Abstract | Novel empirical relationships are introduced to estimate the median, mean and dispersion of strength ratios to cause dynamic instability in oscillators with non-trivial backbones and arbitrary periods. The backbones investigated range from a simple bilinear elastic-negative shape to a trilinear that includes an elastic, a hardening and a negative stiffness segment that terminates at zero strength. Using 72 ground motion records that were recorded on firm soil we calculate the mean, median, 16% and 84% percentiles of the corresponding lateral strength ratios required for the appearance of dynamic instability. Processing of the results shows the influence of the oscillator parameters to the occurrence of dynamic instability: Lengthening the oscillator period, delaying the onset of negative stiffness and decreasing the negative slope are all shown to delay the appearance of collapse. On the other hand, contrary to current engineering intuition, increasing the hardening stiffness while maintaining the same period and coincident negative branches has only a small effect on the onset of instability. Using nonlinear regression, parametric equations are developed that can accurately capture such effects in a simple, easy-to-use formula.