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j ∈ Z , where the sequence { κ$_{j}$ }$_{j}$$_{∈}$$_{Z}$ is a solution of the logistic map 

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By means of the variables x$_{1}$ = x and x$_{2}$ = x $^{′}$, equation (5.9) can be written as a system in the following form, 

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We suppose that the parameter λ in (5.11) is greater than 4 such that the map possesses an invariant Cantor set Λ ⊂ [0 , 1] [16], and it was demonstrated in [1] that for such values of the parameter the map possesses an unpredictable point in Λ . We consider system (5.12) with ζ$_{0}$ ∈ Λ , and the techniques presented in the papers [17]-[19] can be used to prove the existence of a unique unpredictable solution of the system. Moreover, for each natural number p , system (5.12) admits an unstable periodic solution with period 2 p if p is odd and an unstable periodic solution with period p if p is even [20]. The reader is referred to [5, 17, 18, 19, 20, 21] for more information about the dynamics of relay systems. 

In order to demonstrate the chaotic dynamics of (5.12), we make use of the value λ = 4 . 007 in the system and depict the solution corresponding to the initial data x$_{1}$ (0 . 41) = 0 . 6 , x$_{2}$ (0 . 41) = 0 . 5 and ζ$_{0}$ = 0 . 41 in Figure 1. The simulation results seen in Figure 1 confirm the presence of chaos in (5.12). 

It is worth noting that even if the logistic map (5.11) with λ = 4 . 007 has an invariant Cantor set, the chosen initial value of the sequence { ζ$_{j}$ } allows to simulate the solution for 0 . 41 ≤ t ≤ 100 . Due to the instability, simulations of the system cannot be provided for large intervals of time. 

Next, we take into account another Duffing equation, 

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