Iteration of linear functions was the first lab we covered. Towards the end of the lab, I felt like this was a precursor to fractal geometry. After all, the Mandelbrot set is based on an iteration of a quadratic function with the use of complex numbers. Chapter 14 deals with iteration of quadratic functions. This is one step closer to fractal geometry…..and a precursor to chaos theory, both of which are very interesting to me.

Early definitions in the chapter are:

- fixed point: given a function $f(x)$, a point $u$ is a fixed point of $f$ if $f(u)=u$.
- attractor: a fixed point is an
*attractor*when all nearby points move towards it under iteration. - repeller: a fixed point is a
*repeller*when all nearby points move away from it under iteration

Here’s a type of question from the chapter:

Given a function $f(x)=ax(x-1)$, how do various values of $a$ affect fixed points, attractors, repellers, and zeroes. What about changing initial values?

KennyThis is one of the labs on my short list. Chaotic behavior in functions is certainly an interesting one to observe; pinning down the behavior, what more fun could be had?

Speaking of iterating quadratic functions and fractals, interesting things happen when the Mandelbrot set is iterated with cubic and quartic functions…