4 Persistence of chaos The following lemmas, which are concerned with the proximality and frequent separation features of the bounded solutions of (1.6), are needed for the proof of the main theorem of the present section. Lemma 4.1 Suppose that conditions ( C 1) − ( C 4) are valid. If a couple of functions ( x ( t ) , ˜ x ( t )) ∈ A × A is proximal, then the same is true for the couple ( φ$_{x,ζ}$ ( t ) , φ$_{˜}$ $_{x,ζ}$( t )) ∈ B ζ × B ζ for any sequence ζ ∈ Θ . Lemma 4.2 Suppose that conditions ( C 1) − ( C 3) , ( C 5) and ( C 6) are valid. 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 there exist positive numbers ϵ$_{1}$ and Δ such that the couple of functions ( φ$_{x,ζ}$ ( t ) , φ$_{˜}$ $_{x,ζ}$( t )) ∈ B ζ × B ζ is frequently ( ϵ$_{1}$, Δ) − separated for any sequence ζ ∈ Θ . The proofs of Lemma 4.1 and Lemma 4.2 are provided in the Appendix. The main result of the present section is mentioned in the next theorem. Theorem 4.1 Suppose that the conditions ( C 1) − ( C 6) are valid. If ζ ∈ P , then B ζ is a Li-Yorke chaotic set. Proof. Since the set A is Li-Yorke chaotic, there exists a countably infinite set AP 1 ⊂ A of almost periodic solutions. Fix an arbitrary sequence ζ ∈ P , and let us denote by AP ζ 2 the subset of B ζ consisting of bounded solutions φ$_{x,ζ}$ ( t ) of (1.6) such that x ( t ) ∈ AP $_{1}$. According to Theorem 3.1, the elements of AP ζ 2 are the almost periodic solutions of system (1.6). It can be verified using condition ( C 5) that AP ζ 2 is also countably infinite. Now, suppose that S 1 ⊂ A is an uncountable scrambled set. Let us define the set of functions S ζ 2 = { φ$_{x,ζ}$ ( t ) : x ( t ) ∈ S 1 } . The set S ζ 2 ⊂ B ζ is uncountable, and it does not contain any almost periodic solutions in accordance with condition ( C 5) . Since each couple of different functions inside S 1 × S 1 is a Li-Yorke pair, Lemma 4.1 and Lemma 4.2 together imply that the same is true for each couple of different functions inside S ζ 2 × S ζ $_{2}$. Hence, S ζ 2 is a scrambled set. Besides, one can confirm using Lemma 4.2 one more time that each couple of functions inside S ζ 2 × AP ζ 2 is frequently ( ϵ$_{1}$, Δ ) − separated for some positive numbers ϵ$_{1}$ and Δ . Consequently, B ζ is a Li-Yorke chaotic set for each ζ ∈ P . □ Remark 4.1 In Theorem 4.1, we demonstrate that for each fixed sequence ζ ∈ P , system (1.6) admits a Li-Yorke chaotic set B $_{ζ}$. It can be easily shown that B ζ ∩ B η =  whenever ζ and η are different sequences in P . Therefore, there are countably infinite Li-Yorke chaotic sets in the dynamics of (1.6). An illustrative example which supports the theoretical results is presented in the next section based on Duffing oscillators. 8