Now, we state and prove the main theorem of the present section by indicating the extension of chaos in the sense of Li-Yorke for the system (2.3)+(2.5). Theorem 3.1 Let ν$_{β}$ ( t ) be almost periodic. If the generator (2.3) admits a Li-Yorke chaotic set A$_{x}$ , then the replicator (2.5) has Li-Yorke chaotic sets A β y , β ∈ S$_{m}$ . Proof. Assume that the set A$_{x}$ is Li-Yorke chaotic. We need to show that A β y is a Li-Yorke chaotic set as well. First, there is a countably infinite subset P$_{x}$ ⊂ A$_{x}$ consisting from of almost periodic functions. Let us denote Condition$_{(}$$_{C}$$_{4)}$ implies that there is a one-to-one correspondence between the sets P$_{x}$ and P β y . Therefore, P β y is also infinite countable. Furthermore, Lemma 2.2 implies that P β y consists from almost periodic functions. Hence the point (i) in the definition of Li-Yorke chaotic set for A β y is verified. Now, suppose that C$_{x}$ is an uncountable scrambled subset of A$_{x}$ . Let us introduce Condition ( C 4) again implies that there is a one-to-one correspondence between the sets C$_{x}$ and C β y . Therefore, C β y is also uncountable. Under the same condition, it is easy to verify that there does not exist any almost periodic function inside the set C β y . Indeed, assume that y ∈ C β y is almost periodic. Then by (3.16), y ( t ) is uniformly continuous on R . Furthermore, using conditions (C3) and (C5), we see that x ( t ) and ν$_{β}$ ( t ) are uniformly continuous on R as well. So assumption (C1) together with (2.9) give that y $^{′}$( t ) is also uniformly continuous on R . Then by [34], y $^{′}$( t ) is almost periodic, hence by [29], function is almost periodic. Next, condition (C4) implies that h : R n → R n is a homeomorphism (see [15, Theorems (2.7.1), (5.1.4), (5.4.11)]). So by (2.9) we obtain x ( t ) = h − $^{1}$( z ( t $_{))}$, then by [34], x ( t ) is almost periodic which contradicts to x ∈ C$_{x}$ . So any y ∈ C β y is not almost periodic. As a matter of fact we have shown that x ∈ C$_{x}$ is almost periodic if and only if the corresponding y ∈ C β y is almost periodic. Next, since the collection A$_{x}$ is assumed to be chaotic in the sense of Li-Yorke, each couple of functions inside C$_{x}$ × C$_{x}$ is proximal. Thus, Lemma 3.1 implies that the same is valid for each couple inside C β y × C β y . Hence the point (ii) in the definition of Li-Yorke chaotic set for A β y is verified as well. On the other hand, Lemma 3 . 2 implies the existence of positive real numbers ϵ$_{1}$ and Δ such that each couple of functions ( y ( t ) , y ( t )) ∈ C β y × C β y are frequently ( ϵ$_{1}$, Δ ) $_{−}$separated. The same property is 13 z ( t ) = y $^{′}$( t ) − f ( y ( t ) + ν$_{β}$ ( t ) , t ) + f ( ν$_{β}$ ( t ) , t ) C β y = { φ β x ( t $_{)}$( t ) : x ( t ) ∈ C$_{x}$ } . (3.19) P β y = { φ β x ( t $_{)}$( t ) : x ( t ) ∈ P$_{x}$ } ⊂ A β y . (3.18)