content_image/304134.jpg content_image/304131.jpg content_image/304132.jpg
Suppose that D is a scrambled set inside A . Define the set 

<img src='content_image/304134.jpg'>

The set ˜ D is uncountable since the same is true for D . Moreover, the intersection of ˜ D and ˜ R is empty since the intersection of D and R is empty. 

Because each couple of different functions inside D × D is proximal, Lemma 4 . 1 implies the same feature for each couple of different functions inside ˜ D × ˜ D . On the other hand, since each couple of different functions ( x ( t ) , x ( t )) ∈ D × D is frequently ( ϵ$_{0}$, Δ) − separated for some positive numbers ϵ$_{0}$ and Δ , Lemma 4 . 2 implies that each couple of different functions ( y ( t ) , y ( t )) ∈ ˜ D × ˜ D is frequently ( ϵ$_{1}$, Δ) − separated for some positive numbers ϵ$_{1}$ and Δ . Additionally, the frequent separation feature holds also for each pair inside ˜ D × ˜ R . Therefore, the set ˜ D is a scrambled set inside B . Consequently, B is a Li-Yorke chaotic set in accordance with Definition 4.1. □ 

Using the technique of the proof of Theorem 4.1, one can confirm that if the function G ( t, y ) , which is used in the right hand side of system (1 . 1) , is quasi-periodic (periodic) in t uniformly for y ∈ R $^{n}$, then under the conditions of Theorem 4.1, the set B is chaotic in the sense of Li-Yorke with infinitely many quasi-periodic (periodic) motions provided that the same is true for A . 

In the following section, the theoretical results of Section 4 will be supported by simulations. We will show how Theorem 4.1 can be utilized to obtain a Li-Yorke chaotic Duffing oscillator with infinitely many almost periodic motions. In order to construct the Li-Yorke chaotic set A , we will take into account the bounded solutions of a Duffing oscillator perturbed with a relay function whose switching moments are changing chaotically. 

## 5 An Example 

Let us consider the forced Duffing oscillator 

where ν ( t, θ$_{0}$ ) is the relay function defined as 

<img src='content_image/304131.jpg'>

<img src='content_image/304132.jpg'>

