content_image/316506.jpg content_image/316505.jpg content_image/316504.jpg content_image/316507.jpg content_image/316503.jpg
where the relay function v ( t, ζ, λ ) is defined as 

<img src='content_image/316506.jpg'>

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 

<img src='content_image/316505.jpg'>

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 

<img src='content_image/316504.jpg'>

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], 

<img src='content_image/316507.jpg'>

<img src='content_image/316503.jpg'>

