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

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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,

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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

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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

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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