[math-fun] X^2, ln(X), repeat...
Years ago I noticed that if I alternate between hitting the X^2 and the ln(X) buttons on a calculator, the result neither converges nor blows up, but just wanders around, seemingly at random. What, if anything, is known about this sequence? Does it have any fixed points? Any cycles of length N for any N? What is the distribution? The mean value (after the log step) seems to be between 0.15853 and 0.15854. With some values it will blow up. 0 will make it blow up in one step, 1 or -1 will make it blow up in two steps, sqrt(e), -sqrt(e), sqrt(1/e), and -sqrt(1/e) in three steps, etc. The number of such explosive starting values doubles with each step. Is this set of explosive starting values dense, i.e. between any two such values is there always another?
Hello, me too (and probably many other people as well), with cos and 1/x you get the fixed point of cos(x) = x which is 0.7390851332151... but also : with x^2 fives times and log, I found that log(163) = 163/32 very nearly. and then by chance once in 1987 on my hp calculator that exp(Pi) - Pi is very near 20, 19.9990999... I am still wondering why. that log(13981) = 9.5454545439878... which is 105/11 almost. that the sqrt(62) = 7.874007874... sqrt(51)/14 = 0.51010203061020... but for that one , we can find that we have the central binomial coefficients in zig-zag. I am sure, there are others out there. Best regards, Simon Plouffe
Keith's question reminded me a bit of a very interesting sequence, http://oeis.org/A114183, which uses the two operations floor of square root and doubling. So I built a new sequence, by analogy, based on floor of log and squaring, http://oeis.org/A217727. Does every number appear? Neil On Thu, Mar 21, 2013 at 8:26 PM, Keith F. Lynch <kfl@keithlynch.net> wrote:
Years ago I noticed that if I alternate between hitting the X^2 and the ln(X) buttons on a calculator, the result neither converges nor blows up, but just wanders around, seemingly at random. What, if anything, is known about this sequence? Does it have any fixed points? Any cycles of length N for any N? What is the distribution? The mean value (after the log step) seems to be between 0.15853 and 0.15854.
With some values it will blow up. 0 will make it blow up in one step, 1 or -1 will make it blow up in two steps, sqrt(e), -sqrt(e), sqrt(1/e), and -sqrt(1/e) in three steps, etc. The number of such explosive starting values doubles with each step. Is this set of explosive starting values dense, i.e. between any two such values is there always another?
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-- Dear Friends, I have now retired from AT&T. New coordinates: Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
The iteration x := ln(x*x) has repelling length p cycles for all p. Consider any periodic cycle x1, x2, ..., xp = x1. Define the (symbolic) "kneading sequence" k1 k2 ... kp over {0,1}* by ki = 0 if xi>=0, ki = 1 if xi<0. The zero kneading sequence is 0 0 ... 0. Then the periodic cycles are in 1-1 correspondence with nonzero necklaces (a length p necklace is the equivalence class of cyclic shifts of a length p string). So there is one fixed point, with kneading necklace 1. Two period 2 cycles with necklaces {01, 11} - necklace 11 also has period 1, so only one strict period 2 cycle. Two strict period 3 cycles, with necklaces {001, 011}. Three strict period 4 cycles, with necklaces {0001, 0011, 0111}. Six strict period 5 cycles, with necklaces {00001, 00011, 00101, 00111, 01011, 01111}. The inverse iteration for a periodic cycle iterates x_(i+1) = +/-sqrt(e^xi), where the iteration selects the negative sqrt branch for kneading symbol ki=1, and nonnegative branch for ki=0. The nonzero kneading sequences give attracting cycles in the inverse iteration, hence repelling cycles in the original iteration. The zero kneading sequence in the inverse iteration diverges to infinity. The inverse iteration attracting cycles all have positive derivative. I'm not sure what this says about critical points or about other chaotic aspects. Anymore with more dynamical systems knowledge in a position to comment? - Scott
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun- bounces@mailman.xmission.com] On Behalf Of Keith F. Lynch Sent: Thursday, March 21, 2013 5:27 PM To: math-fun@mailman.xmission.com Subject: [math-fun] X^2, ln(X), repeat...
Years ago I noticed that if I alternate between hitting the X^2 and the ln(X) buttons on a calculator, the result neither converges nor blows up, but just wanders around, seemingly at random. What, if anything, is known about this sequence? Does it have any fixed points? Any cycles of length N for any N? What is the distribution? The mean value (after the log step) seems to be between 0.15853 and 0.15854.
With some values it will blow up. 0 will make it blow up in one step, 1 or -1 will make it blow up in two steps, sqrt(e), -sqrt(e), sqrt(1/e), and -sqrt(1/e) in three steps, etc. The number of such explosive starting values doubles with each step. Is this set of explosive starting values dense, i.e. between any two such values is there always another?
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participants (4)
-
Huddleston, Scott -
Keith F. Lynch -
Neil Sloane -
Simon Plouffe