Definition 2.1 ([1]) A point p ∈ X and the trajectory through it are unpredictable if there exist a positive number ϵ$_{0}$ (the sensitivity constant) and sequences { t$_{n}$ } and { τ$_{n}$ } , both of which diverge to infinity, such that lim n $_{→∞}$π ( t$_{n}$, p ) = p and d [ π ( t$_{n}$ + τ$_{n}$, p ) , π ( τ$_{n}$, p )] ≥ ϵ$_{0}$ for each n ∈ N . 

An important point to discuss is the sensitivity or unpredictability. In the famous research studies [2, 3, 10, 11, 12], sensitivity was considered as a property of a system on a certain set of initial data since it compares the behavior of at least couples of solutions. Definition 2.1 allows to formulate unpredictability for a single trajectory. Indicating an unpredictable point p, one can make an error by taking a point π ( t$_{n}$, p ) . Then d [ π ( τ$_{n}$, π ( t$_{n}$, p )) , π ( τ$_{n}$, p )] ≥ ϵ$_{0}$, and this is unpredictability for the motion. Thus, we say about the unpredictability of a single trajectory whereas the former definitions considered the property in a set of motions. 

It was proved in [1] that if p ∈ X is an unpredictable point, then T ( p ) is neither a rest point nor a cycle, and that if a point p ∈ X is unpredictable, then every point of the trajectory T ( p ) is also unpredictable. It is worth noting that the sensitivity constant ϵ$_{0}$ is common for each point on an unpredictable trajectory. 

The dynamics on a set S ⊆ X is sensitive [3, 11] if there exists a positive number ϵ$_{0}$ such that for each u ∈ S and each positive number δ there exist a point u$_{δ}$ ∈ S and a positive number τ$_{δ}$ such that d [ u$_{δ}$, u ] < δ and d [ π ( τ$_{δ}$, u$_{δ}$ ) , π ( τ$_{δ}$, u )] ≥ ϵ$_{0}$. 

A result concerning sensitivity in a quasi-minimal set is given in the next theorem. 

Theorem 2.2 ([1]) The dynamics on Θ$_{p}$ is sensitive if p ∈ X is an unpredictable point. 

Theorem 2.2 mentions the presence of sensitivity in the set Θ$_{p}$ if p is an unpredictable point in X. Ac- cording to Theorem 2.1, if the trajectory T ( p ) of an unpredictable point p ∈ X is contained in a compact subset of X, then Θ$_{p}$ contains an uncountable set of everywhere dense stable P + motions. Additionally, since T ( p ) is dense in Θ$_{p}$ , the transitivity is also valid in the dynamics. Therefore, summarizing the last discussions, we propose a new definition of chaos based on the concept of the unpredictable point. 

## 3 Unpredictable Functions and Chaos 

This section is devoted to the description of unpredictable functions and their connection with chaos. For that purpose the results provided in [4] will be utilized. 

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. 

