consider the solution of (5.13) with initial data u$_{1}$ (0) = 0 . 1 , u$_{2}$ (0) = 0 . 2 . The u$_{1}$ and u$_{2}$ coordinates of the solution are depicted in Figure 2, where one can observe that the represented solution approaches to the asymptotically stable quasi-periodic solution such that the system does not possess chaos. It can be numerically verified that the bounded on R solutions of (5.14) lie inside the compact region R = { ( y$_{1}$, y$_{2}$ ) ∈ R 2 : − 1 . 1 ≤ y$_{1}$ ≤ 4 . 7 , − 5 . 3 ≤ y$_{2}$ ≤ 0 . 5 } . Therefore, it is reasonable to consider the dynamics of (5.14) inside R , and conditions ( C 1) − ( C 4) are valid for system (5.14) with M$_{G}$ = 2 . 0104 , L$_{1}$ = 0 . 006627 , L$_{2}$ $_{= 6}$√$_{2}$$_{,}$ and L$_{3}$ = √$_{2}$$_{.}$ According to the theoretical results of Section 4, the set B consisting of the bounded on R solutions of (5.14) for which ( x$_{1}$ ( t ) , x$_{2}$ ( t )) belongs to A is Li-Yorke chaotic with infinitely many quasi-periodic solutions, which are separated from the motions of the scrambled set. That is, the applied perturbation H ( x$_{1}$ ( t ) , x$_{2}$ ( t )) effects (5.13) in such a way that Li-Yorke chaos takes place in the dynamics of (5.14). In system (5.14) we use the solution of (5.11) that is depicted in Figure 1, and we represent in Figure 3 the solution of (5.14) corresponding to the initial data y$_{1}$ ( t$_{0}$ ) = 1 . 32 , y$_{2}$ ( t$_{0}$ ) = − 2 . 25 , where t$_{0}$ = 0 . 61 . Figure 3 supports the result of Theorem 4.1 such that system (5.14) possesses chaotic motions. Figure 2: The quasi-periodic behavior of system (5.13). 0 50 100 150 −1 0 1 t u 1 0 50 100 150 −1 0 1 t u 2 0 10 20 30 40 50 60 70 80 90 100 0 2 4 t y 1 0 10 20 30 40 50 60 70 80 90 100 −4 −2 0 t y 2 Figure 3: The graphs of the y$_{1}$ and y$_{2}$ coordinates of system (5.14). Now, we will demonstrate that the chaos formation mechanism can be proceeded further. For that 12