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Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation...
bo198214 Wrote:Interesting phenomenon, I guess it has to do with that the function is not strictly increasing in the vicinity of the second fixed point. If you develop the regular half iterate at the left fixed point it gives a non-real function.

If I remember correctly the matrix power approach even yields a non-real solution if applied at 0 between both fixed point *when the function is strictly increasing*.

For complex fixed points, its anyway (mostly) not the case that the regular iteration at one fixed point has the other fixed point as fixed point.

Hmm, what I understand now is the following. While the black line for the f(x)-curve shows the locus for continuous decreasing x, we don't notice, that around the second fixpoint the *iterates* of f(x) oscillate around and converge to the fixpoint. The curve for f°0.5(x) "has to reflect this": two iterates of f°05->one iterate of f(x). There is a (necessary) first crossing of f(x) and f°0.5(x) near the fixpoint (x~ -0.195) and if one looks at the construction-scheme for the continuation of f°0.5(x) it is obvious, that this trajectory must be winding around that of f(x). Well: the *trajectory*. If the curve of f(x) is seen as *trajectory* (for instance via the cobweb-curve), then it lies on a line (the curve of the graph) but is oscillating around and converges to the fixpoint.

So I think now this is a general problem for fractional iterates.
Curious now: the f°05-curve winds around and converges to the fixpoint. But where does the curve appear again for x<-0.5? In the complex plane?


Gottfried Helms, Kassel

Messages In This Thread
RE: Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by Gottfried - 03/03/2009, 12:52 AM

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