Aspects of bifurcation theory for piecewise-smooth, continuous systems

Systems that are not smooth can undergo bifurcations that are forbidden in smooth systems. We review some of the phenomena that can occur for piecewise-smooth, continuous maps and flows when a fixed point or an equilibrium collides with a surface on which the system is not smooth. Much of our understanding of these cases relies on a reduction to piecewise linearity near the border collision. We also review a number of codimension-two bifurcations in which nonlinearity is important.