in the theory of dynamical systems or more specifically, in the theory of differential equations or hybrid systems. The rest of the paper is organized as follows. In the next section, we give the auxiliary results from the paper [1]. Section 3 is concerned with the Bebutov dynamics and the description of unpredictable functions. The existence of unpredictable solutions in a quasi-linear system is considered in Section 4. Section 5 is devoted to an illustrative example. Finally, some concluding remarks are given in Section 6. Throughout the paper, we will denote by R , R $_{+}$, N and Z the sets of real numbers, non-negative real numbers, natural numbers and integers, respectively. Moreover, we will make use of the usual Euclidean norm for vectors and the norm induced by the Euclidean norm for square matrices [7]. Let ( X, d ) be a metric space. A mapping π : R + × X → X is a semi-flow on X [4] if: Suppose that π is a semi-flow on X. A point p ∈ X is stable P + (positively Poisson stable) if there exists a sequence { t$_{n}$ } , t$_{n}$ → ∞ as n → ∞ , such that π ( t$_{n}$, p ) → p as n → ∞ [8]. For a fixed p ∈ X, let us denote by Θ$_{p}$ the closure of the trajectory T ( p ) = { π ( t, p ) : t ∈ R $_{+}$} , i.e., Θ$_{p}$ = T ( p ) . The set Θ$_{p}$ is a quasi-minimal set if the point p is stable P + and T ( p ) is contained in a compact subset of X [8]. It was demonstrated by Hilmy [9] that if the trajectory corresponding to a Poisson stable point p is contained in a compact subset of X and Θ$_{p}$ is neither a rest point nor a cycle, then Θ$_{p}$ contains an uncountable set of motions everywhere dense and Poisson stable. The following theorem can be proved by adapting the technique given in [8, 9]. Theorem 2.1 Suppose that p ∈ X is stable P + and T ( p ) is contained in a compact subset of X. If Θ$_{p}$ is neither a rest point nor a cycle, then it contains an uncountable set of motions everywhere dense and stable P $^{+}$. The description of an unpredictable point and trajectory are as follows. The results of our paper are correct if one considers stable P − (negatively Poisson stable) points for a semi-flow with negative time or both stable P + and stable P − (Poisson stable) points for a flow. The definition of a quasi-minimal set is given for a Poisson stable point in [8]. 2 (iii) π ( t$_{1}$, π ( t$_{2}$, p )) = π ( t$_{1}$ + t$_{2}$, p ) for all t$_{1}$, t$_{2}$ ∈ R + and p ∈ X. (ii) π ( t, p ) is continuous in the pair of variables t and p ; (i) π (0 , p ) = p for all p ∈ X ; 2 Preliminaries