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Finding a super attractive parameter
Let
[dmath]f_a(x) = a x^2 (1-x).[/dmath]
A plot of [imath]f[/imath] together with the line [imath]y=x[/imath] and a cobweb is shown in the figure below. The cobweb seems to illustrate that [imath]f[/imath] has an orbit of period 3.
- Does the orbit appear to be repulsive, neutral, attractive, or super -attractive? How can you tell?
- Write down an equation that the parameter [imath]a[/imath] must satisfy to generate the period 3 orbit shown in the plot.
- Use Newton's method on a computer to find a good decimal approximation to [imath]a[/imath].
