purpose, we take into account the Duffing equation which is equivalent to the system in view of the variables v$_{1}$ = v and v$_{2}$ = v $^{′}$. One can verify that (5.16) admits an asymptotically stable equilibrium point. We perturb system (5.16) with the solutions of (5.14) to constitute the system It is worth noting that (5.17) is in the form of (1.1) with A =    0 1 − 4 − 2    , whose eigenvalues are − 1 ± $_{i}$√$_{3}$$_{.}$ It can be shown in a similar way to system (5.14) that conditions ( C 1) − ( C 4) are valid also for (5.17). According to our results, the collection C of bounded on R solutions of (5.17) for which ( y$_{1}$ ( t ) , y$_{2}$ ( t )) belongs to B is a Li-Yorke chaotic set with infinitely many quasi-periodic motions. Utilizing the solution of (5.14) that is shown in Figure 3 as the perturbation in system (5.17), we depict in Figure 4 the solution of (5.17) with z$_{1}$ ( t$_{0}$ ) = 0 . 37 , z$_{2}$ ( t$_{0}$ ) = − 3 . 71 , where t$_{0}$ = 0 . 61 . Figure 4 reveals that the system (5.17) is Li-Yorke chaotic. v ′′ + 2 v ′ + 4 v + 0 . 0002 v 3 = 0 , (5.15) v ′ 1 = v$_{2}$ v ′ 2 = − 4 v$_{1}$ − 2 v$_{2}$ − 0 . 0002 v 3 $_{1}$, (5.16) z ′ 1 = z$_{2}$ + y$_{1}$ ( t ) + 1 2 sin( y$_{1}$ ( t )) z ′ 2 = − 4 z$_{1}$ − 2 z$_{2}$ − 0 . 0002 z 3 1 + 1 5 y$_{2}$ ( t ) + 1 10 y 3 $_{2}$( t ) . (5.17) 0 10 20 30 40 50 60 70 80 90 100 −2 0 2 t z 1 0 10 20 30 40 50 60 70 80 90 100 −6 −4 −2 0 2 t z 2 Figure 4: The chaotic behavior of system (5.17). To confirm one more time that the 6 − dimensional system (5 . 11) + (5 . 14) + (5 . 17) admits a chaotic 13