Suppose that k$_{1}$ is the smallest natural number satisfying Δ ≤ k$_{1}$. In this case, we have for each k ≥ k$_{1}$ that Hence, for each n ∈ N , the inequality holds, where The theorem is proved. □ In the definition of Devaney chaos, periodic motions constitute a dense subset. However, in our case, instead of periodic motions, Poisson stable motions take place in the dynamics. More precisely, we say that the dynamics on the quasi-minimal set of functions on R is chaotic if the dynamics on it is sensitive, transitive and there exists a continuum of Poisson stable trajectories dense in the quasi-minimal set. Nevertheless, in the framework of chaos there may be infinitely many periodic motions. For instance, the symbolic dynamics of bi-infinite sequences possesses both an uncountable set of non-periodic Poisson stable motions as well as infinitely many cycles [1, 15]. In this part of the paper, we will present an illustrative example. For that purpose, we will make use of coupled Duffing equations such that the first one is forced with a relay function and the second one is perturbed with the solutions of the former. Let us consider the following forced Duffing equation, where the forcing term ν ( t, ζ, λ ) is a relay function defined as In (5.10), the sequence ζ = { ζ$_{j}$ }$_{j}$$_{∈}$$_{Z}$ of switching moments is defined through the equation ζ$_{j}$ = j + κ$_{j}$, 9 ν ( t, ζ, λ ) =     1 . 2 , if ζ$_{2}$$_{j}$ ( λ ) < t ≤ ζ$_{2}$$_{j}$$_{+1}$ ( λ ) , j ∈ Z , 0 . 4 , if ζ$_{2}$$_{j}$ − $_{1}$( λ ) < t ≤ ζ$_{2}$$_{j}$ ( λ ) , j ∈ Z . (5.10) x ′′ + 0 . 68 x ′ + 1 . 6 x + 0 . 008 x 3 = ν ( t, ζ, λ ) , (5.9) ϵ$_{0}$ = 2 − k 1 $^{+1}$Δ ϵ$_{0}$ 4 N m [1 + Δ( ‖ A ‖ + L$_{f}$ )] . d ( φ$_{t}$$_{n}$$_{+}$$_{˜}$ $_{τ$_{n}$}$, φ$_{˜}$ $_{τ$_{n}$}$) ≥ ∞ ∑ k = k 1 2 − $^{k}$ρ$_{k}$ ( φ$_{t}$$_{n}$$_{+}$$_{˜}$ $_{τ$_{n}$}$, φ$_{˜}$ $_{τ$_{n}$}$) ≥ ϵ$_{0}$, ρ$_{k}$ ( φ$_{t}$$_{n}$$_{+}$$_{˜}$ $_{τ$_{n}$}$, φ$_{˜}$ $_{τ$_{n}$}$) ≥ Δ ϵ$_{0}$ 4 N m [1 + Δ( ‖ A ‖ + L$_{f}$ )] . ≥ Δ ϵ$_{0}$ 4 N m [1 + Δ( ‖ A ‖ + L$_{f}$ )] . 5 An Example