the linear system (3.14a), one can easily conclude that the complement is a continuous dynamical system [23]. Thus, to verify the dynamics for the whole system, one need to analyze it in the region G. This set is bounded, consequently for solutions in it conditions ( C 8) and ( C 9) are fulfilled and by Theorem 2.1, they admit B − sequences and continuation property. Consider a function ζ ( y$_{2}$ ) : [ y$_{2}$ ( θ$_{2}$, 0 , Q$_{2}$ ) , y$_{2}$ ( θ$_{1}$, 0 , P )] → [ y$_{2}$ ( θ$_{2}$, 0 , Q$_{2}$ ) , y$_{2}$ ( θ$_{1}$, 0 , P )] such that it is continuously differentiable, satisfies ζ ( y$_{2}$ ) = − R$_{2}$y$_{2}$ in a neighborhood of y$_{2}$ = 0 and is the identity at the boundary points, i.e. ζ ( y$_{2}$ ( θ$_{1}$, 0 , P )) = y$_{2}$ ( θ$_{1}$, 0 , P ) and ζ ( y$_{2}$ ( θ$_{1}$, 0 , P )) = y$_{2}$ ( θ$_{1}$, 0 , P ) . It is easily seen that such function exists. On the basis of this discussion, let us introduce the following system, It is apparent that system (3.16) is equivalent to (3.14) near the orbit of periodic solution Ψ( t ) . That is, they have the same trajectories there. Specifying (2.1) for (3.16), it is easy to obtain that Φ( y$_{1}$, y$_{2}$ ) = ˜ Φ( y$_{1}$, y$_{2}$ ) = ( y$_{1}$ − exp(0 . 00025 π )) y$_{1}$, f ( y$_{1}$, y$_{2}$ ) = ( y$_{2}$, − y$_{1}$ + 0 . 001 y$_{2}$ ) and J ( y ) = ( y$_{1}$, ζ ( y$_{2}$ )) . Now, we will verify that system (3.16) defines a K − smooth discontinuous flow. First, condition ( C 1) is verified since ∇ Φ$_{1}$( y ) = ∇ Φ$_{2}$( y ) = (1 , 0) = 0 , for all y ∈ D. The jump function J ( y ) = ( y$_{1}$, ζ ( y$_{2}$ )) is continuously differentiable function. So, condition ( C 2) is valid. It is true that Γ ∩ ˜ Γ ⊆ ∂ Γ ∩ ˜ ∂ Γ . In- equalities 〈∇ Φ$_{1}$( y ) , f ( y ) 〉 = 〈 (1 , 0) , ( y$_{2}$, − y$_{1}$ + 0 . 001 y$_{2}$ ) 〉 = y$_{2}$ = 0 and 〈∇ Φ$_{2}$( y ) , f ( y ) 〉 = 〈 (1 , 0) , ( y$_{2}$, − y$_{1}$ + 0 . 001 y$_{2}$ ) 〉 = y$_{2}$ = 0 , if y ∈ Γ \ ∂ Γ , validate the condition ( C 4) . Moreover, 〈∇ ˜ Φ$_{1}$( y ) , f ( y ) 〉 = 〈 (1 , 0) , ( y$_{2}$, − y$_{1}$ + 0 . 001 y$_{2}$ ) 〉 = y$_{2}$ = 0 and 〈∇ ˜ Φ$_{2}$( y ) , f ( y ) 〉 = 〈 (1 , 0) , ( y$_{2}$, − y$_{1}$ + 0 . 001 y$_{2}$ ) 〉 = y$_{2}$ = 0 , if y ∈ ˜ Γ \ ∂ ˜ Γ . Conditions ( C 6) and ( C 7) hold as the function ζ is such defined. Thus, conditions ( C 1) − ( C 10) have been verified. Consequently, the system (3.14) defines the K − smooth discontinuous flow for all motions except the grazing ones. To complete the discussion, one need to linearize the system near the grazing solutions. First, we proceed with the linearization around the grazing periodic orbit (3.15). The solution, Ψ( t ) has two discontinuity moments θ$_{1}$ = π 2 and θ$_{2}$ = ω in the interval [0 , ω ] . The corresponding discontinuity points are of ( γ ) − and ( α ) − types, respectively. Next, we will linearize the system at these points. The linearization at the second point exists [1] and the details of this will be analyzed in the next example. This time, we will focus on the grazing point y ∗ . First, we assume that y ( t ) = y ( t, 0 , y ∗ + Δ y ) , Δ y = (Δ y$_{1}$, Δ y$_{2}$ ) is not a grazing solution. Moreover, the solution intersects the line Γ$_{1}$ at time t = ξ near t = θ$_{1}$ as time increases. The meeting point ¯ y = (¯ y$_{1}$, ¯ y$_{2}$ ) = ( y$_{1}$ ( ξ, 0 , ( y ∗ + Δ y )) , y$_{2}$ ( ξ, 0 , ( y ∗ + Δ y )) , is transversal one. It is clear ¯ y$_{1}$ = exp(0 . 00025 π ) and ¯ y$_{2}$ > 0 . In order to find a linearization at the moment t = θ$_{i}$, we use formula (2.3) for y ( t ) , and find 15 y ′ 1 = y$_{2}$, y ′ 2 = − y$_{1}$ + 0 . 001 y$_{2}$, Δ y$_{2}$ | $^{y}$∈ Γ = ζ ( y$_{2}$ ) − y$_{2}$. (3.16)