3 Sensitivity in the response system Fix a point x$_{0}$ from the chaotic attractor of (2.1) and take a solution x ( t ) with x (0) = x$_{0}$. Since we use the solution x ( t ) as a perturbation in (2.3), we call it a chaotic function . Chaotic functions may be irregular as well as regular (periodic and unstable) [3, 21]. We suppose that the response system (2.3) possesses a compact invariant set U ⊂ R 3 for each chaotic solution x ( t ) of (2.1). The existence of such an invariant set can be shown, for example, using Lyapunov functions [19, 22]. One of main ingredients of chaos is sensitivity [3, 5]. Let us describe this feature for both the drive and response systems. System (2.1) is called sensitive if there exist positive numbers ϵ$_{0}$ and Δ such that for an arbitrary positive number δ$_{0}$ and for each chaotic solution x ( t ) of (2.1), there exist a chaotic solution x ( t ) of the same system and an interval J ⊂ [0 , ∞ ) , with a length no less than Δ , such that ‖ x (0) − x (0) ‖ < δ$_{0}$ and ‖ x ( t ) − x ( t ) ‖ > ϵ$_{0}$ for all t ∈ J. For a given solution x ( t ) of (2.1), let us denote by φ$_{x}$$_{(}$$_{t}$$_{)}$ ( t, y$_{0}$ ) the unique solution of (2.3) satisfying the condition φ$_{x}$$_{(}$$_{t}$$_{)}$ (0 , y$_{0}$ ) = y$_{0}$. We say that system (2.3) is sensitive if there exist positive numbers ϵ$_{1}$ and Δ such that for an arbitrary positive number δ$_{1}$, each y$_{0}$ ∈ U , and a chaotic solution x ( t ) of (2.1), there exist y$_{1}$ ∈ U , a chaotic solution x ( t ) of (2.1), and an interval J 1 ⊂ [0 , ∞ ) , with a length no less than Δ , such that ‖ y$_{0}$ − y$_{1}$ ‖ < δ$_{1}$ and ∥ φ$_{x}$$_{(}$$_{t}$$_{)}$ ( t, y$_{0}$ ) − φ$_{x}$$_{(}$$_{t}$$_{)}$ ( t, y$_{1}$ ) ∥ > ϵ$_{1}$ for all t ∈ J $^{1}$. Next theorem confirms that the sensitivity feature remains persistent for (2.2) when it is perturbed with the solutions of the drive system (2.1). This feature is true even if the systems (2.1) and (2.3) are not synchronized in the generalized sense. The proof of Theorem 3.1 is provided in the Appendix. In the next section, we will demonstrate that the response system possesses chaotic motions regardless of the presence of GS. 4 Chaotic dynamics in the absence of generalized synchroniza- tion Let us take into account the drive system (2.1) with the parameter values σ = 10 , r = 28 , b = 8 / 3 such that the system possesses a chaotic attractor [3, 21]. Moreover, we set σ = 10 , r = 60 , b = 8 / 3 and g$_{1}$ ( x$_{1}$, x$_{2}$, x$_{3}$ ) = 2 . 95 x$_{1}$ − 0 . 25 sin x$_{1}$, g$_{2}$ ( x$_{1}$, x$_{2}$, x$_{3}$ ) = 3 . 06 arctan x$_{2}$, g$_{3}$ ( x$_{1}$, x$_{2}$, x$_{3}$ ) = 3 . 12 x$_{3}$ + 1 . 75 e − x 3 in the response system (2.3). The unperturbed Lorenz system (2.2) is also chaotic with the aforementioned values of σ, r, and b [8, 21]. In order to demonstrate the presence of sensitivity in the response system (2.3) numerically, we depict in Figure 1 the projections of two initially nearby trajectories of the coupled system (2 . 1) + (2 . 3) on the 4 Theorem 3.1 The response system (2.3) is sensitive.