Figure 2: The graph of the z$_{1}$ − coordinate of system ( 5.16). One can observe in the figure that system (5.16) admits Li-Yorke chaos. systems (5.12) and (5.16) to set up the system System (5.17) is in the form of (1.6) with h ( x$_{1}$, x$_{2}$ ) =    0 . 7 x 2 1 − 1 . 2 x$_{1}$ arctan( x$_{2}$ )    . It can be numerically verified that the bounded solutions of (5.17) lie inside t he compact region Therefore, the conditions ( C 1) and ( C 2) are valid f or (5.17) with M$_{f}$ = 1 . 50729 and L$_{f}$ = 0 . 0243 . Moreover, the function h ( x$_{1}$, x$_{2}$ ) clearly satisfies the conditions ( C 4) and ( C 5) on the compact region Λ defined by (5.13). The chaos of system (5.16) is pers istent under the applied perturbation such that system (5.17) possesses Li-Yorke chaos with infinitely many almost periodic motions in basis according to Theorem 4.1. Using the solution ( x$_{1}$ ( t ) , x$_{2}$ ( t )) of (5.12) whose first coordinate is represented in Figure 1 in the perturbation, we depict in Figure 3 the y$_{1}$ − coordin ate of the solution of (5.17) corresponding to the initial data ζ$_{0}$ = 0 . 56 , y$_{1}$ ( t$_{0}$ ) = 0 . 43 , y$_{2}$ ( t$_{0}$ ) = 0 . 04 , w here t$_{0}$ = 0 . 56 . Furthermore, Figure 4 shows the trajectory of the same solution on the y$_{1}$ − y$_{2}$ plan e . Both of Figures 3 and 4 support the result of Theorem 4.1 such that Li-Yorke chaos is permanent in the dynamics of (5.16) even if the perturbation h ( x$_{1}$, x$_{2}$ ) is applied. Figure 3: The time series of the y$_{1}$ − coordinate of system (5.17). The figure reveals the persistence of chaos. 11 0 50 100 150 200 250 −0.5 0 0.5 t y 1 y ′ 1 = y$_{2}$ + 0 . 7 x 2 1 − 1 . 2 x$_{1}$, y ′ 2 = − 2 . 5 y$_{1}$ − 3 . 5 y$_{2}$ + 0 . 01 y 3 1 − 1 . 5 c o s( πt ) + ν$_{2}$ ( t, ζ ) + arctan( x$_{2}$ ) . (5.17) R = { ( y$_{1}$, y$_{2}$ ) ∈ R 2 : − 0 . 9 ≤ y$_{1}$ ≤ 0 . 8 , − 0 . 65 ≤ y$_{2}$ ≤ 1 . 2 } . 0 50 100 150 200 250 −0.2 0 0.2 0.4 0.6 t z 1