Now, we will confirm the presence of quasi-periodic motions embedded in the chaotic attractor of (4.26) by stabilizing one of them through a control technique based on the OGY [39] and Pyragas [43] methods. The idea of the control procedure depends on the usage of both the OGY control for the discrete-time dynamics of the logistic map (4.22), which the source of chaotic motions in the forced Duffing equation (4.23), and the Pyragas control for the continuous-time dynamics of (4.26). The simul- taneous usage of both methods will give rise to the stabilization of a quasi-periodic solution of (4.26) since (4.23) and (4.25) admit periodic motions with incommensurate periods. Let us explain briefly the OGY control method for the map (4.22) [46]. Suppose that the parameter λ in the map (4.22) is allowed to vary in the range [3 . 9 − ε, 3 . 9 + ε $_{]}$, where ε is a given small positive number. Consider an arbitrary solution { κ$_{j}$ } , κ$_{0}$ ∈ [0 , 1] , 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 [46], at each iteration step j after the control mechanism is switched on, we consider the logistic map with the parameter value λ = ¯ λ$_{j}$ , where provided that the number on the right hand side of the formula (4 . 27) belongs to the interval [3 . 9 − ε, 3 . 9 + ε ] . In other words, formula (4.27) is valid if the trajectory { κ$_{j}$ } is sufficiently close to the target periodic orbit. Otherwise, we take ¯ λ$_{j}$ = 3 . 9 , so that the system evolves at its original parameter value, and wait until the trajectory { κ$_{j}$ } enters in a sufficiently small neighborhood of the periodic orbit κ ( i $^{)}$, i = 1 , 2 , . . . , p, such that the inequality − ε ≤ 3 . 9 (2 κ ( i ) − 1)( κ$_{j}$ − κ ( i $^{)}$) ζ ( i $^{)}$(1 − ζ ( i $^{)}$) ≤ ε holds. If this is the case, the control of chaos is not achieved immediately after switching on the control mechanism. Instead, there is a transition time before the desired periodic orbit is stabilized. The transition time increases if the number ε decreases [27]. 17 ¯ λ$_{j}$ = 3 . 9 ( 1 + (2 κ ( i ) − 1)( κ$_{j}$ − κ ( i $^{)}$) κ ( i $^{)}$(1 − κ ( i $^{)}$) ) , (4.27) Figure 3: Chaotic trajectories of systems (4.25) and (4.26). The trajectory of (4.25) with z$_{1}$ (0 . 41) = 0 . 12 , z$_{2}$ (0 . 41) = 0 . 013 is represented in blue color, while the trajectory of (4.26) corresponding to u$_{1}$ (0 . 41) = 0 . 12 , u$_{2}$ (0 . 41) = 0 . 013 is shown in red color. One can observe that the systems (4.25) and (4.26) admit different chaotic attractors. −3 −2 −1 0 1 2 3 −6 −5 −4 −3 −2 −1 0 1 2 3 u$_{1}$, z$_{1}$ u 2 , z 2