ϵ, for each t ∈ $_{J}$; (LY2) A couple of functions ( ψ ( t ) , ψ ( t ) ∈ B × B is frequently ( ϵ$_{0}$, Δ) $_{−}$separated if there exist positive real numbers ϵ$_{0}$, Δ and infinitely many disjoint intervals of length not less than $_{Δ}$, such that ∥ ψ ( t ) − ψ ( t ) ∥ > ϵ$_{0}$, for each t from these intervals. A couple of functions ( ψ ( t ) , ψ ( t ) ∈ B × B is a Li$_{−}$Yorke pair if it is proximal and frequently ( ϵ$_{0}$, $_{Δ)}$- separated for some positive numbers ϵ$_{0}$ and Δ . On the other hand, a set C ⊂ B is called a scrambled set if C does not contain any almost periodic function and each couple of different functions inside C × C is a Li$_{−}$Yorke pair. Let us introduce the sets of functions B is called a Li$_{−}$Yorke chaotic set if: ( i ) it admits a countably infinite set of almost periodic functions; ( ii ) there exists an uncountable scrambled subset C ⊂ B ; ( iii ) for any ψ ( t ) ∈ C and any almost periodic ψ ( t ) ∈ B , the pair ( ψ ( t ) , ψ ( t ) is frequently ( ϵ$_{0}$, Δ) $_{−}$separated for some positive real numbers ϵ$_{0}$ and Δ . The next section is devoted for the clarification of the theoretical results for the chaos extension in systems of the form (2 . 3) + (2 . 5) . The present section is devoted for the rigorous proofs for the extension of chaos in the sense of Li$_{−}$Yorke. We start our discussions with the first ingredient, proximality, of Li$_{−}$Yorke chaos. Lemma 3.1 Suppose that conditions ( C 1) − ( C 6) are valid. If a couple ( x ( t ) , ˜ x ( t )) ∈ A$_{x}$ × A$_{x}$ is proximal, then the same is true for the couple ( φ β x ( t $_{)}$( t ) , φ β ˜ x ( t $_{)}$( t ) ∈ A β y × A β y with the corresponding interval J uniform for all β ∈ S$_{m}$ . Proof. Fix an arbitrary small number ϵ > 0 and an arbitrary large number E > 0 . Take ϵ$_{1}$ > 0 and E$_{1}$ > 0 that will be specified later. Then there is a couple of functions ( x ( t ) , ˜ x ( t )) ∈ A$_{x}$ × A$_{x}$ which is proximal with constants $_{ϵ$_{1}$}$, E$_{1}$ and interval J$_{1}$ = [ a$_{1}$, a$_{1}$ + 3 E$_{1}$ ] for some $_{a$_{1}$}$. Let y ( t ) = φ β x ( t $_{)}$( t ) and 9 3 Extension of Chaos A β y = { φ β x ( t $_{)}$( t ) | x ( t ) ∈ A$_{x}$ } , β ∈ S$_{m}$. (2.13)