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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 

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