Using the variables y$_{1}$ = y and y$_{2}$ = y $^{′}$, equation (5.13) can be reduced to the system We perturb (5.14) with the solutions of (5.12), and set up the system System (5.15) is in the form of (4.2), where A =    0 1 − 1 . 8 − 0 . 95    , f ( z$_{1}$, z$_{2}$ ) = (0 , − 0 . 005 z 3 $_{1}$) and g ( t ) = ( x$_{1}$ ( t ) , x$_{2}$ ( t )) . Both eigenvalues of A have real parts − 0 . 475 , and the coefficient of the non-linear term is chosen sufficiently small in absolute value so that the conditions ( C 1) − ( C 3) are valid for (5.15). According to Theorem 4.1, system (5.15) possesses a unique uniformly exponentially stable unpredictable solution, which is uniformly continuous and bounded on R . Figure 2 shows the solution of (5.15) with z$_{1}$ (0 . 41) = 0 . 1 and z$_{2}$ (0 . 41) = 0 . 2 . For the simulation, the solution ( x$_{1}$ ( t ) , x$_{2}$ ( t )) which is represented in Figure 1 is used. Figure 2 supports the result of Theorem 4.1 such that the represented solution behaves irregularly. Next, we will demonstrate the presence of periodic motions in system (5.15) by means of the Ott- Grebogi-Yorke (OGY) control technique [22]. Since the logistic map (5.11) is the main source of the chaotic behavior in the coupled system (5 . 12) + (5 . 15) , we will apply the OGY method to the map. Let us explain briefly the method for the logistic map [23]. Suppose that the parameter λ in (5.11) is allowed to vary in the range [4 . 007 − ε, 4 . 007+ ε ] , where ε is a given small positive number. Consider an arbitrary solution { κ$_{j}$ } , κ$_{0}$ ∈ Λ , of the map and denote by κ ( i $^{)}$, i = 1 , 2 , . . . , p, the target p − periodic orbit to be stabilized. In the OGY control method [23], at each iteration step j after the control mechanism is 11 z ′ 1 = z$_{2}$ + x$_{1}$ ( t ) z ′ 2 = − 1 . 8 z$_{1}$ − 0 . 95 z$_{2}$ − 0 . 005 z 3 1 + x$_{2}$ ( t ) . (5.15) y ′ 1 = y$_{2}$ y ′ 2 = − 1 . 8 y$_{1}$ − 0 . 95 y$_{2}$ − 0 . 005 y 3 $_{1}$. (5.14) Figure 2: Chaotic behavior of system (5.15). The solution ( x$_{1}$ ( t ) , x$_{2}$ ( t )) represented in Figure 1 is utilized as the perturbation in (5.15). The irregularity is observable in both z$_{1}$ and z$_{2}$ coordinates. 0 10 20 30 40 50 60 70 80 90 100 −0.5 0 0.5 1 t z 1 0 10 20 30 40 50 60 70 80 90 100 −2 −1 0 t z 2