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 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 ) . 10 Figure 2: Heteroclinic solution of (4.8). The x$_{1}$$_{−}$ coordinates of φ$_{˜}$ $_{η}$( t ) , φ$_{η}$ 1 ( t ) and φ$_{η}$ 2 ( t ) are represented in blue, red and green colors, respectively. The figure confirms that φ$_{˜}$ $_{η}$( t ) is heteroclinic to the periodic solutions φ$_{η}$ 1 ( t ) , φ$_{η}$ 2 ( t ) . Figure 1: Homoclinic solution of (4.8). The x$_{1}$$_{−}$ coordinates of φ$_{η}$ ( t ) and φ$_{η}$ ∗ ( t ) are shown in blue and red colors, respectively. The figure manifests that φ$_{η}$ ( t ) is homoclinic to φ$_{η}$ ∗ ( t ) . η = { . . . , h 3 $_{2}$( η$_{0}$ ) , h 2 $_{2}$( η$_{0}$ ) , h$_{2}$ ( η$_{0}$ ) , η$_{0}$, F$_{µ}$ ( η$_{0}$ ) , F 2 $_{µ}$( η$_{0}$ ) , F 3 $_{µ}$( η$_{0}$ ) , . . . } , ˜ η = { . . . , h 3 $^{$_{1}$(}$˜ η$_{0}$ ) , h 2 $^{$_{1}$(}$˜ η$_{0}$ ) , h$_{1}$ $^{(}$˜ η$_{0}$ ) , ˜ η$_{0}$, F$_{µ}$ $^{(}$˜ η$_{0}$ ) , F 2 $^{$_{µ}$(}$˜ η$_{0}$ ) , F 3 $^{$_{µ}$(}$˜ η$_{0}$ ) , . . . } , −15 −10 −5 0 5 10 15 20 25 30 0 0.5 1 1.5 t x 1 −20 −15 −10 −5 0 5 10 15 20 25 30 −1 0 1 2 t x 1