content_image/309685.jpg content_image/309691.jpg content_image/309690.jpg
Conditions ( C 1) − ( C 3) are valid for system (4.7) with M$_{f}$ = 0 . 001352 and L$_{f}$ = 0 . 0104 . According to Theorem 2.1, system (4.7) possesses a unique uniformly exponentially stable unpredictable solution. 

To demonstrate the chaotic dynamics of (4.7), let us consider the system 

<img src='content_image/309685.jpg'>

where v ( t ) is the solution of (4.4) represented in Figure 1. The x$_{1}$ and x$_{2}$ − coordinates of the solution of (4.8) corresponding to the initial conditions x$_{1}$ ( t ) = 0 . 18 , x$_{2}$ ( t ) = 0 . 01 , t ∈ [ − 0 . 1 , 0] are shown in Figure 2. The figure supports the result of Theorem 2.1 such that (4.7) possesses an unpredictable solution, and it reveals that the dynamics of (4.8) is chaotic. Moreover, the trajectory of the same solution is depicted in Figure 3, and this simulation also confirms the presence of chaos in system (4.8). 

<img src='content_image/309691.jpg'>

<img src='content_image/309690.jpg'>

