Bebutov dynamical system [14], and consequently, the unpredictable functions and solutions immediately imply the Poincaré chaos presence according to our results in [4]. The second one is the easy verification of the convergence. Thus, the present study is useful for the theory of differential equations as well as chaos researches. For the construction of unpredictable functions we have applied the significant results on the equivalence of discrete dynamics obtained in papers [15, 16]. Let us introduce the following definition. Definition 1.1 A uniformly continuous and bounded function ϕ : R → R m is unpredictable if there exist posi- tive numbers ϵ$_{0}$, δ, and sequences { t$_{n}$ } , { u$_{n}$ } both of which diverge to infinity such that ‖ ϕ ( t + t$_{n}$ ) − ϕ ( t ) ‖ → 0 as n → ∞ on compact subsets of R , and ‖ ϕ ( t + t$_{n}$ ) − ϕ ( t ) ‖ ≥ ϵ$_{0}$ for each t ∈ [ u$_{n}$ − δ, u$_{n}$ + δ ] and n ∈ N . To create Poincaré chaos [4], uniform continuity is not a necessary condition for an unpredictable function ϕ ( t ) , and instead of the condition ‖ ϕ ( t + t$_{n}$ ) − ϕ ( t ) ‖ ≥ ϵ$_{0}$ for each t ∈ [ u$_{n}$ − δ, u$_{n}$ + δ ] and n ∈ N , one can request that ‖ ϕ ( t$_{n}$ + u$_{n}$ ) − ϕ ( u$_{n}$ ) ‖ ≥ ϵ$_{0}$ for each n ∈ N . For the needs of verification of theorems on the existence of unpredictable solutions of differential equations we apply Definition 1.1, but for the future studies the following definition may also be beneficial. Definition 1.2 A continuous and bounded function ϕ : R → R m is unpredictable if there exist positive numbers ϵ$_{0}$, δ, and sequences { t$_{n}$ } , { u$_{n}$ } both of which diverge to infinity such that ‖ ϕ ( t + t$_{n}$ ) − ϕ ( t ) ‖ → 0 as n → ∞ on compact subsets of R , and ‖ ϕ ( t$_{n}$ + u$_{n}$ ) − ϕ ( u$_{n}$ ) ‖ ≥ ϵ$_{0}$ for each n ∈ N . The main object of the present study is the following system of delay differential equations, where τ is a positive number, the eigenvalues of the matrix A ∈ R m × m have negative real parts, f : R m → R m is a continuous function, and g : R → R m is a uniformly continuous and bounded function. Our purpose is to prove that system (1.1) possesses a unique unpredictable solution which is uniformly exponentially stable, provided that the function g ( t ) is unpredictable in accordance with Definition 1.1. In the remaining parts of the paper, we will make use of the usual Euclidean norm for vectors and the norm induced by the Euclidean norm for square matrices [11]. Since the eigenvalues of the matrix A in system (1.1) have negative real parts, there exist numbers K ≥ 1 and ω > 0 such that ∥ e At ∥ ≤ Ke − ωt for t ≥ 0 . The following conditions are required. x ′ ( t ) = Ax ( t ) + f ( x ( t − τ )) + g ( t ) , (1.1) 2 Unpredictable solutions (C1) There exists a positive number M$_{f}$ such that sup x ∈ R m ‖ f ( x ) ‖ ≤ M$_{f}$ ; 2