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 tex2html_wrap_inline71 . 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 tex2html_wrap_inline85 . Clearly, tex2html_wrap_inline85 is an open interval centered at 1/2 and has the property that, if tex2html_wrap_inline89 then F(x) > 1, so tex2html_wrap_inline93 and tex2html_wrap_inline95 . tex2html_wrap_inline85 is the set of points which immediately escape from I. All other points in I remain in I after one iteration of F.

Let tex2html_wrap_inline107 If tex2html_wrap_inline109 , then tex2html_wrap_inline111 , tex2html_wrap_inline113 , and so, as before, tex2html_wrap_inline95 . Inductively, let tex2html_wrap_inline117 . That is,


so that tex2html_wrap_inline121 consists of all points which escape from I at the tex2html_wrap_inline125 iteration. As above, if x lies in tex2html_wrap_inline121 , it follows that the orbit of x tends eventually to tex2html_wrap_inline133 . Since we know the ultimate fate of any point which lies in the tex2html_wrap_inline121 , 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 tex2html_wrap_inline85 is an open interval centered at 1/2, tex2html_wrap_inline149 consists of two closed intervals, tex2html_wrap_inline151 on the left and tex2html_wrap_inline153 on the right.

Note that F maps both tex2html_wrap_inline151 and tex2html_wrap_inline153 , monotonically onto I; F is increasing on tex2html_wrap_inline151 and decreasing on tex2html_wrap_inline153 . Since tex2html_wrap_inline167 , there are a pair of open intervals, one in tex2html_wrap_inline151 and one in tex2html_wrap_inline153 , which are mapped into tex2html_wrap_inline85 by F. Therefore this pair of intervals is precisely the set tex2html_wrap_inline177 .

Now consider tex2html_wrap_inline179 . This set consists of 4 closed intervals and F maps each of them monotonically onto either tex2html_wrap_inline151 or tex2html_wrap_inline153 . Consequently tex2html_wrap_inline187 maps each of them onto 1. Thus, each of the four intervals in tex2html_wrap_inline179 contains an open subinterval which is mapped by tex2html_wrap_inline187 onto tex2html_wrap_inline85 . Therefore, points in these intervals escape from I upon the third iteration of F. This is the set we called tex2html_wrap_inline197 . For later use, we observe that tex2html_wrap_inline199 is alternately increasing and decreasing on these four intervals. It follows that the graph of tex2html_wrap_inline187 must therefore have two ``humps''.

Continuing in this manner we note two facts. First, tex2html_wrap_inline121 consists of tex2html_wrap_inline205 disjoint open intervals. Hence tex2html_wrap_inline207 consists of tex2html_wrap_inline209 closed intervals since


Secondly, tex2html_wrap_inline213 maps each of these closed intervals monotonically onto I. In fact, the graph of tex2html_wrap_inline213 is alternately increasing and decreasing on these intervals. Thus the graph of tex2html_wrap_inline213 has exactly tex2html_wrap_inline205 humps on I, and it follows that the graph of tex2html_wrap_inline225 crosses the line y = x at least tex2html_wrap_inline205 times. This implies that tex2html_wrap_inline225 has at least tex2html_wrap_inline205 fixed points or, equivalently, Per(F) consists of tex2html_wrap_inline205 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 tex2html_wrap_inline251 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 tex2html_wrap_inline255 . Next, remove the middle thirds of the resulting intervals, that is, the pair of intervals tex2html_wrap_inline257 and tex2html_wrap_inline259 . Continue removing middle thirds in this fashion; note that tex2html_wrap_inline205 open intervals are removed at the tex2html_wrap_inline263 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 tex2html_wrap_inline265 . Under a microscope which magnifies this interval by a factor of three, the "piece" of the Cantor set in tex2html_wrap_inline265 looks exactly like the original set. More precisely, the linear map L(x) = 3x maps the portion of the Cantor set in tex2html_wrap_inline265 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 tex2html_wrap_inline273 by a factor of tex2html_wrap_inline275 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 tex2html_wrap_inline285 . The reader may check that tex2html_wrap_inline287 suffices. Hence, for these values of a, there exists tex2html_wrap_inline291 such that tex2html_wrap_inline293 for all tex2html_wrap_inline295 . By the chain rule, it follows that tex2html_wrap_inline297 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 tex2html_wrap_inline307 . Notice that tex2html_wrap_inline309 for all tex2html_wrap_inline311 . Choose n so that tex2html_wrap_inline315 . By the Mean Value Theorem, it then follows that tex2html_wrap_inline317 , which implies that at least one of tex2html_wrap_inline319 or tex2html_wrap_inline321 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 tex2html_wrap_inline331 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 tex2html_wrap_inline337 were isolated, every near point near p must leave I under iteration of F. Such points must belong to some tex2html_wrap_inline331 . Either there is a sequence of endpoints of the tex2html_wrap_inline331 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 tex2html_wrap_inline331 map to 0 and hence are in A. In the latter, we may assume that tex2html_wrap_inline225 maps p to 0 and all other points in a neighborhood of p into the negative real axis. But then tex2html_wrap_inline225 has a maximum at p so that tex2html_wrap_inline367 . By the chain rule, we must have tex2html_wrap_inline369 for some i < n. Hence tex2html_wrap_inline373 , and so tex2html_wrap_inline375 is not in I, contradicting the fact that tex2html_wrap_inline337 .

Hence we have proved

Theorem. If tex2html_wrap_inline287 , 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 tex2html_wrap_inline89 , there is an N such that for all tex2html_wrap_inline391 , we have tex2html_wrap_inline393 . Then almost exactly the same proof applies.

We have now succeeded in understanding the gross behavior of orbits of tex2html_wrap_inline225 when a > 4. Either a point tends to tex2html_wrap_inline133 under iteration of tex2html_wrap_inline71 ,or else its entire orbit lies in a Cantor set A. For points tex2html_wrap_inline295 , 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.

Scott Sutherland
Fri Jan 31 00:06:41 EST 1997