It is well known that overdamped unforced dynamical systems do not oscillate. However, well-designed coupling schemes, together with the appropriate choice of initial conditions, can induce oscillations (corresponding to transitions between the stable steady states of each nonlinear element) when a control parameter exceeds a threshold value. In recent publications [A. Bulsara, Phys. Rev. E 70, 036103 (2004); V. In, ibid. 72, 045104 (2005)], we demonstrated this behavior in a specific prototype system, a soft-potential mean-field description of the dynamics in a hysteretic "single-domain" ferromagnetic sample. These oscillations are now finding utility in the detection of very weak "target" magnetic signals, via their effect on the oscillation characteristics--e.g., the frequency and asymmetry of the oscillation wave forms. We explore the underlying dynamics of a related system, coupled bistable "standard quartic" dynamic elements; the system shows similarities to, but also significant differences from, our earlier work. dc as well as time-periodic target signals are considered; the latter are shown to induce complex oscillatory behavior in different regimes of the parameter space. In turn, this behavior can be harnessed to quantify the target signal.