  As the parameters of a piecewise-smooth system of ODEs are varied, a periodic
orbit undergoes a bifurcation when it collides with a surface where the system
is discontinuous. Under certain conditions this is a grazing-sliding
bifurcation. Near grazing-sliding bifurcations structurally stable dynamics are
captured by piecewise-linear continuous maps. Recently it was shown that maps
of this class can have infinitely many asymptotically stable periodic solutions
of a simple type. Here this result is used to show that at a grazing-sliding
bifurcation an asymptotically stable periodic orbit can bifurcate into
infinitely many asymptotically stable periodic orbits. For an abstract ODE
system the periodic orbits are continued numerically revealing subsequent
bifurcations at which they are destroyed.
