uniformly continuous. Moreover, any system of differential equations which admits uniformly continuous and bounded unpredictable solution has a Poincaré chaos. For differential equations, we say that a solution is an unpredictable one if it is uniformly continuous and bounded on the real axis. Let us consider the system where all eigenvalues of the constant p × p matrix A have negative real parts, the function f : R p → R p is bounded, and the function g : $^{R}$→ R p is a uniformly continous and bounded. Since the eigenvalues of the matrix A have negative real parts, there exist positive numbers K$_{0}$ and ω such that ∥ e At ∥ ≤ K$_{0}$e ωt for t ≥ 0 [18]. The presence of an unpredictable solution in the dynamics of (2.2) is mentioned in the next theorem. Theorem 2.2 ([3]) If g ( t ) is an unpredictable function and the function f ( x ) is Lipschitzian with a sufficiently small Lipschitz constant L$_{f}$ such that K$_{0}$L$_{f}$ − ω < 0 , then system (2.2) possesses a unique uniformly exponentially stable unpredictable solution. In the next two sections we will consider unpredictable functions whose domain consists of all integers, that is, unpredictable sequences . In this section, we will show the presence of an unpredictable point in the symbolic dynamics [5, 19] with a distinguishing feature. Let us consider the space Σ$_{2}$ = { s = ( s$_{0}$s$_{1}$s$_{2}$ . . . ) | s$_{j}$ = 0 or 1 } of infinite sequences of 0 ’s and 1 ’s with the metric where s = ( s$_{0}$s$_{1}$s$_{2}$ . . . ) , t = ( t$_{0}$t$_{1}$t$_{2}$ . . . ) ∈ Σ$_{2}$ . The Bernoulli shift σ : Σ$_{2}$ → Σ$_{2}$ is defined as σ ( s$_{0}$s$_{1}$s$_{2}$ . . . ) = ( s$_{1}$s$_{2}$s$_{3}$ . . . ) . The map σ is continuous and Σ$_{2}$ is a compact metric space [5, 19]. In the book [5], the sequence was considered, which is constructed by successively listing all blocks of 0 ’s and 1 ’s of length n, then length n + 1 , etc. In the proof of the next lemma, an element s ∗∗ = ( s ∗∗ 0 s ∗∗ 1 s ∗∗ 2 . . . ) of Σ$_{2}$ will be constructed in a similar way to s ∗ with the only difference that the order of the blocks will be chosen in a special way. An extension of the constructed sequence to the left hand side will also be provided. 5 s ∗ = ( 0 1 ︸︷︷︸ 1 blocks | 00 01 10 11 ︸ ︷︷ ︸ 2 blocks | 000 001 010 011 . . . ︸ ︷︷ ︸ 3 blocks | . . . ) , (3.3) d ( s, t ) = ∞ ∑ k =0 | s$_{k}$ − t$_{k}$ | 2 k , x $^{′}$( t ) = Ax ( t ) + f ( x ( t )) + g ( t ) , (2.2) 3 An unpredictable sequence of the symbolic dynamics