than unity. Hence, there exists real nonsingular matrix M, which satisfies The remaining part of the proof is same as proof of Lemma 5.1.1 in [17]. □ Throughout this section, we will assume that ( A 4) is valid. That is, the variational system (3.13) consists of m periodic subsystems. For each of these systems, we find the matrix of monodromy, U$_{j}$ ( ω ) and denote corresponding Floquet multipliers by ρ ( j ) i , i = 1 , . . . , n, j = 1 , . . . , m. In the next part of the paper, the following assumption is needed. (A5) ρ ( j ) 1 = 1 and | ρ ( j ) i | < 1 , i = 2 , . . . , n for each j = 1 , . . . , m. Lemma 4.2 Assume that the assumptions (A4) and (A5) are valid. Then, for each j = 1 , . . . , m, the system (3.13) admits a fundamental matrix of the form where P$_{j}$ ∈ P C $^{1}$( $^{R}$, ζ ) is a regular, ω -periodic matrix and H$_{j}$ is an ( n − 1) × ( n − 1) − matrix with all eigenvalues have negative real parts. The proof of Lemma 4.2, can be done similar to that of Lemma 4.1. Theorem 4.1 Assume that conditions ( C 1) − ( C 7) , ( C 10) , and the assumptions ( A 1) − ( A 5) hold. Then ω - periodic solution Ψ( t ) of (2 . 1) is orbitally asymptotically stable and has the asymptotic phase property. Proof. Since of the group property, we may assume Ψ(0) is not a discontinuity point. Then, one can displace the origin to the point Ψ(0) , and the coordinate system can be rotated in such a way that the tangent vector Ψ ′ 0 = Ψ ′ (0) points in the direction of the positive x$_{1}$ axis i.e. the coordinates of this vector are Ψ ′ 0 = (Ψ ′ $_{01}$, 0 , . . . , 0) , Ψ ′ 01 > 0 . Let θ$_{i}$, i ∈ $^{Z}$, be the discontinuity moments of Ψ( t ) . Denote the path of the solution by η = { x ∈ X : x = Ψ( t ) , t ∈ $^{R}$} . There exists a natural number p, such that θ$_{i}$$_{+}$$_{p}$ = θ$_{i}$ + ω for all i. Because of condi- tions ( C 1) − ( C 7) and K − differentiability of Ψ( t ) there exists continuous dependence on initial data and consequently there exists a neighborhood of η such that any solutions which starts in the set will have moments of discontinuity which constitute a B − sequence with difference between neighbors approxi- mately equal to the distance between corresponding neighbor moments of discontinuity of the periodic solution Ψ( t ) . Consequently we can determine variational system for Ψ( t ) , with points of discontinuity θ$_{i}$, i ∈ $^{Z}$. On the basis of discussion in Section 2 . 1 , one can define in the neighborhood of η a B − equivalent system of type (2.4). The variational system of it takes the form $_{M}$− $^{1}$exp(Λ ω ) M =   $^{1}$ 0 0 $_{C$_{1}$}$   . U$_{j}$ ( t ) = P$_{j}$ ( t )[1 , exp( H$_{j}$ω )] , t ∈ $^{R}$, (4.34) 22