## 2 Preliminaries 

Throughout the paper, we will make use of the usual Euclidean norm for vectors and the norm induced by the Euclidean norm for matrices. 

Let ( X, d ) be a metric space, and T refer to either the set of real numbers or the set of integers. Suppose that π : T × X → X is a flow on X , i.e., π (0 , σ ) = σ for all σ ∈ X, π ( t, σ ) is continuous in the pair of variables t and σ, and π ( t$_{1}$, π ( t$_{2}$, σ )) = π ( t$_{1}$ + t$_{2}$, σ ) for all t$_{1}$, t$_{2}$ ∈ T and σ ∈ X [16]. We modified the Poisson stable points to unpredictable points in paper [10] as follows. 

Definition 2.1 A point σ ∈ X and the trajectory through it are unpredictable if there exist a positive number ϵ$_{0}$ (the unpredictability constant) and sequences { t$_{k}$ } and { τ$_{k}$ } , both of which diverge to infinity, such that lim k →∞ π ( t$_{k}$, σ ) = σ and d [ π ( t$_{k}$ + τ$_{k}$, σ ) , π ( τ$_{k}$, σ )] ≥ ϵ$_{0}$ for each k ∈ N . 

To develop the row of periodic, quasiperiodic, and almost periodic oscillations to a new one, we specified in [11]-[13] unpredictability for functions as points of a dynamics. 

Definition 2.2 A uniformly continuous and bounded function ϕ : R → R p is unpredictable if there exist positive numbers ϵ$_{0}$, δ, and sequences { t$_{k}$ } , { τ$_{k}$ } , both of which diverge to infinity, such that ‖ ϕ ( t + t$_{k}$ ) − ϕ ( t ) ‖ → 0 as k → ∞ uniformly on compact subsets of R , and ‖ ϕ ( t + t$_{k}$ ) − ϕ ( t ) ‖ ≥ ϵ$_{0}$ for each t ∈ [ τ$_{k}$ − δ, τ$_{k}$ + δ ] and n ∈ N . 

The last definition can be considered as a more restrictive version of the next two, which will be useful in the future for applications of functional analysis methods in the theory of differential equations. 

Definition 2.3 A continuous and bounded function ϕ : R → R p is unpredictable if there exist positive numbers ϵ$_{0}$, δ, and sequences { t$_{k}$ } , { τ$_{k}$ } , both of which diverge to infinity, such that ‖ ϕ ( t + t$_{k}$ ) − ϕ ( t ) ‖ → 0 as k → ∞ uniformly on compact subsets of R and ‖ ϕ ( t$_{k}$ + τ$_{k}$ ) − ϕ ( τ$_{k}$ ) ‖ ≥ ϵ$_{0}$ for each k ∈ N . 

Definition 2.4 A continuous and bounded function ϕ : R → R p is unpredictable if there exist positive numbers ϵ$_{0}$, δ, and sequences { t$_{k}$ } , { τ$_{k}$ } , both of which diverge to infinity, such that ‖ ϕ ( t$_{k}$ ) − ϕ (0) ‖ → 0 as k → ∞ and ‖ ϕ ( t$_{k}$ + τ$_{k}$ ) − ϕ ( τ$_{k}$ ) ‖ ≥ ϵ$_{0}$ for each k ∈ N . 

The following definition of an unpredictable sequence was first mentioned in paper [13] as an analogue for Definition 2.2. 

Definition 2.5 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$_{n}$$_{+}$$_{ζ}$$_{k}$ − ϕ$_{n}$ ‖ → 0 uniformly as k → ∞ for each n in bounded intervals of integers and ‖ ϕ$_{ζ}$$_{k}$$_{+}$$_{η}$$_{k}$ − ϕ$_{η}$$_{k}$ ‖ ≥ ϵ$_{0}$ for each k ∈ N . 

