For all values of $r > r_0 \simeq 3.5699...$, the topological entropy of the logistic map is strictly positive [1]. That means that there is an uncountable set of points whose orbit accumulates on a compact set that is not countable, ie there is an uncountable set of points which are chaotic in your sense. 

Note that the term "chaotic" is often misused. When it comes to the logistic family, the term "chaotic parameter" is often used with a different meaning, namely the existence of an non-uniformly hyperbolic SRB measure attracting a set of points of full Lebesgue measure. It is known that the set of such parameters has density going to one as the parameter goes to $4$, but it is also known that its complement is open and dense in the interval $[0,4]$.

It is in general pretty hard to determine if a given point for a given parameter has an orbit converging to a periodic orbit. On the other hand, it is usually feasible to exhibit a parameter in a given interval and an initial value in a given interval whose orbit has a specific behavior.

For your particular case, I would be surprised if an answer can be given because for the parameter 3.8 there are infinitely many periodic points and your initial value may fall on such a point after a while, but due to rounding errors, this would not be caught by the computer. Or instead a non periodic orbit may end up in a loop due to such errors. 

[1] the graph of the entropy as a function of the parameter appears for example in the 2015 article of Bruin and Van Strien  https://www.ams.org/journals/jams/2015-28-01/S0894-0347-2014-00795-5/
Monotonicity of the entropy goes back to work of Douady (1995) and Milnor, Thurston (1988).