Bifurcations in impact systems

A regular approach to stability and bifurcation analysis in systems with mechanical collisions is proposed. It is based upon explicit formulas, expressing general solution matrices in terms of derivations of active forces as well as reaction ones. It is shown that the phenomenon of grazing impact, which was known to be a discontinuous bifurcation, can be regularized owing to the appropriate impact rule, which differs from the usual one. This results in a new classification of grazing bifurcations. In short, given periodic motion does not have to disappear: it might survive after such bifurcation and even preserve stability. A similar conclusion is valid with respect to bifurcation in systems with symmetry, though classifying conditions have another form. Mechanical examples are considered: linear oscillator with one or two stops and rigid block under periodic excitation.