  We consider the systematic force on a heavy probe induced by interaction with
an overdamped diffusive medium where particles undergo a rotating force around
a fixed center. The stiffness matrix summarizes the stability of the probe
around that center, where the induced force vanishes. We prove that the
introduction of the rotational force in general enhances the stability of that
point (and may turn it from unstable to stable!), starting at second-order in
the nonequilibrium amplitude. When the driving is further enhanced the
stabilization occurs for a wide range of rotation profiles and the induced
stiffness converges to a universal expression proportional to the average
mechanical stiffness. The model thus provides a rigorous example of
stabilization of a fixed point due to contact with a nonequilibrium medium and
beyond linear order around equilibrium.
