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Making use of the solution y ( t ) = ( y$_{1}$ ( t ) , y$_{2}$ ( t ) , y$_{3}$ ( t )) of (2.3) corresponding to the initial conditions x$_{1}$ (0) = 9 . 47 , x$_{2}$ (0) = 3 . 29 , x$_{3}$ (0) = 34 . 49 , y$_{1}$ (0) = 10 . 67 , y$_{2}$ (0) = − 8 . 06 , y$_{3}$ (0) = 71 . 89 , the largest Lyapunov exponent of (4.5) is evaluated as 0 . 7693 . In other words, the response (2.3) possesses a positive conditional Lyapunov exponent, and this corroborates the absence of GS for the coupled systems (2 . 1) + (2 . 3) . 

The next section is devoted to the presence of periodic motions embedded in the chaotic attractor of the response system. 

## 5 Periodic solutions of the response system 

To demonstrate the existence of periodic motions embedded in the chaotic attractor of (2.3), we will take into account (2.1) with the parameter values in such a way that the system exhibits a period-doubling cascade [23, 24]. 

Consider the drive system (2.1) in which σ = 10 , b = 8 / 3 and r is a parameter [13, 21]. For the values of r between 99 . 98 and 100 . 795 the system possesses two symmetric stable periodic orbits such that one of them spirals round twice in x$_{1}$ > 0 and once in x$_{1}$ < 0 , whereas another spirals round twice in x$_{1}$ < 0 and once in x$_{1}$ > 0 . The book [21] calls such periodic orbits as x $^{2}$y and y $^{2}$x, respectively. More precisely, “ x ” is written every time when the orbit spirals round in x$_{1}$ > 0 , while “ y ” is written every time when it spirals round in x$_{1}$ < 0 . As the value of the parameter r decreases towards 99 . 98 a period- doubling bifurcation occurs in (2.1) such that two new symmetric stable periodic orbits ( x $^{2}$yx$^{2}$y and y $^{2}$xy$^{2}$x ) appear, and the previous periodic orbits lose their stability [13, 21]. According to Franceschini [13], system (2.1) undergoes infinitely many period-doubling bifurcations at the parameter values 99 . 547 , 99 . 529 , 99 . 5255 and so on. The sequence of bifurcation parameter values accumulates at r$_{∞}$ = 99 . 524 . For values of r smaller than r$_{∞}$ , infinitely many unstable periodic orbits take place in the dynamics of 

