one can verify for t ≤ θ$_{k}$$_{0}$ that Therefore, The last inequality implies that sup $^{t}$≤θ$_{k}$$_{0}$ ‖ φ$_{η}$ ( t ) − φ$_{ζ}$ ( t ) ‖ < ϵ. Consequently, and φ$_{η}$ ( t ) belongs to W $^{u}$( φ$_{ζ}$ ( t )) . □ The main result of the present paper is mentioned in the following theorem, which can be proved by using the results of Lemma 3.1 and Lemma 3.2. Theorem 3.1 Under the conditions ( C 1) − ( C 7) , the following assertions are valid. The next section is devoted to examples concerning homoclinic and heteroclinic motions in an im- pulsive Duffing equation. Let us take into account the impulsive Duffing equation 8 4 Examples x ′′ + 0 . 2 x ′ + 0 . 81 x + 0 . 001 x 3 = 0 . 7 cos ( 2 π 3 t ) + g ( t, ζ ) , t = θ$_{k}$, Δ x | t = θ$_{k}$ = − 0 . 12 x + 0 . 09 + ζ$_{k}$, Δ x $^{′}$| t = θ$_{k}$ = − 0 . 12 x ′ + 0 . 015 sin( x ) , (4.6) lim $^{t}$→−∞ ‖ φ$_{η}$ ( t ) − φ$_{ζ}$ ( t ) ‖ = 0 , ( 1 − N L$_{f}$ ω − N L$_{J}$ 1 − e − ωθ ) sup $^{t}$≤θ$_{k}$$_{0}$ ‖ φ$_{η}$ ( t ) − φ$_{ζ}$ ( t ) ‖ ≤ N ϵ λ ( 1 ω + 1 1 − e − ωθ ) . ‖ φ$_{η}$ ( t ) − φ$_{ζ}$ ( t ) ‖ < ∫ t −∞ N e − ω ( $^{t}$−s ) ( L$_{f}$ ‖ φ$_{η}$ ( s ) − φ$_{ζ}$ ( s ) ‖ + ϵ λ ) ds + ∑ −∞ <θ$_{k}$<t $_{N e}$− ω ( $^{t}$−θ$_{k}$ ) ( L$_{J}$ ‖ φ$_{η}$ ( θ$_{k}$ ) − φ$_{ζ}$ ( θ$_{k}$ ) ‖ + ϵ λ ) ≤ N ω ( L$_{f}$ sup $^{t}$≤θ$_{k}$$_{0}$ ‖ φ$_{η}$ ( t ) − φ$_{ζ}$ ( t ) ‖ + ϵ λ ) + N 1 − e − ωθ ( L$_{J}$ sup $^{t}$≤θ$_{k}$$_{0}$ ‖ φ$_{η}$ ( t ) − φ$_{ζ}$ ( t ) ‖ + ϵ λ ) . (i) If η ∈ Θ is homoclinic to ζ ∈ Θ , then φ$_{η}$ ( t ) ∈ A is homoclinic to φ$_{ζ}$ ( t ) ∈ A ; (ii) If η ∈ Θ is heteroclinic to ζ $^{1}$, ζ2 ∈ Θ , then φ$_{η}$ ( t ) ∈ A is heteroclinic to φ$_{ζ}$ 1 ( t ) , φ$_{ζ}$ 2 ( t ) ∈ A ; (iii) If Θ is hyperbolic, then the same is true for A .