We investigate analytically and numerically a basic model of driven Brownian motion with a velocity-dependent friction coefficient in nonlinear viscoelastic media featured by a stress plateau at intermediate shear velocities and profound memory effects. For constant force driving, we show that nonlinear oscillations of a microparticle velocity and position emerge by a Hopf bifurcation at a small critical force (first dynamical phase transition), where the friction's nonlinearity seems to be wholly negligible. They also disappear by a second Hopf bifurcation at a much larger force value (second dynamical phase transition). The bifurcation diagram is found in an analytical form confirmed by numerics. Surprisingly, the particles' inertial and the medium's nonlinear properties remain crucial even in a parameter regime where they were earlier considered entirely negligible. Depending on the force and other parameters, the amplitude of oscillations can significantly exceed the size of the particles, and their period can span several time decades, primarily determined by the memory time of the medium. Such oscillations can also be thermally excited near the edges of dynamical phase transitions. The second dynamical phase transition combined with thermally induced stochastic limit cycle oscillations leads to a giant enhancement of diffusion over the limit of vast driving forces, where an effective linearization of stochastic dynamics occurs.
Keywords: Hopf bifurcations and emerging nonlinear oscillations; memory effects; nonlinear Brownian motion; stochastic limit cycle and force-induced giant diffusion.