Influence of a square-root singularity on the behaviour of piecewise smooth maps

We consider a one-dimensional continuous piecewise smooth nonlinear map with square-root singularity. For particular choice of the parameters, this map represents the normal form of discrete-time representation of impact oscillator near grazing bifurcation. Owing to the nonlinearity, the map exhibits smooth and nonsmooth bifurcations very closely related to each other. Our main aim is to study how the nonlinearity influences the known results of nonsmooth bifurcations and the interaction between smooth and nonsmooth bifurcations considering the whole parameter space. We also explain the organizing source of several atypical bifurcation phenomena using the concept of multi-parametric bifurcation.