content_image/316874.jpg content_image/316873.jpg content_image/316872.jpg content_image/316871.jpg
one can verify for t ≤ θ$_{k}$$_{0}$ that 

<img src='content_image/316874.jpg'>

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 )) . □ 

<img src='content_image/316873.jpg'>

<img src='content_image/316872.jpg'>

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. 

(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 . 

The next section is devoted to examples concerning homoclinic and heteroclinic motions in an im- pulsive Duffing equation. 

## 4 Examples 

Let us take into account the impulsive Duffing equation 

<img src='content_image/316871.jpg'>

