B$_{1}$ =     $^{O$_{2}$}$$^{,}$ if (N1) is valid ,    − 1 0 0 . 001 − 1 . 8 exp(0 . 00025 π ) 0    , if (N2) is valid . The matrix D (1) 1 = O$_{2}$ is for near solutions of (3.15) which are in the region where Z in, see Fig. 1, and do not intersect the curve of discontinuity Γ$_{1}$ . The matrix is for near solutions of (3.15), which intersects the curve of discontinuity Γ$_{1}$ . They start in the subregion, where the point Y is placed. Thus, the linearization for Ψ( t ) at the grazing point exists. Moreover, since another point of discontinuity (0 , exp(0 . 0005 π )) is not grazing, the linearization at the point exist as well as linearization at points of continuity [1, 40]. Consequently, there exist linearization around Ψ( t ) . To verify condition ( A 3) , consider a near solution y ( t ) = y ( t, 0 , ¯ y ) to Ψ( t ) , where ¯ y = (0 , ¯ y$_{2}$ ) , ¯ y$_{2}$ > Ψ$_{2}$(0) = 1 , which satisfy the condition ( N 1) . It is true that θ$_{i}$$_{+1}$ − θ$_{i}$ = π 2 = ω 2 . The first coordinate of the near solution is y$_{1}$ ( t ) = ¯ y exp(0 . 0005 t ) sin( t ) and y$_{1}$ ( ω $_{2}$) = y$_{1}$ ( π 2 ) = ¯ y exp(0 . 00025 π ) > exp(0 . 00025 π ) = Ψ$_{1}$( ω 2 ) . Thus, the meeting moment of near solution y ( t ) with the surface of discontinuity is less than ω $_{2}$. So, it implies that 0 < τ ( y ) < π 2 − ϵ for a small number ϵ if the first coordinate of ¯ y is close to exp(0 . 00025 π ) . This validates condition ( A 3) . Now, Lemma 3.1 proves the condition ( C ) . Now, let us consider the point (0 , 0) . We have that 〈∇ Φ((0 , 0)) , f ((0 , 0)) 〉 = 〈 (1 , 0) , (0 , 0) 〉 = 0 . That is the origin is a grazing point. In the same time it is a fixed point of the system. For this particular grazing point, we can find the linearization directly. Indeed, all the near solutions satisfy the linear impulsive system, Consider a solution x ( t ) = x ( t, 0 , x$_{0}$ ) , where x$_{0}$ = ( x 0 $_{1}$, x 0 $_{2}$) = (0 , 0) with moments of discontinuity Now, on the basis of the discussion made above, one can obtain the bivalued matrix of coefficients for the grazing point as The continuity of the derivatives in a neighborhood of y ∗ implies that the function W is differentiable at the grazing point y = y ∗ , and the condition ( A 2) is valid. D (2) 1 =    − 1 0 0 . 001 − 1 . 8 exp(0 . 00025 π ) $_{0}$   $_{x}$′ 1 = x$_{2}$, $_{x}$′ 2 = − x$_{1}$ + 0 . 001 x$_{2}$, Δ x$_{2}$ | x$_{1}$ =0 = − (1 + R$_{2}$ ) x$_{2}$. (3.27) 18