content_image/321613.jpg content_image/321614.jpg content_image/321612.jpg content_image/321616.jpg content_image/321611.jpg
written as 

where D =    − 2 0 0 0 − 3 0 0 0 − 1   $^{,}$ and 

<img src='content_image/321613.jpg'>

<img src='content_image/321614.jpg'>

System (6.13) admits an unpredictable solution ψ ( t ) in accordance with Theorem 5.1. Therefore, Corollary 5.1 implies that ψ ( t ) = P ψ ( t ) is an unpredictable solution of (6.11). 

To demonstrate the chaotic behavior, we depict in Figure 2 the x$_{2}$ − coordinate of the solution of (6.11) corresponding to the initial data x$_{1}$ ( ζ$_{0}$ ) = 0 . 17 , x$_{2}$ ( ζ$_{0}$ ) = 0 . 51 , x$_{3}$ ( ζ$_{0}$ ) = 0 . 48 , ζ$_{0}$ = 0 . 4 . The coefficient µ = 3 . 86 and the initial value ζ$_{0}$ = 0 . 4 are considered for shadowing in [22]. The chaotic behavior is also valid in the remaining coordinates, which are not just pictured here. Moreover, Figure 3 shows the 3 − dimensional chaotic trajectory of the same solution. It is worth noting that the chaotic solutions of (6.11) take place inside the compact region 

<img src='content_image/321612.jpg'>

Figures 2 and 3 support the results of Theorem 5.1 and Corollary 5.1 such that the represented solution behaves chaotically. 

<img src='content_image/321616.jpg'>

Now, we will demonstrate the extension of unpredictable solutions and Poincaré chaos. For that purpose, we consider the system 

<img src='content_image/321611.jpg'>

