One can confirm under the conditions ( C 1) − ( C 7) that for a fixed sequence ζ ∈ Θ , the bounded solution φ$_{ζ}$ ( t ) attracts all other solutions of (1.1), i.e., ‖ x ( t ) − φ$_{ζ}$ ( t ) ‖ → 0 as t → ∞ for any solution x ( t ) of (1.1). Moreover, for each ζ ∈ Θ . In this section, first of all, we will describe the stable, unstable and hyperbolic sets as well as the homoclinic and heteroclinic motions for both system (1.1) and the discrete map (1.2). These definitions were introduced in the papers [3, 7]. After that the existence of homoclinic and heteroclinic motions in the dynamics of (1.1) will be proved. Consider the set Θ described in the previous section once again. The stable set of a sequence ζ ∈ Θ is defined as and the unstable set of ζ is The set Θ is called hyperbolic if for each ζ ∈ Θ the stable and unstable sets of ζ contain at least one element different from ζ. A sequence η ∈ Θ is homoclinic to another sequence ζ ∈ Θ if η ∈ W $^{s}$( ζ ) ∩ W $^{u}$( ζ ) . Moreover, η ∈ Θ is heteroclinic to the sequences ζ 1 ∈ Θ , ζ 2 ∈ Θ , η = ζ $^{1}$, η = ζ $^{2}$, if η ∈ W $^{s}$( ζ $^{1}$) ∩ W $^{u}$( ζ $^{2}$) . On the other hand, let us denote by A the set consisting of all bounded on R solutions of sys- tem (1.1). A bounded solution φ$_{η}$ ( t ) ∈ A belongs to the stable set W $^{s}$( φ$_{ζ}$ ( t )) of φ$_{ζ}$ ( t ) ∈ A if ‖ φ$_{η}$ ( t ) − φ$_{ζ}$ ( t ) ‖ → 0 as t → ∞ . 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 → −∞ . We say that A is hyperbolic if for each φ$_{ζ}$ ( t ) ∈ A 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 ) ∈ A is homoclinic to another solution φ$_{ζ}$ ( t ) ∈ A if φ$_{η}$ ( t ) ∈ W $^{s}$( φ$_{ζ}$ ( t )) ∩ W $^{u}$( φ$_{ζ}$ ( t )) , and φ$_{η}$ ( t ) ∈ A is heteroclinic to the bounded solutions φ$_{ζ}$ 1 ( t ) , φ$_{ζ}$ 2 ( t ) ∈ A , 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 )) . In what follows, we will denote by i (( a, b )) the number of the terms of the sequence { θ$_{k}$ } , k ∈ Z , which belong to the interval ( a, b ) , where a and b are real numbers such that a < b. It is worth noting that i (( a, b )) ≤ 1 + b − a θ . 4 W $^{u}$( ζ ) = { η ∈ Θ | ‖ η$_{k}$ − ζ$_{k}$ ‖ → 0 as k → −∞} . W $^{s}$( ζ ) = { η ∈ Θ | ‖ η$_{k}$ − ζ$_{k}$ ‖ → 0 as k → ∞} , sup $^{t}$∈ $_{R}$‖ φ$_{ζ}$ ( t ) ‖ ≤ N ( M$_{f}$ + M$_{F}$ ω + M$_{J}$ + M$_{F}$ 1 − e − ωθ ) 3 Homoclinic and heteroclinic motions