The recent advances for an archetypal smooth and discontinuous oscillator

The so called archetypal smooth and discontinuous (SD) oscillator with irrational nonlinearity is a simple massspring system constrained to a straight line by a geometrical parameter alpha which is the dimensionless distance to the fixed point. The typical phenomenon of this oscillator is the dynamics transition from smooth to discontinuous depending on the smooth changing of the geometrical parameter, which has attracted a quite mount of investigations on the complex nonlinear behaviours of the SD oscillator since it was firstly proposed and appeared in Physical Review E74 (4)(2006)046218. The present work provides a comprehensive review of stateof-the-art researches on the SD oscillator begining with the complex nonlinear dynamics under the smooth and discontinuous cases, including the fundamental dynamical characteristics of the unperturbed system, perturbed bifurcations, chaotic motions and also the coexistence of multiple atrractors. Then, the work lists several extended oscillators with irrational type of nonlinear restoring forces based upon the SD oscillator. Finally, the work details the applications of the SD oscillator in engineering fields especially in vibration isolation and energy harvesting by means of the features of negative stiffness and the multiple stabilities. This review work shows the importance of irrational nonlinear restoring forces controlled by geometrical parameters in the engineering structures designing. The concluding remarks suggest further promising directions, such as the dynamics near local and global bifurcation with high co-dimension caused by the increasing geometrical parameters, the construction of universal unfoldings and irrational elliptic function for the situation of multiple geometrical parameters, the design of novel model with geometrical nonlinearity for engineering application and the improvement of experimental method for oscillators with irrational nonlinearity.