∂W$_{i}$ ( x ( ξ, 0 , x ∗ + Δ x )) ∂x 0 1 =   $^{−}$$^{√$_{2 + 0}$$_{.}$$_{22}$}$ $_{2}$√$_{2}$ √$_{2 + 0}$$_{.}$$_{22}$ $_{2}$√$_{2}$    . (4.52) ∂W$_{i}$ ( x ( ξ, 0 , x ∗ + Δ x )) ∂x 0 2 =    √$_{2 + 0}$$_{.}$$_{22}$ $_{2}$√$_{2}$ − √$_{2 + 0}$$_{.}$$_{22}$ $_{2}$√$_{2}$    . (4.53) W$_{ix}$ ( x ∗ ) =   $^{−}$$^{√$_{2 + 0}$$_{.}$$_{22}$}$ $_{2}$√$_{2}$ √$_{2 + 0}$$_{.}$$_{22}$ $_{2}$√$_{2}$ √$_{2 + 0}$$_{.}$$_{22}$ $_{2}$√$_{2}$ − √$_{2 + 0}$$_{.}$$_{22}$ $_{2}$√$_{2}$    . (4.54) Using similar method with that of the first one, the second derivative can be computed as Combining (4.52) and (4.53) , we can obtain the following matrix for the linearization at the grazing point x ∗ , It is appearant that the matrix W$_{ix}$ ( x ∗ ) is continuous with respect to its arguments, since it is constant if the point x ∗ +Δ x is not from the orbit of the grazing solution. Since of the limit procedure, it is the same constant for all points of the grazing solution. Thus, the Jacobian is constant matrix in a neighborhood of the grazing point and condition ( A 2) is valid. Now, let us check the validity of the condition ( A 3) . Consider a near solution x ( t ) = x ( t, 0 , ¯ x ) , to the grazing cycle Ψ( t ) , where ¯ x = (0 , ¯ x$_{2}$ ) , ¯ x$_{2}$ > Ψ$_{2}$(0) = 1 . So, the near solution x ( t ) satisfies the condition ( N 1) . For the grazing periodic solution, it is true that θ$_{i}$$_{+1}$ − θ$_{i}$ = 2 π = ω. The grazing solution Ψ( t ) = x ( t, 0 , (0 , 1)) , touches the line of discontinuity Γ at t = ω $_{8}$. The first coordinate of the near solution is x$_{1}$ ( t ) = ¯ x$_{2}$ sin( t ) , and x$_{1}$ ( ω 8 ) = ¯ x$_{2}$ sin( ω 8 ) = ¯ x$_{2}$ √$_{2}$ > Ψ$_{1}$( ω 8 ) = 1 √$_{2}$ . Consequently, the near solution x ( t ) meets the line of discontinuity Γ before the moment ω $_{8}$. This implies that 0 < τ ( x ) < π 4 − ϵ, for a small positive ϵ whenever x$_{1}$ ( t ) is close to 1 √$_{2}$ . Thus, the condition ( A 3) is valid and Lemma 3.1 proves condition ( C ) . In the light of the above discussion, the bivalued matrix of coefficients for the grazing point is easily obtained as 29