Definition 2.5 is of main use in the present paper. It is requested by the method of the proof. Nevertheless, in future analyses, there may be needs for another definition, which can be considered as a direct specification of Definition 2.1 as well as an analogue of Definition 2.4. Definition 2.6 A bounded sequence { ϕ$_{n}$ } , n ∈ Z , in R p is called unpredictable if there exist a positive number ϵ$_{0}$ and sequences { ζ$_{k}$ } , { η$_{k}$ } , k ∈ N , of positive integers, both of which diverge to infinity, such that ‖ ϕ$_{ζ}$$_{k}$ − ϕ$_{0}$ ‖ → 0 , as k → ∞ , and ‖ ϕ$_{ζ}$$_{k}$$_{+}$$_{η}$$_{k}$ − ϕ$_{η}$$_{k}$ ‖ ≥ ϵ$_{0}$ for each k ∈ N . The topologies in Definitions 2.2 and 2.5 are metrizable [16]. Consequently, the existence of an unpredictable sequence in the sense of Definition 2.5 indicates the presence of Poincaré chaos [10]. In what follows, an unpredictable sequence and an unpredictable solution are understood as mentioned in Definition 2.5. In this paper, we will consider the following discrete equation, where n ∈ Z , A ∈ R p × p is a nonsingular matrix, f : R p → R p is a continuous function, and { ψ$_{n}$ } , n ∈ Z , is an unpredictable sequence. The following assumptions on equation (2.1) are required. According to the results of [17], if conditions ( C 1) − ( C 3) hold, then equation (2.1) possesses a unique bounded solution { φ$_{n}$ } , n ∈ Z , which satisfies the relation One can show under the same conditions that the bounded solution attracts all other solutions of (2.1). More precisely, the inequality is valid for all n ≥ n$_{0}$ , where { x$_{n}$ } , n ∈ Z , is a solution of (2.1) with x$_{n}$$_{0}$ = x 0 for some integer n$_{0}$ and x 0 ∈ R $^{p}$. ‖ x$_{n}$ − φ$_{n}$ ‖ ≤ ( ‖ A ‖ + L$_{f}$ ) ( n − n 0 ) ∥ x 0 − φ$_{n}$$_{0}$ ∥ φ$_{n}$ = n ∑ j = −∞ A n − j [ f ( φ$_{j}$ − $_{1}$) + ψ$_{j}$ − $_{1}$] . (2.2) x$_{n}$$_{+1}$ = Ax$_{n}$ + f ( x$_{n}$ ) + ψ$_{n}$, (2.1) (C1) There exists a positive number M$_{f}$ such that sup x ∈ R p ‖ f ( x ) ‖ ≤ M$_{f}$ ; (C2) There exists a positive number L$_{f}$ such that ‖ f ( x ) − f ( y ) ‖ ≤ L$_{f}$ ‖ x − y ‖ for all x , y ∈ R $^{p}$; (C3) ‖ A ‖ + L$_{f}$ < 1 . 3