implies for s ≥ s 0 that Applying the Gronwall-Bellman Lemma for piecewise continuous functions [6] to the last inequality, one can obtain that Therefore, we have for t ≥ t $^{0}$, t ∈ T $_{0}$, that Consequently, ‖ y ( t ) − ϕ$_{ζ}$ ( t ) ‖ → 0 as t → ∞ , t ∈ T $_{0}$. □ In the next section, we will deal with the presence of chaos in system (1.1). The map (1.2) is called Li-Yorke chaotic on Λ if [11, 13, 29, 35, 36]: (i) For every natural number p$_{0}$, there exists a p$_{0}$ − periodic point of F in Λ; (ii) There is an uncountable set S ⊂ Λ , the scrambled set, containing no periodic points, such that for every ζ $^{1}$, ζ2 ∈ S with ζ 1 = ζ $^{2}$, we have lim sup $^{k}$→∞ ∥ F $^{k}$( ζ $^{1}$) − F $^{k}$( ζ $^{2}$) ∥ > 0 and lim inf $^{k}$→∞ ∥ F $^{k}$( ζ $^{1}$) − F $^{k}$( ζ $^{2}$) ∥ = 0; (iii) For every ζ 1 ∈ S and a periodic point ζ 2 ∈ Λ , we have lim sup $^{k}$→∞ ∥ F $^{k}$( ζ $^{1}$) − F $^{k}$( ζ $^{2}$) ∥ > 0 . $^{k}$→∞ Let us denote by Θ the set of all sequences ζ = { ζ$_{k}$ } , k ∈ Z , obtained by equation (1.2). A pair of sequences ζ = { ζ$_{k}$ } , ˜ ζ = { ˜ ζ$_{k}$ } ∈ Θ is proximal if lim inf $^{k}$→∞ ∥ ζ$_{k}$ − ˜ ζ$_{k}$ ∥ = 0 . Moreover, the pair is frequently separated if lim sup $^{k}$→∞ ∥ ζ$_{k}$ − ˜ ζ$_{k}$ ∥ > 0 . We say that a pair ϕ$_{ζ}$ ( t $_{)}$, ϕ$_{˜}$ $_{ζ}$( t ) of bounded solutions of (1.1) is proximal if for an arbitrary small real number ϵ > 0 and arbitrary large natural number E, there exists an integer m such that ∥ ϕ$_{ζ}$ ( t ) − ϕ$_{˜}$ $_{ζ}$( t ) ∥ < ϵ for all t ∈ [ $^{θ$_{2}$$_{m}$}$− $_{1}$, θ$_{2(}$$_{m}$$_{+}$$_{E}$$_{)}$ ] ∩ T $_{0}$. On the other hand, the pair ϕ$_{ζ}$ ( t $_{)}$, ϕ$_{˜}$ $_{ζ}$( t ) is frequently ( ϵ$_{0}$, $_{Δ)}$-separated if there exist numbers ϵ$_{0}$ > 0 , Δ > 0 and infinitely many disjoint intervals J$_{q}$ ⊂ T $_{0}$, q ∈ N , each with a length no less than Δ , such that ∥ ϕ$_{ζ}$ ( t ) − ϕ$_{˜}$ $_{ζ}$( t ) ∥ > ϵ$_{0}$ for each t from these intervals. Furthermore, a pair ϕ$_{ζ}$ ( t $_{)}$, ϕ$_{˜}$ $_{ζ}$( t ) of solutions of (1.1) is called a Li-Yorke pair if it is proximal and frequently ( ϵ$_{0}$, $_{Δ)}$-separated for some positive numbers ϵ$_{0}$ and Δ . 6 + ∑ s $^{0}$≤ s$_{k}$<s X ( s, s$_{k}$ +) [ f ( ψ − $^{1}$( s$_{k}$ ) , x ( s$_{k}$ ) − f ( ψ − $^{1}$( s$_{k}$ ) , φ$_{ζ}$ ( s$_{k}$ ) )] δ$_{k}$ ‖ x ( s ) − φ$_{ζ}$ ( s ) ‖ ≤ N e − λ ( $^{s}$−s $^{0}$) ∥ y$_{0}$ − φ$_{ζ}$ ( s $^{0}$) ∥ + ∫ s s 0 N L$_{f}$ e − λ ( $^{s}$−r ) ‖ x ( r ) − φ$_{ζ}$ ( r ) ‖ dr + ∑ s $^{0}$≤ s$_{k}$<s N L$_{f}$ ¯ δe − λ ( $^{s}$−s$_{k}$ ) ‖ x ( s$_{k}$ ) − φ$_{ζ}$ ( s$_{k}$ ) ‖ . ‖ x ( s ) − φ$_{ζ}$ ( s ) ‖ ≤ N (1 + N L$_{f}$ ¯ δ ) p ∥ y$_{0}$ − φ$_{ζ}$ ( s $^{0}$) ∥ e $^{[}$− λ + NL$_{f}$ + p ln(1+ NL$_{f}$ ¯ δ ) /ψ ( ω )]( $^{s}$−s $^{$^{0}$)}$, s ≥ s $^{0}$. ‖ y ( t ) − ϕ$_{ζ}$ ( t ) ‖ ≤ N (1 + N L$_{f}$ ¯ δ ) p ∥ y$_{0}$ − ϕ$_{ζ}$ ( t $^{0}$) ∥ e $^{[}$− λ + NL$_{f}$ + p ln(1+ NL$_{f}$ ¯ δ ) /ψ ( ω )]( ψ ( t $^{)}$− ψ ( t $^{$^{0}$))}$. 4 The chaotic dynamics