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: To provide a rigorous study of the subject, we introduce the following set of functions, 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. 8 B = { φ$_{x}$$_{(}$$_{t}$$_{)}$ ( t ) | x ( t ) ∈ A } . (4.7) 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 .