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Linear complementarity problems provide a powerful framework to modelnonsmooth phenomena in a variety of real-world applications. In dynamicalcontrol systems, they appear coupled to a linear input-output system in theform of linear complementarity systems. Mimicking the program that led to thefoundation of bifurcation theory in smooth maps, we introduce a novel notion ofequivalence between linear complementarity problems that sets the basis for atheory of bifurcations in a large class of nonsmooth maps, including, but notrestricted to, steadystate bifurcations in linear complementarity systems. Ourdefinition exploits the rich geometry of linear complementarity problems andleads to constructive algebraic conditions for identifying and classifying thenonsmooth singularities associated with nonsmooth bifurcations. We thoroughlyillustrate our theory on an extended applied example, the design of bistabilityin an electrical network, and a more theoretical one, the identification andclassification of all possible equivalence classes in two-dimensional linearcomplementarity problems.

Equivalence of Linear Complementarity Problems: Theory and Application to Nonsmooth Bifurcations