Bifurcations and periods in chaos with HP50G

[Thomas Klemm]

From WolframAlpha I have the following exact values with \(a=1.25\) for a solution of:

\(a-(a-(a-(a-x^2)^2)^2)^2=x\)

x = 1/2 (1 - sqrt(2)) ≈ -0.207107
x = 1/2 (1 + sqrt(2)) ≈ 1.20711
x = 1/2 (-1 - sqrt(6)) ≈ -1.72474
x = 1/2 (sqrt(6) - 1) ≈ 0.724745

If you iterate the following program you will notice that only two of them are attractive.

Code:

00 { 12-Byte Prgm }
01 XEQ 01
02▸LBL 01
03 XEQ 00
04▸LBL 00
05 RCL 00
06 X<>Y
07 X↑2
08 -
09 END

Examples

1.25 STO 00

1.20711
R/S R/S R/S …

It slowly converges to the exact value.

0.724745
R/S R/S R/S …

After only a few iteration it apparently starts converging to the other fixed-point as well.
Similarly for the other two values.

The derivative of the 4th iteration of the function \(a - x^2\) decides whether it is attractive.
A fixed-point \(x_0\) is attractive if \(|f\,'(x_{0})|<1\).
I'm leaving the verification up to you.