content_image/311548.jpg content_image/311547.jpg content_image/311546.jpg
(2.7) satisfy the relation 

It can be verified by using the last equation that 

<img src='content_image/311548.jpg'>

<img src='content_image/311547.jpg'>

where K = max ( i,j ) K$_{ij}$. Benefiting from the Gronwall-Bellman Lemma for piecewise continuous functions [77], we obtain for s ≥ s 0 that 

<img src='content_image/311546.jpg'>

Condition ( C 6) implies that ‖ y ( s ) − φ$_{ζ}$ ( s ) ‖ → 0 as s → ∞ . Hence, ‖ x ( t ) − ϕ$_{ζ}$ ( t ) ‖ → 0 as t → ∞ , t ∈ T $_{0}$. 

□ 

The presence of Li-Yorke chaos in SICNN (1.2) will be investigated in the next section. 

## 4 Li-Yorke chaos 

The map (1.3) is called Li-Yorke chaotic on Λ if [27, 29]: (i) For every natural number p$_{0}$, there ex- ists a p$_{0}$ $_{−}$periodic point of F in Λ; (ii) There is an uncountable set S ⊂ Λ , the scrambled set, con- taining no periodic points, such that for every ξ$_{1}$ = { ξ ij 1 } , ξ$_{2}$ = { ξ ij 2 } ∈ S with ξ$_{1}$ ̸ = ξ$_{2}$, we have lim sup k →∞ ∥ F $^{k}$( ξ$_{1}$ ) − F k ( ξ$_{2}$ ) ∥ > 0 and lim inf k →∞ ∥ F $^{k}$( ξ$_{1}$ ) − F $^{k}$( ξ$_{2}$ ) ∥ = 0; (iii) For every ξ$_{1}$ ∈ S and a periodic point ξ$_{2}$ ∈ Λ , we have lim sup k →∞ ∥ F $^{k}$( ξ$_{1}$ ) − F $^{k}$( ξ$_{2}$ ) ∥ > 0 . Denote by Θ the set of all sequences ζ = { ζ$_{k}$ }$_{k}$$_{∈}$$_{Z}$ obtained by equation (1.3). We say that a pair 

of sequences ζ = { ζ$_{k}$ }$_{k}$$_{∈}$$_{Z}$ , ˜ ζ = { ˜ ζ$_{k}$ } k ∈ Z ∈ Θ is proximal if lim inf k →∞ ∥ ζ$_{k}$ − ˜ ζ$_{k}$ ∥ = 0 , and it is frequently separated if lim sup k →∞ ∥ ζ$_{k}$ − ˜ ζ$_{k}$ ∥ > 0 . A ij 

Let be the collection of all bounded solutions ϕ$_{ζ}$ ( t ) = { ϕ ζ ( t ) } of (1.2) such that ζ ∈ Θ . A pair ϕ$_{ζ}$ ( t ) , ϕ$_{˜}$ $_{ζ}$( t ) ∈ A is proximal if for an arbitrary small positive number ϵ and an arbitrary large natural number E, there exists an integer k$_{0}$ such that ∥ ϕ$_{ζ}$ ( t ) − ϕ$_{˜}$ $_{ζ}$( t ) ∥ < ϵ for all t ∈ [ θ$_{2}$$_{k}$$_{0}$$_{−}$$_{1}$, θ$_{2(}$$_{k}$$_{0}$$_{+}$$_{E}$$_{)}$ ] ∩ T $_{0}$. On the other hand, the pair ϕ$_{ζ}$ ( t ) , ϕ$_{˜}$ $_{ζ}$( t ) ∈ A is frequently ( ϵ$_{0}$, ϵ$_{1}$ $_{)}$-separated if there exist numbers ϵ$_{0}$ > 0 , ϵ$_{1}$ > 0 and infinitely many disjoint intervals J$_{q}$ ⊂ T $_{0}$, q ∈ N , each with a length no less than ϵ$_{1}$ such that 

