The solution ¯ x ( t ) has a linerization with respect to solution x$_{0}$ ( t ) if the condition ( A ) is valid and, moreover, if the point x$_{0}$ ( θ$_{i}$ ) is of ( α ) − or ( β ) − type, then the condition ( B ) is fulfilled. For the case x$_{0}$ ( θ$_{i}$ ) is of ( γ ) − type the condition ( C ) is true. The solution x$_{0}$ ( t ) is K − differentiable with respect to the initial value x$_{0}$ on [0 , b ] if for each solution ¯ x ( t ) with sufficiently small Δ x the linearization exists. The functions u$_{i}$ ( t ) and ν$_{ij}$ depend on Δ x and uniformly bounded on a neighborhood of x$_{0}$. It is easy to see that the differentiability implies B − continuous dependence on solutions to initial data. Define the map ζ ( t, x ) as ζ ( t, x ) = x ( t, 0 , x ) , for x ∈ D. A K -smooth discontinuous flow is a map ζ ( t, x ) : $^{R}$× D → D, which satisfies the following properties: In [1], it was proved that if the conditions of Theorem 2.1 and (C1)-(C10) are fulfilled, then system (2.1) defines a B -smooth discontinuous flow [1] if there is no grazing points for the dynamics. It is easy to observe that the B -smooth discontinuous flow is a subcase of the K -smooth discontinuous flow. In the next section, we will construct a variational system for (2.1) in the neighborhood of grazing orbits. That is, we will assume that some of the discontinuity points are ( γ ) − type points. Linearization around a solution and its stability will be taken into account. Thus, analysis of the discontinuous dynamical systems with grazing points will be completed. The object of this section is to verify K − differentiability of the grazing solution. Consider a grazing solution x$_{0}$ ( t ) = x ( t, 0 , x$_{0}$ ) , x$_{0}$ ∈ D, of (2.1). We will demonstrate that one can write the variational system for the solution x$_{0}$ ( t ) as follows: 3 Linearization around grazing orbits and discontinuous dynam- ics u ′ = A ( t ) u, Δ u | t = θ$_{i}$ = B$_{i}$u ( θ$_{i}$ ) , (3.9) (I) The group property: (i) ζ (0 , x ) : D → D is the identity; (ii) ζ ( t, ζ ( s, x )) = ζ ( t + s, x ) is valid for all t, s ∈ $^{R}$and x ∈ D. (II) ζ ( t, x ) ∈ P C $^{1}$( $^{R}$) for each fixed x ∈ D. (III) ζ ( t, x ) is K -differentiable in x ∈ D on [ a, b ] ⊂ $^{R}$for each a, b such that the discontinuity points of ζ ( t, x ) are interior points of [ a, b ] . 9