This work overviews and continues recent investigations of the authors [
] on the dynamical integrity of nonlinear mechanical systems and other oscillators. In fact, it has been realized that attractors must be paralleled by uncorrupted basins for safe practical applications [
]. Eroded basins constitute a critical state for the structure corresponding to its impending failure, which can be instantaneous in the presence of perturbations or uncertainties. On the other hand, using the structure only when erosion is prevented may be too much conservative, because it can actually survive safely well above the relevant threshold if the erosion is not sharp.