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## 6 Control of chaos 

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In this part of the paper, we will present a numerical technique to control the chaos of system (5.17). For that purpose, we will make use of the OGY control method [25] applied to the logistic map (1.5), which is the main source of chaos in the unidirectionally coupled system (5.12)+(5.17). 

Let us briefly explain the OGY algorithm for the logistic map (1.5) [25, 37]. Denote by κ ( j $^{)}$, j = 1 , 2 , . . . , p, the target p − periodic orbit of (1.5) with µ = 3 . 9 to be stabilized. Suppose that the parameter µ in (1.5) is allowed to vary in the range [3 . 9 − ε, 3 . 9 + ε ] , where ε is a given small positive number. In the OGY control method [37], after the control mechanism is switched on, we consider (1.5) with the parameter value µ = µ$_{i}$, where 

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provided that the number on the right-hand side of the formula (6.18) belongs to the interval [3 . 9 − ε, 3 . 9 + ε ] . Here, the sequence { κ$_{i}$ } , i ≥ 0 , satisfying κ$_{0}$ ∈ [0 , 1] is an arbitrary solution of the map 

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Formula (6.18) is valid only if the trajectory { κ$_{i}$ } is sufficiently close to the target periodic orbit κ ( j $^{)}$, j = 1 , 2 , . . . , p . Otherwise, we take µ$_{i}$ = 3 . 9 in (6.19) so that the system evolves at its original parameter value, and wait until the trajectory { κ$_{i}$ } enters in a sufficiently small neighborhood of the periodic orbit such that the inequality − ε ≤ 3 . 9 (2 κ ( j ) − 1)( κ$_{i}$ − κ ( j $^{)}$) κ ( j $^{)}$(1 − κ ( j $^{)}$) ≤ 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 [9]. 

Consider the sequence ζ = { ζ$_{i}$ } , i ≥ 0 , generated through the equation ζ$_{i}$ = 1 . 05 i + κ$_{i}$ , where { κ$_{i}$ } , κ$_{0}$ ∈ [0 , 1] , is a solution of (6.19). To control the chaos of (5.17), we replace the sequence ζ = { ζ$_{i}$ } 

