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 $_{x}$′ $_{1}$( t ) = x$_{2}$ ( t ) , $_{x}$′ $_{2}$( t ) = − 1 . 5 x$_{1}$ ( t ) − 4 x$_{2}$ ( t ) − 0 . 02 x 2 $_{1}$( t − 0 . 1) + v ( t ) , (4.8) 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). Figure 2: The time series of x$_{1}$ and x$_{2}$ − coordinates of system (4.8). Chaotic behavior in both coordinates is observable in the figure. Figure 3: The trajectory of system (4.8). The figure manifests that the dynamics of (4.8) is chaotic. 10 0.15 0.2 0.25 −0.04 −0.02 0 0.02 0.04 x$_{1}$ x 2 0 50 100 150 0.15 0.2 0.25 t x 1 0 50 100 150 −0.04 −0.02 0 0.02 0.04 t x 2