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, 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. Next, we will demonstrate the chaotic behavior of a multidimensional system of differential equations. Let us take into account the system 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 12 ν ( t ) =     ( − 1 , 1 , 2) , if ζ$_{2}$$_{j}$ < t ≤ ζ$_{2}$$_{j}$$_{+1}$, j ∈ $^{Z}$, (3 , 1 , − 1) , if ζ$_{2}$$_{j}$ − 1 < t ≤ ζ$_{2}$$_{j}$ , j ∈ $^{Z}$. (6.12) x $^{′}$( t ) = Ax ( t ) + ν ( t ) , (6.11) Figure 1: Chaotic behavior in equation (6.9). The figure confirms that Poincaré chaos takes place in the dynamics of equation (6.9). 0 10 20 30 40 50 60 70 80 90 100 −0.2 0 0.2 0.4 t x ψ ( t ) = t ∫ −∞ e − 3( t − s ) / $^{2}$ν ( s ) ds