We use a set of qualitatively different models of coupled oscillators (genetic, membrane, Ca-metabolism, and chemical oscillators) to study dynamical regimes in the presence of small detuning. In particular, we focus on a distinct oscillation quenching mechanism, the oscillation death phenomenon. Using bifurcation analysis in general, we demonstrate that under strong coupling via slow variable detuning can eliminate standard oscillatory solutions from a large region of the parameter space, establishing the dominance of oscillation death. We argue furthermore that the oscillation death dominance effect provides a reliable dynamical control mechanism in the general case of N coupled oscillators.