solution φ ( t, φ ( t$_{1}$, x $^{0}$)) = φ ( t + t$_{1}$, x $^{0}$) has its initial value in S $^{j}$, consequently, satisfies (4.48). In the light of the B − equivalence, the corresponding solution x ( t ) , x (0) = φ (0) − Ψ(0) , of (4.35) satisfies the property that for all ϵ > 0 , there exists T ( ϵ ) such that x ( t ) is in an ϵ − neighborhood of Ψ( t ) for t ∈ [ T ( ϵ ) , ∞ ) . That is, the solution Ψ( t ) is orbitally asymptotically stable and there exists an asymptotical phase. □ Definitions of the orbital stability and an asymptotic phase as well as theorem of orbital stability for non-grazing periodic solutions are also presented in [44]. In our paper, we suggest the orbital stability theorem for grazing periodic solutions, its proof and formulate the definitions for the stability. They are different in many aspects from those provided in [44]. It is valuable that they also valid, if the solution is non-grazing. To shed light on our theoretical results, we will present the following examples. Example 4.1 We continue with the system presented in Examples 3.1 and 3.2. In Example 3.1, we verified that system (3.16)defines a K − smooth discontinuous flow in the plane and the variational system (3.29)+(3.30) around the grazing periodic solution, Ψ( t ) is approved. Using systems (3.29) and (3.30), one can evaluate the Floquet multipliers as ρ (1) 1 = 1 , ρ (1) 2 = 0 . 8551 , ρ (2) 1 = 1 and ρ (2) 2 = 0 . This verifies condition ( A 5) . The conditions ( C 1) − ( C 7) and ( C 10) are validated and the assumptions (A4) and (A5) verified. By using Theorem 4.1, we can assert that the solution, Ψ( t ) is orbitally asymptotically stable. The stability is illustrated in Fig. 4. The red one is for a trajectory of the discontinuous periodic solution (3.15) of (3.14) and the blue one is for the near solution of (3.14) with initial value y$_{0}$ = (0 . 8 , 1 . 2) . It can be observed from Fig. 4 that the blue trajectory approaches the red one as time increases. Example 4.2 (A periodic solution with a non-axial grazing). We will take into account the following Figure 4: The red discontinuous cycle of (3.14) axially grazes Γ at (0 . 00025 π ) , 0) and (0 , − exp( − 0 . 0005 π )) is an ( α ) -type point. The blue arcs are of the trajectory with initial value (0 . 8 , 1 . 2) . It can be observed that it approaches the grazing one as time increases. 27 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 −1.5 −1 −0.5 0 0.5 1 1.5 y$_{1}$(t) y 2 (t)