case, there exists a compact region Λ ⊂ R m such that the trajectories of all solutions that belong to A is inside Λ . An example of such a system was provided in [19] with a theoretical discussion. By means of the assumption on the matrix A, one can confirm the existence of positive numbers K and α such that ‖ e $^{At}$‖ ≤ Ke − $^{αt}$, t ≥ 0 . In the remaining parts of the paper, Θ will stand for the set of all sequences ζ = { ζ$_{i}$ } , i ∈ Z , generated by equation (1.4). Since the value of µ in (1.5) is fixed between 3 . 84 and 4 so that the map is chaotic in the sense of Li-Yorke, the map (1.5) possesses a periodic orbit with period p for each natural number p [15]. We will denote by P ⊂ Θ the countably infinite set of all sequences ζ = { ζ$_{i}$ } , i ∈ Z , generated by (1.4) in which { κ$_{i}$ } is a periodic solution of (1.5). The following assumptions are required: (C1) There exists a positive number M$_{f}$ such that sup y ∈ R $^{n}$,t ∈ $_{R}$‖ f ( y, t ) ‖ ≤ M$_{f}$ ; (C2) There exists a positive number L$_{f}$ < α/K such that ‖ f ( y$_{1}$, t ) − f ( y$_{2}$, t )) ‖ ≤ L$_{f}$ ‖ y$_{1}$ − y$_{2}$ ‖ for all y$_{1}$, y$_{2}$ ∈ R $^{n}$, t ∈ R ; (C3) There exists a positive number ζ such that ζ$_{i}$$_{+1}$ − ζ$_{i}$ ≥ ζ for each ζ = { ζ$_{i}$ } ∈ Θ and i ∈ Z ; (C4) There exists a positive number L$_{1}$ such that ‖ h ( x$_{1}$ ) − h ( x$_{2}$ ) ‖ ≤ L$_{1}$ ‖ x$_{1}$ − x$_{2}$ ‖ for all x$_{1}$, x$_{2}$ ∈ Λ ; (C5) There exists a positive number L$_{2}$ such that ‖ h ( x$_{1}$ ) − h ( x$_{2}$ ) ‖ ≥ L$_{2}$ ‖ x$_{1}$ − x$_{2}$ ‖ for all x$_{1}$, x$_{2}$ ∈ Λ ; (C6) There exists a positive number M$_{F}$ such that sup x ∈ Λ ,t ∈ $_{R}$‖ F ( x, t ) ‖ ≤ M$_{F}$ . Under the conditions ( C 1) − ( C 3) , one can confirm using the results of papers [19, 29] that the relay system (1.2) is Li-Yorke chaotic with infinitely many almost periodic motions in basis for the values of the parameter µ between 3 . 84 and 4 . We refer the reader to [32] for further information about the dynamics of relay systems. Under the same conditions, for a given solution x ( t ) ∈ A of (1.1) and a sequence ζ ∈ Θ , system (1.6) possesses a unique solution φ$_{x,ζ}$ ( t ) which is bounded on the whole real axis [33], and this solution satisfies the relation φ$_{x,ζ}$ ( t ) = ∫ t −∞ e A ( t − s $^{)}$[ f ( φ$_{x,ζ}$ ( s ) , s ) + ν ( s, ζ ) + h ( x ( s ))] ds. (2.8) One can confirm that sup t ∈ R ‖ y ( t ) ‖ ≤ M for all y ( t ) ∈ B ζ and η ∈ Θ , where M = K α ( m + M$_{f}$ + M$_{h}$ ) , m = max {‖ m$_{0}$ ‖ , ‖ m$_{1}$ ‖} , and M$_{h}$ = max x ∈ Λ ‖ h ( x ) ‖ . 5 For a fixed sequence ζ ∈ Θ and a fixed solution x ( t ) ∈ A , the bounded solution φ$_{x,ζ}$ ( t ) attracts all other solutions of (1.6) such that the inequality ‖ φ$_{x,ζ}$ ( t ) − y$_{x,ζ}$ ( t ) ‖ ≤ K ‖ φ$_{x,ζ}$ ( t$_{0}$ ) − y$_{0}$ ‖ e ( KL$_{f}$ − α )( t − t 0 ) is satisfied for t ≥ t$_{0}$, where y$_{x,ζ}$ ( t ) is a solution of (1.6) with y$_{x,ζ}$ ( t$_{0}$ ) = y$_{0}$ for some y$_{0}$ ∈ R n and t$_{0}$ ∈ R . To provide a theoretical discussion for the persistence of chaos, for each sequence ζ ∈ Θ , let us introduce the set B ζ consisting of all bounded solutions φ$_{x,ζ}$ ( t ) of system (1.6) in which x ( t ) belongs to A .