set is Poincaré chaotic, and there are infinitely many unpredictable sequences in the set. The last observation will be applied in the next section to construct an unpredictable function. In this part of the paper, we will provide an example of an unpredictable function benefiting from the dynamics of the logistic map (4.4). Let us fix two different points d$_{1}$ and d$_{2}$ in R $^{p}$, and suppose that γ is a positive number. Take a se- quence { k$_{n}$ } of positive integers satisfying both of the inequalities 2 − k$_{n}$ ≤ 1 2 n and e − γk$_{n}$ ≤ γ 4 ‖ d$_{1}$ − d$_{2}$ ‖ n for each n ∈ $^{N}$. Fix µ ∈ [3 + (2 / 3) 1 / $^{2}$, 4] and a sequence { δ$_{n}$ } of positive numbers such that δ$_{n}$ ≤ 1 12 ‖ d$_{1}$ − d$_{2}$ ‖ nk$_{n}$ , n ∈ $^{N}$. In a similar way to the items ( i ) and ( ii ) of Theorem 4.1, one can verify that there exist a positive number ϵ$_{0}$, a sequence { i$_{n}$ } of even positive integers, a sequence { j$_{n}$ } of positive integers, and a solution { η$_{n}$ } , n ∈ $^{Z}$, of the logistic map (4.4) such that the inequalities and hold for each n ∈ $^{N}$. It is easy to observe that the constructed sequence { η$_{n}$ } is unpredictable and consequently, it generates a quasi-minimal set and Poincaré chaos similar to that of the last section. Now, consider the function φ : $^{R}$→ R p defined as where the function ν ( t ) is defined as and the sequence { ζ$_{j}$ } , j ∈ $^{Z}$, of switching moments is defined through the equation ζ$_{j}$ = j + η$_{j}$ for each j, in which { η$_{j}$ } is the solution of (4.4) satisfying (5.5) and (5.6). The function φ ( t ) is bounded such that sup t ∈ R ‖ φ ( t ) ‖ ≤ max {‖ d$_{1}$ ‖ , ‖ d$_{2}$ ‖} γ . Moreover, φ ( t ) is uniformly continuous since its derivative is bounded. In the proof of the following theorem, we will denote by ̂ ( a, b ] the oriented interval such that ̂ ( a, b ] = 8 ν ( t ) =     d$_{1}$, if ζ$_{2}$$_{j}$ < t ≤ ζ$_{2}$$_{j}$$_{+1}$, j ∈ $^{Z}$, d$_{2}$, if ζ$_{2}$$_{j}$ − 1 < t ≤ ζ$_{2}$$_{j}$, j ∈ $^{Z}$, φ ( t ) = t ∫ −∞ e − γ ( t − s $^{)}$ν ( s ) ds, (5.7) | η$_{i}$$_{n}$$_{+}$$_{j}$$_{n}$ − η$_{j}$$_{n}$ | ≥ ϵ$_{0}$ (5.6) | η$_{i}$$_{n}$$_{+}$$_{r}$ − η$_{r}$ | ≤ δ$_{n}$, r = − 2 k$_{n}$, − 2 k$_{n}$ + 1 , . . . , k$_{n}$ − 1 , (5.5) 5 An unpredictable function