Moreover, if ( C 6) additionally holds, then by using the Gronwall’s Lemma for piecewise continuous functions (Akhmet, 2010b; Samoilenko & Perestyuk, 1995) one can confirm that the bounded solution ϕ$_{ζ}$ ( t ) attracts all other solutions of (1.1) for a fixed sequence ζ, that is, ‖ y ( t ) − ϕ$_{ζ}$ ( t ) ‖ → 0 as t → ∞ , t ∈ T$_{0}$ , where y ( t ) , y ( t $^{0}$) = y$_{0}$, is a solution of (1.1) for some t 0 ∈ T$_{0}$ and y$_{0}$ ∈ R $^{n}$. In the next section, we will investigate the existence of homoclinic and heteroclinic motions in the dynamics of (1.1). Denote by H the set of all sequences ζ = { ζ$_{k}$ }$_{k}$$_{∈}$$_{Z}$ obtained by the discrete map (1.2). The following definitions are adapted from the papers (Akhmet, 2008, 2010a). The stable set of a sequence ζ ∈ H is defined as and the unstable set of ζ is The set H is called hyperbolic if for each ζ ∈ H the stable and unstable sets of ζ contain at least one element different from ζ. A sequence η ∈ H is homoclinic to another sequence ζ ∈ H if η ∈ W $^{s}$( ζ ) ∩ W $^{u}$( ζ ) . Moreover, η ∈ H is heteroclinic to the sequences ζ $^{1}$, ζ2 ∈ H , η = ζ $^{1}$, η = ζ $^{2}$, provided that η ∈ W $^{s}$( ζ $^{1}$) ∩ W $^{u}$( ζ $^{2}$) . Suppose that D is the set consisting of all solutions ϕ$_{ζ}$ ( t ) , ζ ∈ H , of (1.1), which are bounded on T$_{0}$ . A solution ϕ$_{η}$ ( t ) ∈ D belongs to the stable set W $^{s}$( ϕ$_{ζ}$ ( t )) of ϕ$_{ζ}$ ( t ) ∈ D if ‖ ϕ$_{η}$ ( t ) − ϕ$_{ζ}$ ( t ) ‖ → 0 as t → ∞ , t ∈ T$_{0}$ . Besides, ϕ$_{η}$ ( t ) is an element of the unstable set W $^{u}$( ϕ$_{ζ}$ ( t )) of ϕ$_{ζ}$ ( t ) provided that ‖ ϕ$_{η}$ ( t ) − ϕ$_{ζ}$ ( t ) ‖ → 0 as t → −∞ , t ∈ T$_{0}$ . We say that D is hyperbolic if for each ϕ$_{ζ}$ ( t ) ∈ D the sets W $^{s}$( ϕ$_{ζ}$ ( t )) and W $^{u}$( ϕ$_{ζ}$ ( t )) contain at least one element different from ϕ$_{ζ}$ ( t ) . A solution ϕ$_{η}$ ( t ) ∈ D is homoclinic to another solution ϕ$_{ζ}$ ( t ) ∈ D if ϕ$_{η}$ ( t ) ∈ W $^{s}$( ϕ$_{ζ}$ ( t )) ∩ W $^{u}$( ϕ$_{ζ}$ ( t )) , and$_{ϕ}$$_{η}$$_{(}$$_{t}$$_{)}$ ∈ D is heteroclinic to the bounded solutions$_{ϕ}$$_{ζ}$ 1 ( t ) , ϕ$_{ζ}$ 2 ( t ) ∈ D , ϕ$_{η}$ ( t ) = ϕ$_{ζ}$ 1 ( t ) , ϕ$_{η}$ ( t ) = ϕ$_{ζ}$ 2 ( t ) , if ϕ$_{η}$ ( t ) ∈ W $^{s}$( ϕ$_{ζ}$ 1 ( t )) ∩ W $^{u}$( ϕ$_{ζ}$ 2 ( t )) . The following theorem can be proved by using the reduction technique to impulsive differential equa- tions, which was introduced in (Akhmet & Turan, 2006). Theorem 3.1 Under the conditions ( C 1) − ( C 6) , the following assertions are valid: (i) If η ∈ H is homoclinic to ζ ∈ H , then ϕ$_{η}$ ( t ) ∈ D is homoclinic to ϕ$_{ζ}$ ( t ) ∈ D ; 4 W $^{u}$( ζ ) = { η = { η$_{k}$ }$_{k}$$_{∈}$$_{Z}$ ∈ H : ‖ η$_{k}$ − ζ$_{k}$ ‖ → 0 as k → −∞ } . W $^{s}$( ζ ) = { η = { η$_{k}$ }$_{k}$$_{∈}$$_{Z}$ ∈ H : ‖ η$_{k}$ − ζ$_{k}$ ‖ → 0 as k → ∞ } , 3 Homoclinic and Heteroclinic Motions