content_image/301940.jpg content_image/301939.jpg content_image/301937.jpg content_image/301936.jpg
In (6.10), the sequence { ζ$_{j}$ } is defined as ζ$_{j}$ = j + η$_{j}$, j ∈ $^{Z}$, and { η$_{j}$ } is the unpredictable sequence determined in Section 5 for the map (4.4) with µ = 3 . 91 . 

According to Theorem 5.1, 

<img src='content_image/301940.jpg'>

is a globally asymptotically stable unpredictable solution of (6.9). We represent a solution of (6.9) corresponding to the initial data x ( ζ$_{0}$ ) = 0 . 3 , ζ$_{0}$ = 0 . 4 in Figure 1. The choice of the coefficient µ = 3 . 91 and the initial value ζ$_{0}$ = 0 . 4 is approved by the shadowing analysis in paper [22]. The simulation seen in Figure 1 supports the result of Theorem 5.1 such that the equation (6.9) behaves chaotically. 

<img src='content_image/301939.jpg'>

Next, we will demonstrate the chaotic behavior of a multidimensional system of differential equations. 

Let us take into account the system 

<img src='content_image/301937.jpg'>

where x ( t ) = ( x$_{1}$ ( t ) , x$_{2}$ ( t ) , x$_{3}$ ( t )) ∈ R $^{3}$, A =    − 3 0 − 1 − 2 1 − 2 − 2 4 − 4   $^{,}$ and the function ν : $^{R}$→ R 3 is defined as 

Similarly to equation (6.10), in (6.12), the sequence { ζ$_{j}$ } of switching moments is defined through the equation ζ$_{j}$ = j + η$_{j}$, where { η$_{j}$ } is the unpredictable sequence determined in Section 5 for the map (4.4) with µ = 3 . 86 . 

By means of the transformation y = P − $^{1}$x, where P =    1 2 1 0 1 − 1 − 1 0 − 2    , system (6.11) can be 

<img src='content_image/301936.jpg'>

