content_image/327295.jpg
that ‖ x ( t ) − x ( t ) ‖ < ϵ for all t ∈ J. Besides, a couple of functions ( x ( t ) , x ( t )) ∈ A × A is frequently ( ϵ$_{0}$, Δ) − separated if there exist numbers ϵ$_{0}$ > 0 , Δ > 0 and infinitely many disjoint intervals each with a length no less than Δ such that ‖ x ( t ) − x ( t ) ‖ > ϵ$_{0}$ for each t from these intervals. It is worth noting that the numbers ϵ$_{0}$ and Δ depend on the functions x ( t ) and x ( t ) . 

We say that a couple of functions ( x ( t ) , x ( t )) ∈ A × A is a Li-Yorke pair if they are proximal and frequently ( ϵ$_{0}$, Δ) − separated for some positive numbers ϵ$_{0}$ and Δ . 

The definition of a Li-Yorke chaotic set with infinitely many almost periodic motions is as follows. 

Definition 4.1 A is called a Li-Yorke chaotic set with infinitely many almost periodic motions if: 

i. There exists a countably infinite set R ⊂ A of almost periodic functions; 

ii. There exists an uncountable set D ⊂ A , the scrambled set, such that the intersection of D and R is empty and each couple of different functions inside D × D is a Li-Yorke pair; 

iii. For any function x ( t ) ∈ D and any almost periodic function x ( t ) ∈ R , the couple ( x ( t ) , x ( t )) is frequently ( ϵ$_{0}$, Δ) − separated for some positive numbers ϵ$_{0}$ and Δ . 

To provide a rigorous study of the subject, we introduce the following set of functions, 

<img src='content_image/327295.jpg'>

The following assertions, which are about the proximality and frequent separation features for system (1.1), can be proved in a similar way to Lemma 6 . 1 and Lemma 6 . 2 [32]. 

Lemma 4.1 Under the conditions ( C 1) − ( C 3) , if a couple of functions ( x ( t ) , x ( t )) ∈ A × A is proximal, then the same is true for the couple ( φ$_{x}$$_{(}$$_{t}$$_{)}$ ( t ) , φ$_{x}$$_{(}$$_{t}$$_{)}$ ( t ) ∈ B × B . 

Lemma 4.2 Assume that the conditions ( C 1) , ( C 2) and ( C 4) hold. If a couple of functions ( x ( t ) , x ( t )) ∈ A × A is frequently ( ϵ$_{0}$, Δ) − separated for some positive numbers ϵ$_{0}$ and Δ , then the couple of functions ( φ$_{x}$$_{(}$$_{t}$$_{)}$ ( t ) , φ$_{x}$$_{(}$$_{t}$$_{)}$ ( t ) ∈ B × B is frequently ( ϵ$_{1}$, Δ) − separated for some positive numbers ϵ$_{1}$ and Δ . 

The main result of the present section is mentioned in the following theorem. 

Theorem 4.1 Suppose that the conditions ( C 1) − ( C 4) are fulfilled. If the set A is chaotic in the sense of Definition 4.1, then the same is true for the set B . 

Proof. Let R be the set of almost periodic functions inside A . If x ( t ) belongs to R , then the function G ( t, y ) + H ( x ( t )) in (1.1) is almost periodic in t uniformly for y ∈ R n [67]. In this case, the bounded solution φ$_{x}$$_{(}$$_{t}$$_{)}$ ( t ) of (1.1) is almost periodic [67]. Consider the set ˜ R = { φ$_{x}$$_{(}$$_{t}$$_{)}$ ( t ) | x ( t ) ∈ R } . According to condition ( C 4) there is a one-to-one correspondence between R and ˜ R . Since the set R is countably infinite, the set ˜ R is also countably infinite. 

