system (1.1), which satisfies the following relation, Moreover, for a fixed x ( t ) , if y ( t ) is a solution of (1.1) with y (0) = y$_{0}$ ∈ R $^{n}$, then the inequality ∥ φ$_{x}$$_{(}$$_{t}$$_{)}$ ( t ) − y ( t ) ∥ ≤ N ∥ φ$_{x}$$_{(}$$_{t}$$_{)}$ (0) − y$_{0}$ ∥ e ( NL$_{1}$ − ω ) t holds for t ≥ 0 so that ∥ φ$_{x}$$_{(}$$_{t}$$_{)}$ ( t ) − y ( t ) ∥ → 0 as t → ∞ . In other words, the bounded on R solution φ$_{x}$$_{(}$$_{t}$$_{)}$ ( t ) attracts all other solutions of (1.1). In this section, we will deal with the presence of chaos in the dynamics of (1.1) through a cascade of almost periodic solutions. For that purpose, we will take advantage of a period-doubling cascade [3, 21, 22, 64, 65] of a system which will be used as the source for the perturbation function x ( t ) in (1.1). We will understand chaos in system (1.1) as the presence of sensitivity, which is the main ingredient of chaos [1, 13, 66], and the existence of infinitely many unstable almost periodic solutions in a bounded region. Let us consider the system where µ is a parameter and the function F : R × R m × R → R m is continuous in all of its arguments. Suppose that there exists a positive number T such that F ( t + T, x, µ ) = F ( t, x, µ ) for all t ∈ R , x ∈ R m and µ ∈ R . We assume that system (3.4) admits a period-doubling cascade [3, 21, 22], that is, there exist a finite number µ ∞ and a sequence of period-doubling bifurcation values { µ$_{j}$ } , j ∈ N , satisfying µ$_{j}$ → µ ∞ as j → ∞ such that for each j as the parameter µ increases or decreases through µ$_{j}$, system (3 . 4) undergoes a period-doubling bifurcation, which gives rise to the formation of a new periodic solution with a twice period of the former one. At the parameter value µ = µ ∞ , there exist infinitely many unstable periodic solutions of (3.4) all lying in a bounded region. The perturbation function x ( t ) in (1.1) will be provided by system (3.4) with µ = µ ∞ , that is, it will be a solution of the system It is worth noting that the results of the present section are valid even if we replace the non- 5 3 Chaos Through a Cascade of Almost Periodic Solutions φ$_{x}$$_{(}$$_{t}$$_{)}$ ( t ) = ∫ t −∞ e A ( t − s $^{)}$[ G ( s, φ$_{x}$$_{(}$$_{t}$$_{)}$ ( s )) + H ( x ( s ))] ds. (2.3) x ′ = F ( t, x, µ ) , (3.4) x ′ = F ( t, x, µ ∞ ) . (3.5)