  Linear complementarity problems provide a powerful framework to model
nonsmooth phenomena in a variety of real-world applications. In dynamical
control systems, they appear coupled to a linear input-output system in the
form of linear complementarity systems. Mimicking the program that led to the
foundation of bifurcation theory in smooth maps, we introduce a novel notion of
equivalence between linear complementarity problems that sets the basis for a
theory of bifurcations in a large class of nonsmooth maps, including, but not
restricted to, steadystate bifurcations in linear complementarity systems. Our
definition exploits the rich geometry of linear complementarity problems and
leads to constructive algebraic conditions for identifying and classifying the
nonsmooth singularities associated with nonsmooth bifurcations. We thoroughly
illustrate our theory on an extended applied example, the design of bistability
in an electrical network, and a more theoretical one, the identification and
classification of all possible equivalence classes in two-dimensional linear
complementarity problems.
