where the relay function v ( t, ζ, λ ) is defined as In (4.21), the sequence ζ = { ζ$_{j}$ }$_{j}$$_{∈}$$_{Z}$ , ζ$_{0}$ ∈ [0 , 1] , of switching moments is defined through the equation ζ$_{j}$ = j + κ$_{j}$, j ∈ Z , and the sequence { κ$_{j}$ }$_{j}$$_{∈}$$_{Z}$ is a solution of the logistic map The interval [0 , 1] is invariant under the iterations of (4.22) for the values of λ between 1 and $_{4}$, and the map possesses Li-Yorke chaos for λ = 3 . 9 [35]. Making use of the new variables x$_{1}$ = x and x$_{2}$ = x $^{′}$, one can reduce equation (4.20) to the system According to the results of [2], for each ζ$_{0}$ ∈ [0 , 1] , system (4.23) with λ = 3 . 9 possesses a bounded on R solution, and the collection A$_{x}$ consisting of all such bounded on R solutions is Li-Yorke chaotic. Moreover, for each natural number $_{m}$, system (4.23) admits unstable periodic solutions with periods 2 m. The reader is referred to [2, 3, 4, 5, 7, 9] for more information about the dynamics of relay systems. Figure 1 shows the solution of system (4.23) with λ = 3 . 9 , ζ$_{0}$ = 0 . 41 corresponding to the initial data x$_{1}$ (0 . 41) = 0 . 8 , x$_{2}$ (0 . 41) = 0 . 7 . The simulation results seen in Figure 1 confirm the presence of chaos in (4.23). Next, let us consider the following Duffing equation [13], 15 z ′′ + 0 . 15 z ′ − 0 . 5 z (1 − z $^{2}$) = 0 . 2 sin(0 . 9 t ) . (4.24) x ′ 1 = x$_{2}$ x ′ 2 = − 1 . 4 x$_{1}$ − 0 . 82 x$_{2}$ − 0 . 01 x 3 $_{1}$+ v ( t, ζ, λ ) . (4.23) κ$_{j}$$_{+1}$ = λκ$_{j}$ (1 − κ$_{j}$ ) . (4.22) v ( t, ζ, λ ) =     $^{0}$$^{.}$$^{3}$$^{,}$ if ζ$_{2}$$_{j}$ ( λ ) < t ≤ ζ$_{2}$$_{j}$$_{+1}$ ( λ ) , j ∈ Z , 1 . 9 , if ζ$_{2}$$_{j}$$_{−}$$_{1}$ ( λ ) < t ≤ ζ$_{2}$$_{j}$ ( λ ) , j ∈ Z . (4.21) 0 50 100 150 0 0.5 1 1.5 t x 1 0 50 100 150 −1 0 1 t x 2 Figure 1: The chaotic behavior of system (4.23).