Does the logistic map have an attractor for a particular value of the parameter?

Question

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. 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 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:

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?

Answer 1

The answer to your question is yes, the logistic map definitely has an attractor. To show this for general maps/dynamical systems, we can use a direct numerical study, delve into analytical arguments which might not be entirely complete, or a combination of both.

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An attractor is a set in the phase space of the dynamical system or map which attracts a certain portion of the trajectories or orbits (and doesn't let go). The other conditions you mention are just to eliminate any arbitrariness in definition and some pathological cases.

For any map, you can see whether it has an attractor by letting the orbit start at random points, run a large number of iterations and then check if the orbits converged to one or more stable orbits. This is exactly how bifurcation diagrams are made and the points are the ones at which you find your random orbits after a long time. So in a certain sense bifurcation diagrams are "attractor graphs". See the bifurcation diagram of the logistic map:

In this graph, every $r = r_0$ slice corresponds to an attractor at $r_0$. You can verify that for small $r$ the attractors is just a point which then doubles frequencies in a cascade. The positions of a number of these fixed points and their doubling can be found explicitly as shown in this MathWorld article on the logistic map. I.e., in some cases we can even find the attractor analytically.

For higher $r$ the matter is more delicate - it has been shown by Mitchel J. Feigenbaum in 1979 that the cascade of bifurcations for such a map diverges and eventually creates a Cantor-like attractor. We can for example find the fractal dimension of this set (Grassberger 1981), but as far as I know we cannot generally fully specify it's position by analytical means. However, the attractor definitely exists.

Answer 2

The attractors here form a series $A_n$ such that $A_n=\big\{x:f^n(x)=x, f^i(x)\neq x~\forall ~i<n,x\in \mathbb R\big\}$, where $f(x)=3.9x(1-x)$. One can easily see that these satisfy the first criteria.

As for the second criteria, this need not be the case. From this¹ paper (which specifically discusses one dimensional maps):

Let $x^*$ be a fixed point of $f$. Then the basin of attraction (or the stable set) $W^s(x^*)$ is defined as $$W^s(x^*)=\{x:\lim\limits_{n\to\infty}f^n(x)=x^*\}$$

Note the lack of requirement for the basin to be a neighborhood.

If you go a bit further on the Wikipedia page, we see:

Many other definitions of attractor occur in the literature. For example, some authors require that an attractor have positive measure (preventing a point from being an attractor), others relax the requirement that $B(A)$ be a neighborhood

With this in mind, we can make it satisfy the second criteria.

This is pretty simple: we find the set of numbers $B_n={x:f(x)=a_n ~\forall~ a_n \in A_n}$. This need not be the entire basin, we just need a portion to prove that $A_n$ is an attractor. Of course, we need to prove that $\exists b_n\in B_n$ that is not an element of $A_n$. This can be done by noticing that $f(x)=a$ has 2 roots $\forall~a\in(0,1)$. Since the roots of $f(x)=a_n$ cannot overlap for different $a_n$ (otherwise the two $a_n$'s would be equal), we get $2n$ distinct values for which $f(x)\in A_n$. Since $A_n$ has only $n$ elements, we have some extra numbers which can be considered to be part of the basin of attraction.

^{1. Elaydi, S., & Sacker, R. J. (2003). The basin of attraction of one-dimensional maps.}