content_image/327355.jpg content_image/327356.jpg content_image/327357.jpg content_image/327358.jpg
purpose, we take into account the Duffing equation 

which is equivalent to the system 

<img src='content_image/327355.jpg'>

<img src='content_image/327356.jpg'>

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 

<img src='content_image/327357.jpg'>

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. 

<img src='content_image/327358.jpg'>

To confirm one more time that the 6 − dimensional system (5 . 11) + (5 . 14) + (5 . 17) admits a chaotic 

