  The motion of Brownian particles in nonlinear baths, such as, e.g.,
viscoelastic fluids, is of great interest. We theoretically study a simple
model for such bath, where two particles are coupled via a sinusoidal
potential. This model, which is an extension of the famous Prandtl Tomlinson
model, has been found to reproduce some aspects of recent experiments, such as
shear-thinning and position oscillations [J. Chem. Phys. {\bf 154}, 184904
(2021)]. Analyzing this model in detail, we show that the predicted behavior of
position oscillations agrees qualitatively with experimentally observed trends;
(i) oscillations appear only in a certain regime of velocity and trap stiffness
of the confining potential, and (ii), the amplitude and frequency of
oscillations increase with driving velocity, the latter in a linear fashion.
Increasing the potential barrier height of the model yields a rupture
transition as a function of driving velocity, where the system abruptly changes
from a mildly driven state to a strongly driven state. The frequency of
oscillations scales as $(v_0-v_0^*)^{1/2}$ near the rupture velocity $v_0^*$,
found for infinite trap stiffness. Investigating the (micro-)viscosity for
different parameter ranges, we note that position oscillations leave their
signature by an additional (mild) plateau in the flow curves, suggesting that
oscillations influence the micro-viscosity. For a time-modulated driving, the
mean friction force of the driven particle shows a pronounced resonance
behavior, i.e, it changes strongly as a function of driving frequency. The
model has two known limits: For infinite trap stiffness, it can be mapped to
diffusion in a tilted periodic potential. For infinite bath friction, the
original Prandtl Tomlinson model is recovered. We find that the flow curve of
the model (roughly) crosses over between these two limiting cases.
