It is well known that overdamped and unforced dynamical systems do not oscillate. However, well-designed coupling schemes, together with the appropriate choice of initial conditions, can induce oscillations when a control parameter exceeds a threshold value. We demonstrate this effect in a specific system, a soft-potential mean-field description of the dynamics in a (hysteretic) single-domain ferromagnetic sample. Using a specific (unidirectional, with cyclic boundary conditions) coupling scheme, together with nonidentical initial conditions, one can cause the coupled system of N elements (N odd) to oscillate when the coupling coefficient is swept through a critical value. The ensuing oscillations could find utility in the detection of very weak "target" signals, via their effect on the oscillation characteristics.