the corresponding linearization of Ψ$_{0}$( t ) is the system (5.78) . The monodromy matrix of (5.78) can be evaluated as It can be easily observed that the discontinuous solution intersects the line x$_{1}$ = 0 . For this reason, one can specify the first coordinate of the initial value as ζ$_{1}$ = ζ 0 1 ≡ 0 . In the light of these discussions and the formula (5.58) , the following equations are obtained: ∣ $_{2}$, 0) ∂ T ∂ S $_{1}$( ω, 0 , ζ 0 $_{2}$, 0) ∂ζ$_{2}$ ∂ S $_{2}$( ω, 0 , ζ 0 $_{2}$, 0) ∂ T ∂ S $_{2}$( ω, 0 , ζ 0 $_{2}$, 0) ∂ζ$_{2}$ ∣ = ∣ ∣$^{0}$ − 0 . 0006 1 0 . 0009 ∣ = − 0 . 0006 = 0 . (5.86)    0 . 939 − 0 . 00052 − 0 . 000427 1    . (5.84) S $_{1}$( T , 0 , ζ$_{2}$, µ ) = x$_{1}$ ( T , 0 , ζ$_{2}$, µ ) = 0 , S $_{2}$( T , 0 , ζ$_{2}$, µ ) = x$_{2}$ ( T , 0 , ζ$_{2}$, µ ) − ζ$_{2}$ = 0 . (5.85) We have obtained regular behavior in dynamics near grazing orbits by the Poincar ` e small parameter analysis. Nevertheless, outside the attractors irregular phenomena may be observed. In Figure 9, the solutions of the system (5.69) with parameter µ = − 0 . 2 are depicted through simula- tions. The red arcs are the trajectory of the system (5.69) with initial value (0 . 7 , 0 . 05) and the blue arcs For each fixed µ = 0 , solutions near to the periodic ones intersect the line of discontinuity Γ transver- sally once during the time approximately equal to the period. That is, the smoothness which is requested for the application of the Poincar ` e condition is valid, since the smoothness for the grazing point has already been verified. It is clear that there can not be another solutions with period close to 2 π. Thus, one can make the following conclusion. The original system (5.69) admits two orbitally stable periodic solutions, continuous and discontinuous, if µ < 0 . There is a single orbitally stable continuous solution (grazing) if µ = 0 . Additionally, there is a unique discontinuous orbitally stable periodic solution for positive values of the parameter. Consequently, grazing bifurcation of cycles appears for the system with small parameter. Thus, condition ( A 6) holds. Then, utilizing Theorem 5.1, it is easy to conclude that for sufficiently small µ there exists a unique periodic solution of the system (5.69) with a period ≈ 2 π. It is true that for positive as well as negative µ. Moreover, these solutions are orbitally asymptotically stable because of the continuous dependence of solutions on parameter and initial values and they meet the discontinuity line transversally. Then, taking the derivative of the system (5.85) with respect to T and ζ$_{2}$, and calculating it at T = ω, ζ$_{2}$ = ζ 0 2 = 0 , and for µ = 0 , the following determinant is obtained 39