Thread Rating:
  • 1 Vote(s) - 5 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation...
#1
Hi -

I'm considering the properties of the regular iteration after fixpoint-shift.
To have easier conditions I look at the function f(x) = x^2 - 0.5 and its iterates.

It has the fixpoints and .

I compute the half-iterate f°0.5(x) by translating f(x) -> g(x - x_a) + x_a and determine the powerseries for g°0.5(x)

The result is interesting; the curve for the half-iterate meets the first fixpoint, but misses the second. Instead, the series diverges in that region.
However, using the more reliable results of f°0.5(x) at other values of x, I can construct a bit of continuation, which gives one winding around (xb,xb).

Naively, I'd expected that the function crossed both fixpoint but it seems, that this needed a completely different powerseries, and thus a modified procedure.

I'm without an idea currently, how to proceed here. Does anyone have a comment?

Here are two plots:

a) an overview. Integer iterates f°-1(x), f°0, f°1,f°2, f°3,f°4 and the regular f°0.5 iteration based on the powerseries representation

   



b) a detail, f°0.5 and f°1 in the vicinity of (xb,xb) and a continuation based on more reliable results of f°0.5 at other values of x

   

Gottfried
Gottfried Helms, Kassel
Reply


Messages In This Thread
Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by Gottfried - 03/02/2009, 02:50 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  (Again) fixpoint outside Period tommy1729 2 2,610 02/05/2017, 09:42 AM
Last Post: tommy1729
  Polygon cyclic fixpoint conjecture tommy1729 1 2,331 05/18/2016, 12:26 PM
Last Post: tommy1729
  The " outside " fixpoint ? tommy1729 0 1,571 03/18/2016, 01:16 PM
Last Post: tommy1729
  2 fixpoint pairs [2015] tommy1729 0 1,868 02/18/2015, 11:29 PM
Last Post: tommy1729
  [2014] The secondary fixpoint issue. tommy1729 2 3,795 06/15/2014, 08:17 PM
Last Post: tommy1729
  Simple method for half iterate NOT based on a fixpoint. tommy1729 2 3,520 04/30/2013, 09:33 PM
Last Post: tommy1729
  Iteration exercises: Lucas-Lehmer-test and Schröder-function Gottfried 0 2,761 04/04/2012, 06:17 AM
Last Post: Gottfried
  Iteration series: Different fixpoints and iteration series (of an example polynomial) Gottfried 0 2,967 09/04/2011, 05:59 AM
Last Post: Gottfried
  Fractional iteration of x^2+1 at infinity and fractional iteration of exp bo198214 10 17,271 06/09/2011, 05:56 AM
Last Post: bo198214
  2 fixpoint failure tommy1729 1 2,956 11/13/2010, 12:25 AM
Last Post: tommy1729



Users browsing this thread: 1 Guest(s)