autonomous system (3.5) with the autonomous equation where F : R m × R → R m is a continuous function in all of its arguments. We suppose that system (3.5) ((3.6)) admits a chaotic attractor, let us say a set in R m for (3.6). Fix x$_{0}$ from the attractor, and take a solution x ( t ) of (3.6) with x (0) = x$_{0}$. Since x$_{0}$ is a member of the chaotic attractor, we will call x ( t ) a chaotic solution [1, 3, 13, 66]. There exists a compact set Λ ⊂ R m such that the trajectories of the chaotic solutions of (3.5) ((3.6)) lie inside Λ for all t. If we denote M$_{H}$ = max x ∈ Λ ‖ H ( x ) ‖ , then one can verify that sup t ∈ R ∥ φ$_{x}$$_{(}$$_{t}$$_{)}$ ( t ) ∥ ≤ N ( M$_{G}$ + M$_{H}$ ) ω for each chaotic solution x ( t ) of (3.5) ((3.6)). The following conditions are needed throughout the section. The next subsection is concerned with the extension of sensitivity. Let us describe the sensitivity feature for system (3.5) as well as its replication by system (1.1). System (3.5) is called sensitive if there exist positive numbers ϵ$_{0}$ and Δ such that for an arbitrary positive number δ$_{0}$ and for each chaotic solution x ( t ) of (3.5), there exist a chaotic solution x ( t ) of the same system, t$_{0}$ ∈ R and an interval J ⊂ [ t$_{0}$, ∞ ) with a length no less than Δ such that ‖ x ( t$_{0}$ ) − x ( t$_{0}$ ) ‖ < δ$_{0}$ and ‖ x ( t ) − x ( t ) ‖ > ϵ$_{0}$ for all t ∈ J. We say that system (1.1) replicates the sensitivity of (3.5) if there exist positive numbers ϵ$_{1}$ and Δ such that for an arbitrary positive number δ$_{1}$ and for each chaotic solution x ( t ) of (3.5), there exist a chaotic solution x ( t ) of (3.5), t$_{0}$ ∈ R and an interval J 1 ⊂ [ t$_{0}$, ∞ ) with a length no less than Δ such that ∥ φ$_{x}$$_{(}$$_{t}$$_{)}$ ( t$_{0}$ ) − φ$_{x}$$_{(}$$_{t}$$_{)}$ ( t$_{0}$ ) ∥ < δ$_{1}$ and ∥ φ$_{x}$$_{(}$$_{t}$$_{)}$ ( t ) − φ$_{x}$$_{(}$$_{t}$$_{)}$ ( t ) ∥ > ϵ$_{1}$ for all t ∈ J $^{1}$. The next assertion is about the sensitivity feature of system (1.1). Lemma 3.1 Under the conditions ( C 1) − ( C 5) , system (1.1) replicates the sensitivity of (3.5). We omit the proof of Lemma 3.1 since it can be proved in a very similar way to Lemma 5 . 1 [32]. We will handle the presence of chaos through a cascade of almost periodic motions in system (1.1) in the following subsection. 6 3.1 Extension of Sensitivity (C3) There exists a positive number L$_{2}$ such that ‖ H ( x$_{1}$ ) − H ( x$_{2}$ ) ‖ ≤ L$_{2}$ ‖ x$_{1}$ − x$_{2}$ ‖ for all x$_{1}$, x$_{2}$ ∈ Λ; (C4) There exists a positive number L$_{3}$ such that ‖ H ( x$_{1}$ ) − H ( x$_{2}$ ) ‖ ≥ L$_{3}$ ‖ x$_{1}$ − x$_{2}$ ‖ for all x$_{1}$, x$_{2}$ ∈ Λ; (C5) There exists a positive number L$_{4}$ such that ‖ F ( t, x$_{1}$, µ ∞ ) − F ( t, x$_{2}$, µ ∞ ) ‖ ≤ L$_{4}$ ‖ x$_{1}$ − x$_{2}$ ‖ for all t ∈ R , x$_{1}$, x$_{2}$ ∈ Λ . x ′ = F ( x, µ ∞ ) , (3.6)