We shall continue in the next ingredient of Li-Yorke chaos in the following lemma. Lemma 3.2 Suppose that conditions ( C 1) − ( C 6) are valid. If a couple ( x ( t ) , x ( t )) ∈ A$_{x}$ × A$_{x}$ is frequently ( ϵ$_{0}$, Δ) − separated for some positive numbers ϵ$_{0}$ and Δ , then the couple ( φ β x ( t $_{)}$( t ) , φ β x ( t $_{)}$( t ) ∈ A β y × A β y is frequently ( ϵ$_{1}$, Δ) − separated for some positive numbers ϵ$_{1}$ and Δ uniformly for β ∈ S$_{m}$ (see (3.17) and (3.18) ). Proof. Let ( x ( t ) , x ( t )) ∈ A$_{x}$ × A$_{x}$ be a couple of functions frequently ( ϵ$_{0}$, Δ) − separated for some ϵ$_{0}$ > 0 and Δ > $_{0}$. In that case, one can find a sequence { Δ$_{l}$ } , satisfying Δ$_{l}$ ≥ Δ , l ∈ N , and a sequence d$_{l}$ ∈ R such that for each l ∈ N the inequality ‖ x ( t ) − x ( t ) ‖ > ϵ$_{0}$ is satisfied for t ∈ J$_{l}$ = [ d$_{l}$, d$_{l}$ + Δ$_{l}$] and J$_{l}$ ∩ J$_{m}$ =  whenever l = m. Set y ( t ) = φ β x ( t $_{)}$( t ) and y ( t ) = φ β x ( t $_{)}$( t $_{)}$. Now, let the sequence { b$_{l}$ } be the midpoints of the intervals J$_{l}$, that is b$_{l}$ = d$_{l}$ + Δ$_{l}$ / 2 for each l ∈ N , and consider J 1 l = [ b$_{l}$, b$_{l}$ + Δ$_{l}$ / 4] ⊂ $_{J$_{l}$}$. Next, for any t ∈ J 1 l , we have that which implies since for any s ∈ J 1 l . Consequently, we obtain that which implies 11 max t ∈ J 1 l ‖ y ( t ) − y ( t ) ‖ ≥ L$_{1}$ Δ$_{l}$ ϵ$_{0}$ 8 + ( P$_{1}$ + P$_{2}$r$_{1}$ )Δ$_{l}$ . (3.15) max t ∈ J 1 l ‖ y ( t ) − y ( t ) ‖ ≥ L$_{1}$ Δ$_{l}$ 4 ϵ$_{0}$ − ( 1 + ( P$_{1}$ + P$_{2}$r$_{1}$ ) Δ$_{l}$ 4 ) max t ∈ J 1 l ‖ y ( t ) − y ( t ) ‖ , ‖ f ( y ( s ) + ν$_{β}$ ( s ) , s ) − f ( y ( s ) + ν$_{β}$ ( s ) , s ) ‖ ≤ ( P$_{1}$ + P$_{2}$ max {‖ y ( s ) ‖ , ‖ y ( s ) ‖} ) ‖ y ( s ) − y ( s ) ‖ ≤ ( P$_{1}$ + P$_{2}$r$_{1}$ ) max t ∈ J 1 l ‖ y ( t ) − y ( t ) ‖ max t ∈ J 1 l ‖ y ( t ) − y ( t ) ‖ ≥ ‖ y ( b$_{l}$ + Δ$_{l}$ / 4) − y ( b$_{l}$ + Δ$_{l}$ / 4) ‖ ≥ L$_{1}$ Δ$_{l}$ ϵ$_{0}$ 4 − ‖ y ( b$_{l}$ ) − y ( b$_{l}$ ) ‖ − ( P$_{1}$ + P$_{2}$r$_{1}$ ) Δ$_{l}$ 4 max t ∈ J 1 l ‖ y ( t ) − y ( t ) ‖ , y ( t ) − y ( t ) = ( y ( b$_{l}$ ) − y ( b$_{l}$ )) + t ∫ b l ( f ( y ( s ) + ν$_{β}$ ( s ) , s ) − f ( y ( s ) + ν$_{β}$ ( s ) , s )) ds + t ∫ b l ( h ( x ( s )) − h ( x ( s ))) ds,