Figure 2: Chaotic trajectory of system (2.3). The figure manifests the persistence of chaos. holds, where y ( t, x$_{0}$, y$_{0}$ ) , y ( t, x$_{0}$, y$_{0}$ ) denote the solutions of (2.3) with the initial data y (0 , x$_{0}$, y$_{0}$ ) = y$_{0}$, y (0 , x$_{0}$, y$_{0}$ ) = y$_{0}$ and the same solution x ( t ) , x (0) = x$_{0}$, of (2.1). A numerical technique that can be used to analyze coupled systems for the presence or absence of GS is the auxiliary system approach [12]. We will make use of this technique for the coupled system (2.1)+(2.3). For that purpose, we consider the auxiliary system which is an identical copy of (2.3). Using the initial data x$_{1}$ (0) = 4 . 43 , x$_{2}$ (0) = − 2 . 27 , x$_{3}$ (0) = 30 . 81 , y$_{1}$ (0) = 3 . 09 , y$_{2}$ (0) = 4 . 98 , y$_{3}$ (0) = 46 . 21 , z$_{1}$ (0) = 7 . 69 , z$_{2}$ (0) = 6 . 25 , z$_{3}$ (0) = 52 . 65 , we depict in Figure 3 the projection of the stroboscopic plot of the 9 − dimensional system (2 . 1)+(2 . 3)+(4 . 4) on the y$_{2}$ − z$_{2}$ plane. In the simulation, the first 200 iterations are omitted. Since the stroboscopic plot does not take place on the line z$_{2}$ = y$_{2}$, the systems (2.1) and (2.3) are unsynchronized. Hence, the response system (2.3) exhibits chaotic behavior even if GS does not occur in the systems (2.1) and (2.3). In order to demonstrate the absence of GS one more time by evaluating the conditional Lyapunov exponents [8, 9, 11], we consider the following variational system for (2.3), ˙ ξ$_{1}$ = − 10 ξ$_{1}$ + 10 ξ$_{2}$ ˙ ξ$_{2}$ = ( − y$_{3}$ ( t ) + 60) ξ$_{1}$ − ξ$_{2}$ − y$_{1}$ ( t ) ξ$_{3}$ ˙ ξ$_{3}$ = y$_{2}$ ( t ) ξ$_{1}$ + y$_{1}$ ( t ) ξ$_{2}$ − (8 / 3) ξ$_{3}$. (4.5) ˙ z$_{1}$ = − 10 z$_{1}$ + 10 z$_{2}$ + 2 . 95 x$_{1}$ ( t ) − 0 . 25 sin( x$_{1}$ ( t )) ˙ z$_{2}$ = − z$_{1}$z$_{3}$ + 60 z$_{1}$ − z$_{2}$ + 3 . 06 arctan( x$_{2}$ ( t )) ˙ z$_{3}$ = z$_{1}$z$_{2}$ − 8 3 z$_{3}$ + 3 . 12 x$_{3}$ ( t ) + 1 . 75 e − x 3 ( t $^{)}$, (4.4) 6 −40 −20 0 20 40 −40 −20 0 20 40 40 50 60 70 80 90 y$_{2}$ y$_{1}$ y 3