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ζ = { ζ$_{k}$ } of (4.7) the corresponding bounded solution φ$_{ζ}$ ( t ) of (4.8) is also periodic. 

Consider the map (4.7) with µ = 3 . 9 . It was demonstrated in [8] that the orbit 

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where η$_{0}$ = 1 / 3 . 9 , is homoclinic to the fixed point η ∗ = 2 . 9 / 3 . 9 of (4.7). Denote by φ$_{η}$ ( t ) and φ$_{η}$ ∗ ( t ) the bounded solutions of (4.8) corresponding to η and η ∗ , respectively. One can conclude by using Theorem 3.1 that φ$_{η}$ ( t ) is homoclinic to the periodic solution φ$_{η}$ ∗ ( t ) . Figure 1 shows the graphs of the x$_{1}$ − coordinates of φ$_{η}$ ( t ) and φ$_{η}$ ∗ ( t ) . In the figure, the solution φ$_{η}$ ( t ) is represented in blue color, while φ$_{η}$ ∗ ( t ) is represented in red color. Figure 1 reveals that φ$_{η}$ ( t ) is homoclinic to φ$_{η}$ ∗ ( t ) , i.e., ‖ φ$_{η}$ ( t ) − φ$_{η}$ ∗ ( t ) ‖ → 0 as t → ±∞ . 

Now, we set µ = 4 in equation (4.7). According to [8], the orbit 

where ˜ η$_{0}$ = 1 / 4 , is heteroclinic to the fixed points η 1 = 3 / 4 and η 2 = 0 of (4.7). Suppose that φ$_{˜}$ $_{η}$( t ) , φ$_{η}$ 1 ( t ) and φ$_{η}$ 2 ( t ) are the bounded solutions of (4.8) corresponding to ˜ η, η 1 and η $^{2}$, respectively. Theorem 3.1 implies that φ$_{˜}$ $_{η}$( t ) is heteroclinic to the periodic solutions φ$_{η}$ 1 and φ$_{η}$ 2 . Figure 2 shows the graphs of the x$_{1}$ − coordinates of φ$_{˜}$ $_{η}$( t ) , φ$_{η}$ 1 ( t ) and φ$_{η}$ 2 ( t ) in blue, red and green colors, respectively. The figure supports Theorem 3.1 such that φ$_{˜}$ $_{η}$( t ) converges to φ$_{η}$ 1 ( t ) as time increases and converges to φ$_{η}$ 2 ( t ) as time decreases, i.e., φ$_{˜}$ $_{η}$( t ) is heteroclinic to φ$_{η}$ 1 ( t ) , φ$_{η}$ 2 ( t ) . 

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