equations. We utilize the topology of uniform convergence on any compact subset of the real axis to introduce the unpredictable functions. More precisely, the Bebutov dynamical system [8] has been applied. Ad- ditionally, using the same dynamics we have introduced Poisson functions. All these make our duty of incorporating chaos investigation to theory of differential equations initiated in papers [9]-[14] and in the book [15] seems to proceed in the correct way. The main goal of this paper is the construction of a concrete unpredictable function and its application to differential equations. To give the procedure we start with unpredictable sequences as motions of symbolic dynamics and the logistic map. Then an unpredictable function is determined as an improper convolution integral with a relay function. Finally, we demonstrated in examples unpredictable functions as solutions of differential equations. Let ( X, d ) be a metric space and π : $^{T$_{+}$}$× X → X, where T$_{+}$ is either the set of non-negative real numbers or the set of non-negative integers, be a semi-flow on X, i.e., π (0 , x ) = x for all x ∈ X, π ( t, x ) is continuous in the pair of variables t and x, and π ( t$_{1}$, π ( t$_{2}$, x )) = π ( t$_{1}$ + t$_{2}$, x ) for all t$_{1}$, t$_{2}$ ∈ $^{T$_{+}$}$, x ∈ X. A point x ∈ X is called positively Poisson stable (stable P $^{+}$) if there exists a sequence { t$_{n}$ } satisfying t$_{n}$ → ∞ as n → ∞ such that lim n $_{→∞}$π ( t$_{n}$, x ) = x [16]. For a given point x ∈ X, let us denote by Θ$_{x}$ the closure of the trajectory T ( x ) = { π ( t, x ) : t ∈ $^{T$_{+}$}$} , i.e., Θ$_{x}$ = T ( x ) . The set Θ$_{x}$ is a quasi-minimal set if the point x is stable P + and T ( x ) is contained in a compact subset of X [16]. It was demonstrated by Hilmy [17] that if the trajectory corresponding to a Poisson stable point x is contained in a compact subset of X and it is neither a rest point nor a cycle, then the quasi-minimal set contains an uncountable set of motions everywhere dense and Poisson stable. The following theorem can be proved by adapting the technique given in [16, 17]. Theorem 2.1 ([2]) Suppose that x ∈ X is stable P + and T ( x ) is contained in a compact subset of X. If Θ$_{x}$ is neither a rest point nor a cycle, then it contains an uncountable set of motions everywhere dense and stable P $^{+}$. The definitions of an unpredictable point and unpredictable trajectory are as follows. Definition 2.1 ([2]) A point x ∈ X and the trajectory through it are unpredictable if there exist a positive number ϵ$_{0}$ (the unpredictability constant) and sequences { t$_{n}$ } and { τ$_{n}$ } , both of which diverge to infinity, such that lim n $_{→∞}$π ( t$_{n}$, x ) = x and d ( π ( t$_{n}$ + τ$_{n}$, x ) , π ( τ$_{n}$, x )) ≥ ϵ$_{0}$ for each n ∈ $^{N}$. Markov [16] proved that a trajectory stable in both Poisson and Lyapunov (uniformly) senses must be an almost periodic one. Since Definition 2.1 implies instability, an unpredictable motion cannot be 3 2 Preliminaries