In the relay function (5.9), the sequence { θ$_{i}$ } , i ∈ Z , of switching moments is defined through the equation θ$_{i}$ = i + κ$_{i}$, in which the sequence { κ$_{i}$ } , κ$_{0}$ ∈ [0 , 1] , is generated by the logistic map such that where λ is a parameter. The interval [0 , 1] is invariant under the iterations of (5.10) for the values of λ between 0 and 4 [68]. For λ = 3 . 9 , the map (5.10) is Li-Yorke chaotic [2], and the family { ν ( t, θ$_{0}$ ) } , θ$_{0}$ ∈ [0 , 1] , is chaotic in the sense of Li-Yorke with infinitely many periodic motions [29]. Using the variables x$_{1}$ = x and x$_{2}$ = x $^{′}$, equation (5.8) can be written as a system in the form According to the results of the paper [29], the set A consisting of the bounded on R solutions of (5.11) with λ = 3 . 9 is Li-Yorke chaotic such that it possesses infinitely many periodic solutions with periods 2 p, p ∈ N , which are separated from the motions of the scrambled set. The chaoticity of the switching moments { θ$_{i}$ } gives rise to the presence of chaos in system (5.11). The reader is referred to [28, 29, 30, 31, 33, 34, 38] for further information about the dynamics of relay systems. In what follows, we will take into account system (5.11) with λ = 3 . 9 . To illustrate the chaotic dynamics of (5.11), let us consider the solution of the system with the initial data x$_{1}$ ( t$_{0}$ ) = 0 . 24 , x$_{2}$ ( t$_{0}$ ) = 0 . 16 , where t$_{0}$ = 0 . 61 . The simulation results are shown in Figure 1, where one can observe the chaotic behavior of (5.11). The sequence { θ$_{i}$ } of switching moments is used with θ$_{0}$ = 0 . 61 in the simulation. It is worth noting that the bounded on R solutions of (5.11) lie inside the compact region Λ = { ( x$_{1}$, x$_{2}$ ) ∈ R 2 : 0 ≤ x$_{1}$ ≤ 0 . 9 , − 1 ≤ x$_{2}$ ≤ 1 . 3 } . Figure 1: Li-Yorke chaos in system (5.11). 0 10 20 30 40 50 60 70 80 0 0.5 1 t x 1 0 10 20 30 40 50 60 70 80 −1 0 1 t x 2 10 κ$_{i}$$_{+1}$ = λκ$_{i}$ (1 − κ$_{i}$ ) , (5.10) x ′ 1 = x$_{2}$ x ′ 2 = − 11 2 x$_{1}$ − x$_{2}$ − 0 . 0003 x 3 $_{1}$+ sin(2 πt ) + ν ( t, θ$_{0}$ ) . (5.11)