Background:

Currently I am studying a course on non-linear dynamics. We have been studying about
attractors only intuitively, so I do not have a definition for an attractor. Let me give you a couple of examples.

Consider a problem where a ball is slipping in a parabolic valley. It is clear
intuitively and quantitatively that the lowermost point of the parabola is the
attractor for it. 

Say we are talking about a [Lorenz system][2]. We may not guess intuitively that
there is an attractor, a priori, but using simulations for different values of
initial conditions, we may qualitatively (and maybe quantitatively too) compare the trajectories for different initial conditions and convince ourselves that we have an attractor.

Problem:

Consider the [logistic map](http://en.wikipedia.org/wiki/Logistic_map) given as 

$$x_n = ax_{n-1}{(1-x_{n-1})}.$$

Let us fix a value of the parameter where we "know" that the logistic map exhibits
chaos. Say $a=3.9$.

Let us take the definition for the term attractor from [Wikipedia][1]:

An attractor is a subset A of the phase space characterized by the following three
conditions:

 - $A$ is forward invariant under $f$: if $a$ is an element of $A$ then so is
$f(t,a)$, for all $t > 0$.
 - There exists a neighborhood of $A$, called the basin of attraction for $A$ and
denoted $B(A)$, which consists of all points $b$ that "enter $A$ in the limit $t →
∞$". More formally, $B(A)$ is the set of all points $b$ in the phase space with the
following property: For any open neighborhood $N$ of $A$, there is a positive
constant $T$ such that $f(t,b) ∈ N$ for all real $t > T$.
 - There is no proper subset of $A$ having the first two properties.

How might we demonstrate/convince ourselves of/disprove the statement that the logistic map does have an attractor for a particular choice of the parameter at which the map exhibits chaos?
    


  [1]: http://en.wikipedia.org/wiki/Attractor
  [2]: http://en.wikipedia.org/wiki/Lorenz_attractor