Figure 3: The stabilization of the 2 − periodic solution of (5.15) corresponding to the fixed point 3 . 007 / 4 . 007 of the logistic map (5.11). The value ε = 0 . 095 is used and the control is switched on at t = ζ$_{20}$. 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 (5 . 16) belongs to the interval [4 . 007 − ε, 4 . 007+ ε ] . In other words, formula (5.16) is valid if the trajectory { κ$_{j}$ } is sufficiently close to the target periodic orbit. Otherwise, we take ¯ λ$_{j}$ = 4 . 007 , 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 ≤ 4 . 007 (2 κ ( i ) − 1)( κ$_{j}$ − κ ( i $^{)}$) κ ( i $^{)}$(1 − κ ( 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 [24]. Figure 3 shows the stabilization of an unstable 2 − periodic solution of (5.15). Here, the OGY control method is used around the fixed point 3 . 007 / 4 . 007 of the logistic map (5.11), and the simulation is performed for the initial data x$_{1}$ (0 . 41) = 0 . 6 , x$_{2}$ (0 . 41) = 0 . 5 , z$_{1}$ (0 . 41) = 0 . 1 , z$_{2}$ (0 . 41) = 0 . 2 , ζ$_{0}$ = 0 . 41 . The control is switched on at t = ζ$_{20}$ and the value ε = 0 . 095 is utilized. One can confirm that even if the control is switched on at t = ζ$_{20}$ there is a transition time before the stabilization such that the control becomes dominant approximately at t = 76 . Figure 3 reveals that the OGY control technique is appropriate for the stabilization of the unstable periodic motions of system (5.15). On the other hand, Figure 4 shows the simulation result for (5.15) when the OGY method is applied around the 2 − periodic orbit κ (1) ≈ 0 . 34459 , κ (2) ≈ 0 . 90497 of (5.11). The represented solution corre- sponds again to the initial data x$_{1}$ (0 . 41) = 0 . 6 , x$_{2}$ (0 . 41) = 0 . 5 , z$_{1}$ (0 . 41) = 0 . 1 , z$_{2}$ (0 . 41) = 0 . 2 , ζ$_{0}$ = 0 . 41 . The value ε = 0 . 072 is used and the control is switched on at t = ζ$_{25}$. The presence of a transition time before the stabilization is observable in Figure 4 such that the control becomes dominant approximately at t = 46 . One can observe that the stabilized 2 − periodic solutions seen in Figure 3 and Figure 4 are 12 ¯ λ$_{j}$ = 4 . 007 ( 1 + (2 κ ( i ) − 1)( κ$_{j}$ − κ ( i $^{)}$) κ ( i $^{)}$(1 − κ ( i $^{)}$) ) , (5.16) 0 50 100 150 −0.5 0 0.5 1 t z 1 0 50 100 150 −2 −1 0 t z 2