z ( θ$_{k}$ +) − z ( θ$_{k}$ ) = Y$_{j}$ ( θ$_{k}$ +) a + θ$_{k}$ ∫ 0 G ( j ) 1 ( θ$_{k}$ + , s ) r ( s, z ( s )) ds − ∞ ∫ θ$_{k}$ G ( j ) 2 ( θ$_{k}$ + , s ) r ( s, z ( s )) ds + ∑ $^{0}$≤ θ$_{i}$<θ$_{k}$ G ( j ) 1 ( θ$_{k}$ + , θ$_{i}$ +) q$_{i}$ ( z ( θ$_{i}$ +)) − ∑ θ$_{k}$<θ$_{i}$$^{<}$∞$^{G}$$^{(}$$^{j}$$^{)}$ 2 ( θ$_{k}$ + , θ$_{i}$ +) q$_{i}$ ( z ( θ$_{i}$ +)) − Y$_{j}$ ( θ$_{k}$ ) a − θ$_{k}$ ∫ 0 G ( j ) 1 ( θ$_{k}$, s ) r ( s, z ( s )) ds + ∞ ∫ θ$_{k}$ G ( j ) 2 ( θ$_{k}$, s ) r ( s, z ( s )) ds − ∑ $^{0}$≤ θ$_{i}$<θ$_{k}$ G ( j ) 1 ( θ$_{k}$, θ$_{i}$ +) q$_{i}$ ( z ( θ$_{i}$ )) + ∑ $^{θ$_{k}$}$≤θ$_{i}$$^{<}$∞$^{G}$$^{(}$$^{j}$$^{)}$ 2 ( θ$_{k}$, θ$_{i}$ +) q$_{i}$ ( z ( θ$_{i}$ +)) = D ( j ) k z ( θ$_{k}$ ) + q$_{k}$ ( z ( θ$_{k}$ )) . z ( j $^{)}$(0 , a ( j $^{)}$) = Y$_{j}$ (0) a ( j ) − ∞ ∫ 0 G ( j ) 2 (0 , s ) r ( s, z ( s )) ds − ∑ 0 <θ$_{k}$$^{<}$∞$^{G}$$^{(}$$^{j}$$^{)}$ 2 (0 , θ$_{k}$ +) q$_{k}$ ( z ( θ$_{k}$ )) = P$_{j}$ (0) a ( j ) − P$_{j}$ (0)    1 0 0 $^{O$_{n}$}$− 1    ∞ ∫ 0 $_{P}$− 1 j ( s ) r ( s, z ( s )) ds − ∑ 0 <θ$_{k}$$^{<}$∞ $_{P}$− 1 j ( s ) q$_{k}$ ( z ( θ$_{k}$ )) . x$_{1}$ + n ∑ i =2 c j $_{i}$x$_{i}$ − h$_{j}$ ( x$_{2}$, . . . , x$_{n}$ ) = 0 , (4.49) The above discussion proves that z ( j $^{)}$( t, a ) , j = 1 , , 2 , . . . , m, are bounded solutions of system (4.35). We will determine the initial values of bounded solutions in terms of ( n − 1) parameters a ( j ) 2 , . . . , a ( j ) n , j = 1 , , 2 , . . . , m. Denote a ( j ) = [0 , a j $_{2}$, a j $_{3}$, . . . , a j $_{n}$] . By using (4.41), we obtain In the way utilized in [17], one can show that the coordinates of the initial value ( x$_{1}$, . . . , x$_{n}$ ) ∈ D of the solution z ( j ) satisfy the equation where h$_{j}$ ∈ C $^{1}$, j = 1 , , 2 , . . . , m. One can see that equation (4.49) determines ( n − 1) dimensional hypersurfaces S j ⊂ D, j = 1 , 2 , . . . , m, in a neighborhood of the origin such that each solution which starts at the surface satisfies inequality (4.48). From the analytical representation, it follows that the equation of the tangent space of S j at the origin is described by the equation x$_{1}$ + n ∑ i =2 c j $_{i}$x$_{i}$ and the first coordinate of the gradient of the left hand side in (4.49) is unity. Moreover, the path η intersects S j transversely. This and condition ( A 4) imply that the path of every solution φ ( t ) near Ψ( t ) intersects one of the manifolds S $^{j}$, j = 1 , 2 , . . . , m, at some ¯ t ∈ [0 , 2 ω ] . Because of the continuous dependence on initial values, a δ ( ϵ ) > 0 exists for a given ϵ > 0 , such that if dist ( x $^{0}$, η$_{δ}$ ) < δ ( ϵ ) , then the solution φ ( t, x $^{0}$) is defined on [0 , 2 ω ] , and dist ( φ ( t, x $^{0}$) , η ) < ϵ ≤ ϵ$_{1}$ for t ∈ [0 , 2 T ] . Therefore, the path of φ ( t, x $^{0}$) intersects S j for some j = 1 , 2 , . . . , m and t$_{1}$ ∈ [0 , 2 ω ] . The 26