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where η$_{0}$ = 1 / 3 . 9 , is homoclinic to the fixed point η ∗ = 2 . 9 / 3 . 9 of (4.4). Denote by ϕ$_{η}$ ( t ) and ϕ$_{η}$ ∗ ( t ) the bounded solutions of (4.3) corresponding to η and η $^{∗}$, respectively. Theorem 3.1 implies that ϕ$_{η}$ ( t ) is homoclinic to ϕ$_{η}$ ∗ ( t ) . Notice that the bounded solution ϕ$_{η}$ ∗ ( t ) is $_{8}$-periodic. We depict in Figure 2 the y$_{2}$ coordinates of ϕ$_{η}$ ( t ) and ϕ$_{η}$ ∗ ( t ) . Figure 2 supports the result of Theorem 3.1 such that ϕ$_{η}$ ( t ) is homoclinic to ϕ$_{η}$ ∗ ( t ) . 

On the other hand, the orbit 

where the function γ$_{2}$ is defined as γ$_{2}$ ( υ ) = 1 2 ( 1 − √ 1 − 4 υ 3 . 9 ) and η$_{0}$ = 1 / 3 . 9 , is heteroclinic to the fixed points η ∗ = 2 . 9 / 3 . 9 and η ∗∗ = 0 of (4.4) according to the results of (Avrutin et al., 2015). Suppose that ϕ$_{η}$ ( t ) , ϕ$_{η}$ ∗ ( t ) and ϕ$_{η}$ ∗∗ ( t ) are the bounded solutions of (4.3) corresponding to the sequences η, η ∗ and η $^{∗∗}$, respectively. One can confirm by using Theorem 3.1 that the solution ϕ$_{η}$ ( t ) is heteroclinic to ϕ$_{η}$ ∗ ( t ) , ϕ$_{η}$ ∗∗ ( t ) . The y$_{2}$ coordinates of ϕ$_{η}$ ( t ) , ϕ$_{η}$ ∗ ( t ) and ϕ$_{η}$ ∗∗ ( t ) are represented in Figure 3, which reveals that ‖ ϕ$_{η}$ ( t ) − ϕ$_{η}$ ∗ ( t ) ‖ → 0 as t → ∞ , t ∈ T$_{0}$ and ‖ ϕ$_{η}$ ( t ) − ϕ$_{η}$ ∗∗ ( t ) ‖ → 0 as t → −∞ , t ∈ T$_{0}$ . 

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