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?