The Logistic Map for a>4
adapted from section 1.6 of
An Introduction to Chaotic Dynamical Systems, by
Robert Devaney
We now consider the behavior of
For the remainder of this section, we will usually drop the subscript a
and write F instead of
. As before, all of the interesting dynamics
of F occur in the unit interval I = [0,1]. Note that, since a > 4, the
maximum value of F is larger than one. Hence certain points leave
I after one iteration of F. Denote the set of such points by
.
Clearly,
is an open interval centered at 1/2 and has the property that,
if
then F(x) > 1, so
and
.
is the set of points which
immediately escape from I. All other points in I remain in I after one
iteration of F.
Let
If
, then
,
, and so, as before,
.
Inductively, let
.
That is,
so that consists of all points which escape from I at the
iteration. As above, if x lies in
, it
follows that the orbit of x tends eventually to
. Since we
know the ultimate fate of any point which lies in the
, it
remains only to analyze the behavior of those points which never
escape from I, i.e., the set of points which lie in
Let us denote this set by A. Our first question is: what precisely is this set of points? To understand A, we describe more carefully its recursive construction.
Since is an open interval centered at 1/2,
consists of two
closed intervals,
on the left and
on the right.
Note that F maps both and
, monotonically onto I; F is
increasing on
and decreasing on
.
Since
, there are a pair of open intervals, one in
and one in
, which are mapped into
by F.
Therefore this pair of intervals is precisely the set
.
Now consider . This set consists of 4 closed
intervals and F maps each of them monotonically onto either
or
.
Consequently
maps each of them onto 1. Thus, each of
the four intervals in
contains an open subinterval which
is mapped by
onto
. Therefore, points in these intervals escape
from I upon the third iteration of F. This is the set we called
. For
later use, we observe that
is alternately increasing and decreasing on
these four intervals. It follows that the graph of
must therefore
have two ``humps''.
Continuing in this manner we note two facts. First, consists of
disjoint open intervals. Hence
consists of
closed intervals since
Secondly, maps each of these closed intervals monotonically onto
I. In fact, the graph of
is alternately increasing and
decreasing on these intervals. Thus the graph of
has exactly
humps on I, and it follows that the graph of
crosses the line
y = x at least
times. This implies that
has at least
fixed points or, equivalently, Per(F) consists of
points in I.
Clearly, the structure of A is much more complicated when a > 4 than
when a < 3.
The construction of A is reminiscent of the construction of the Middle Thirds Cantor set: A is obtained by successively removing open intervals from the "middles" of a set of closed intervals. See section 4.1 of Alligood, Sauer, & Yorke and/or Neal Carothers' Cantor Set web pages for more details.
Definition:
A set is a Cantor set if it is a closed, totally disconnected, and
perfect subset of an interval. A set is totally disconnected if it contains
no intervals; a set is perfect if every point in it is an accumulation point
or limit point of other points in the set.
Example: The Middle-Thirds Cantor Set.
This is the classical example of a Cantor set. Start with [0,1] but
remove the open "middle third," i.e. the interval . Next, remove the middle thirds of the resulting intervals,
that is, the pair of intervals
and
. Continue removing middle thirds in this fashion;
note that
open intervals are removed at the
stage of this
process. Thus, this procedure is entirely analogous to our construction
above.
Remark. The Middle-Thirds Cantor Set is an example of a fractal.
Intuitively, a fractal is a set which is self-similar under magnification.
In the Middle-Thirds Cantor Set, suppose we look only at those points which
lie in the left-hand interval . Under a microscope which
magnifies this interval by a factor of three, the "piece" of the Cantor set
in
looks exactly like the original set. More precisely,
the linear map L(x) = 3x maps the portion of the Cantor set in
homeomorphically onto the entire set.
This process does not stop at the first level: one may magnify any piece of
the Cantor set contained in an interval by
a factor of
to obtain the original set.
To guarantee that our set A is a Cantor set, we need an additional
hypothesis on a. Suppose a is large enough so that | F'(x)| > 1 for
all . The reader may check that
suffices. Hence, for these values of a, there exists
such
that
for all
. By the chain rule, it
follows that
as well. We claim that A contains
no intervals. Indeed, if this were so, we could choose two distinct point
x and y in A with the closed interval
. Notice that
for all
. Choose n so
that
. By the Mean Value Theorem, it then follows that
, which implies that at least one
of
or
lies outside of I. This is a contradiction, and so
A is totally disconnected.
Since A is a nested intersection of closed intervals, it is closed. We
now prove that A is perfect. First note that any endpoint of an is
in A: indeed, such points are eventually mapped to the fixed point at 0,
and so they stay in I under iteration. Now if a point
were
isolated, every near point near p must leave I under iteration of F.
Such points must belong to some
. Either there is a sequence of
endpoints of the
converging to p, or else all points in a deleted
neighborhood of p are mapped out of I by some power of F. In the former
case, we are done as the endpoints of the
map to 0 and hence are in
A. In the latter, we may assume that
maps p to 0 and all other
points in a neighborhood of p into the negative real axis. But then
has a maximum at p so that
. By the chain rule, we must
have
for some i < n. Hence
, and so
is not in I, contradicting the fact that
.
Hence we have proved
Theorem. If , then A is a Cantor set.
Remark. The theorem is true for a > 4, but the proof is more
delicate. Essentially, all we need is that for each , there is an
N such that for all
, we have
. Then almost
exactly the same proof applies.
We have now succeeded in understanding the gross behavior of orbits of
when a > 4. Either a point tends to
under iteration of
,or else its entire orbit lies in a Cantor set A.
For points
, we see the same complicated type of behavior that we
have seen for a=4: there are periodic points of all periods, but even
more: we can symbolically specify a behavior by a string of Rs and Ls, and
such an orbit necessarily exists. We will return to this issue later.