mark

Three logistic functions are graphed below. One has an attractive fixed point, one has an attractive orbit of period 2, and one has an attractive orbit of period 4. Which one is which and why?

An archived instance of discourse for discussion in a Fractal Introductory Colloquium.

mark

hcary

The second function has an attractive fixed point, because it has the smallest absolute value of the derivative at the point of intersection with $y=x$. The first function has an attractive orbit of period 2, because it has the second largest absolute value of the derivative at the point of intersection. The third function has an attractive orbit of period four, with the largest absolute value of the derivative.