  Discontinuous dynamical systems with grazing solutions are discussed. The
group property, continuation of solutions, continuity and smoothness of motions
are thoroughly analyzed. A variational system around a grazing solution which
depends on near solutions is constructed. Orbital stability of grazing cycles
is examined by linearization. Small parameter method is extended for analysis
of neighborhoods of grazing orbits, and grazing bifurcation of cycles is
observed in an example. Linearization around an equilibrium grazing point is
discussed. The mathematical background of the study relies on the theory of
discontinuous dynamical systems [1]. Our approach is analogous to that one of
the continuous dynamics analysis and results can be extended on functional
differential, partial differential equations and others. Appropriate
illustrations with grazing limit cycles and bifurcations are depicted to
support the theoretical results.
