An archived instance of a Chaos and Fractals forum

# Assignment on super-attracting parameters

pikenber

I am having issues with the Sage code portion of this. I keep plugging in my function and getting roots that are way too negative for any iteration to happen. Am I just typing in the code wrong?

Here is the Cobweb Plot of the smallest root:

mark

Remember the following theorem:

A periodic orbit is super-attracting iff it contains a critical point.

Now, consider that last line of code:

F(c,0).roots(ring=RR)


Mathematically, that finds the real solutions of f_c^5(0) = 0, which should be the values of c where 0 maps back to itself after five iterates. The reason we cared about zero in the sample code for the quadratic family f_c(x)=x^2+c is because it’s the critical point. but zero is not the critical point of your family. Once you have your critical point (say x_0), you should solve f_c^5(x_0)=x_0 using code like

(F(c,x0)-x0).roots(ring=RR)


One other point: I think I should have asked you to use the parameter of smallest absolute value in your cobweb plot.