almost periodic. In particular, it is neither an equilibrium, nor a cycle. Based on unpredictable points, a new chaos definition was provided in the paper [2] as follows. Definition 2.2 ([2]) The dynamics on the quasi-minimal set Θ$_{x}$ is called Poincaré chaotic if x is an unpredictable point. It is worth noting that Poincaré chaos admits properties similar to the ingredients of Devaney chaos [5]. In the paper [2], it was proved that if x is an unpredictable point, then the dynamics on Θ$_{x}$ is sensitive. That is, there exists a positive number ˜ ϵ$_{0}$ such that for each x$_{1}$ ∈ Θ$_{x}$ and for each positive number δ there exist a point x$_{2}$ ∈ Θ$_{x}$ and a positive number ¯ t such that d ( x$_{1}$, x$_{2}$ ) < δ and d ( f ( ¯ t, x$_{1}$ ) , f ( ¯ t, x$_{2}$ )) ≥ ˜ ϵ$_{0}$. Besides, since the trajectory T ( x ) is dense in Θ$_{x}$ , transitivity is also present in the dynamics. Moreover, according to Theorem 2.1, there exists a continuum of stable P + orbits in Θ$_{x}$ . In our previous paper [2] we gave the definition of Poincaré chaos for flows, but in this paper we give the definition for semi-flows, since the discussion in the paper [2] is valid also for the latter case. Let us denote by C ( $^{R}$) the set of continuous functions defined on $^{R}$with values in R $^{m}$, and assume that C ( $^{R}$) has the topology of uniform convergence on compact sets, i.e., a sequence { h$_{k}$ } in C ( $^{R}$) is said to converge to a limit h if for every compact set U ⊂ $^{R}$the sequence of restrictions { h$_{k}$ | U } converges to { h | U } uniformly. One can define a metric ρ on C ( $^{R}$) as [8] where h$_{1}$, h$_{2}$ belong to C ( $^{R}$) and Let us define the mapping π : $^{R$_{+}$}$× C ( $^{R}$) → C ( $^{R}$) by π ( t, h ) = h$_{t}$, where h$_{t}$ ( s ) = h ( t + s ) . The mapping π is a semi-flow on C ( $^{R}$) , and it is called the Bebutov dynamical system [8]. Using the Bebutov dynamical system, we give the descriptions of a Poisson function and an unpre- dictable function in the next definitions. Definition 2.3 A Poisson function is a Poisson stable point of the Bebutov dynamical system. Definition 2.4 ([3]) An unpredictable function is an unpredictable point of the Bebutov dynamical sys- tem. It is clear that an unpredictable function is a Poisson function. According to Theorem III. 3 [8], a motion π ( t, h ) lies in a compact set if h is a bounded and uniformly continuous function. Therefore, an unpredictable function h determines Poincaré chaos in the Bebutov dynamical system if it is bounded and 4 ρ$_{k}$ ( h$_{1}$, h$_{2}$ ) = min { 1 , sup s ∈ [ − k,k ] ‖ h$_{1}$ ( s ) − h$_{2}$ ( s ) ‖ } , k ∈ $^{N}$. ρ ( h$_{1}$, h$_{2}$ ) = ∞ ∑ k =1 2 − $^{k}$ρ$_{k}$ ( h$_{1}$, h$_{2}$ ) , (2.1)