its faces by a movi ng belt. This system is governed by a third-order quasi-linear parabolic-hy perbolic partial differential equation subject to complicated boundary cond itions. We use a refined version of the Hopf bifurcation theorem to show th at this problem admits periodic solutions for the belt speed near critical values, with the number of such solutions increasing as the viscosity of th e layer decreases. To verify the hypotheses of this theorem, we must devote considerable effort to the tricky analysis of how the spectrum of the line arized problem depends on the belt speed and material parameters. Our analy sis is complemented with both computations and a proof of the topological f act that the computed disposition of the eigenvalues does not omit any othe rs. We complete our study with a perturbation analysis, justified by the Ho pf bifurcation theorem.