Feigenbaum-number



#5

Hello,

know anybody, what was the program, what used by Feigenbaum originally(!), to calculate this number on his HP65?

Thanks for answering!

Csaba


#6

I don't have the program. From the book "Chaos. Making a New Science" by James Gleick, I can tell you the equations he was working with.

First he worked with x(t+1)=r·x·[1-x(t)]. A quadratic equation. y=rx-rx^2.

You start with a value for "r" and a value for "x" and the result is the next value for "x" if you keep iterating finally you get always the same value. With some pairs of values you end in an alternation between two values. He searched for the values of "r" that produced this duplication.

Samples of values r=2.7 x=0.02 that ends in 0.6296. r=3.5 x1=0.850 x2=0.3828 x3=0.8269 x4=0.5009, in this case there is a double duplication so we end with four x values.

Then he repeated the process with the following equation
x(t+1)=r·sin PI·x(t). Where PI is 3.14159... as you may supposse.

I hope that can help you a little.

#7

I don't think the exact HP-65 program is available, but it can probably be reconstructed from the equations he worked with (see J. Miranda's post) and the very detailed (and to the point) info on this URL:

Period Doubling Ad Infinitum

Best regards from V.

#8

Thanks for your answers! I read that book some years ago, and I wrote a little program in PASCAL. The next prg is the HP28C version:

FEIG:

<< (2.63,0) PMIN (4,1) PMAX CLLCD 2.63 4 FOR R .3 'X'
STO 1 20 START R NEWX NEXT 1 12 START R NEWX R X R->C
PIXEL NEXT .01 STEP >>

NEWX:

<< -> R << R X * 1 X - * 'X' STO >> >>

Press 'FEIG' to start. Running time is about 12 minutes. (Slow, but nice... ;) )

The greatest problem, how to isolate the bifurcation points on the diagram...

Csaba


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