The recent advances for an archetypal smooth and discontinuous oscillator

•The complex nonlinear dynamics characteristics of the archetypal smooth and discontinuous (SD) oscillator including co-dimension two bifurcation problems with a geometrical parameter and the perturbed dynamics.•The extended dynamics of coupled SD oscillator, rotating oscillator and also the self-excited oscillator, and so on, with both smooth and discontinuous behavior depending on the geometrical parameters.•Wide engineering applications especially in low- and ultra-low frequency vibration isolation and emerging harvesting by using the selected negative stiffness property and multiple stabilities.