content_image/304070.jpg content_image/304069.jpg content_image/304068.jpg content_image/304067.jpg
(ii) If η ∈ H is heteroclinic to ζ $^{1}$, ζ2 ∈ H , then ϕ$_{η}$ ( t ) ∈ D is heteroclinic to ϕ$_{ζ}$ 1 ( t ) , ϕ$_{ζ}$ 2 ( t ) ∈ D ; 

(iii) If H is hyperbolic, then the same is true for D . 

The next section is devoted to a numerical example, which supports the results of Theorem 3.1. 

## 4 An Example 

Consider the following system, 

<img src='content_image/304070.jpg'>

where t belongs to the time scale T$_{0}$ = ⋃ ∞ k = $_{−∞}$[ θ$_{2}$$_{k}$$_{−}$$_{1}$, θ$_{2}$$_{k}$ ] , θ$_{k}$ = 4 k + (5 + ( − 1) $^{k}$) / 2 and g ( t, ζ ) = ζ$_{k}$ for t ∈ [ θ$_{2}$$_{k}$$_{−}$$_{1}$, θ$_{2}$$_{k}$ ] , k ∈ Z , such that ζ = { ζ$_{k}$ }$_{k}$$_{∈}$$_{Z}$ , ζ$_{0}$ ∈ [0 , 1] , is a sequence generated by the logistic map 

<img src='content_image/304069.jpg'>

where F ( υ ) = 3 . 9 υ (1 − υ ) . It is worth noting that the map (4.4) is chaotic in the sense of Li-Yorke (Li & Yorke, 1975), and the unit interval [0 , 1] is invariant under the iterations of the map (Hale & Koçak, 1991). 

System (4.3) is in the form of (1.1), where 

<img src='content_image/304068.jpg'>

The conditions ( C 1) − ( C 6) are valid for system (4.3) with N = 2 . 62 , λ = 1 / 12 , M$_{f}$ = 0 . 0503 , L$_{f}$ = 0 . 003 , ¯ κ = 5 and ¯ δ = 3 . 

Since t + 8 ∈ T$_{0}$ whenever t ∈ T$_{0}$ and f ( t + 8 , y$_{1}$, y$_{2}$ ) = f ( t, y$_{1}$, y$_{2}$ ) for all t ∈ T$_{0}$ , y$_{1}$, y$_{2}$ ∈ R , system (4.3) is Li-Yorke chaotic according to the results of (Akhmet & Fen, 2015). Moreover, one can verify that if ζ = { ζ$_{k}$ }$_{k}$$_{∈}$$_{Z}$ is a $_{p}$-periodic solution of (4.4) for some natural number p, then the corresponding bounded solution ϕ$_{ζ}$ ( t ) of (4.3) is 8 $_{p}$-periodic. The y$_{1}$ and y$_{2}$ coordinates of the solution of (4.3) with y$_{1}$ (0) = 3 , y$_{2}$ (0) = − 2 , ζ$_{0}$ = 0 . 292 is shown in Figure 1. It is seen in Figure 1 that the solution behaves chaotically. 

Now, we will demonstrate that homoclinic and heteroclinic motions take place in the chaotic dynamics of (4.3). Consider the function γ$_{1}$ ( υ ) = 1 2 ( 1 + √ 1 − 4 υ 3 . 9 ) . It was mentioned in (Avrutin et al., 2015) that the orbit 

<img src='content_image/304067.jpg'>

