Tag Archives: iteration

Iterated Linear Maps in the Plane

Iterated Linear Maps in the Plane.   That sounds complex, doesn’t it?   In fact, it’s something we are already very familiar with.  At the beginning of the semester we investigated repeated iteration of a single linear equation $f(x)= ax+b$.   This time we are investigating the repeated iteration of a matrix, which is a nice way to write a system of equations.

We can take the matrix $A$= $\left( \begin{array}{cc} a & b \\ c & d \\\end{array} \right)$

and initial values $x_0, y_0$ placed into a vector $x_n$ = $\left( \begin{array}{c} x_0 \\ y_0 \end{array} \right)$

We then investigate repeated iterations of $Ax_n = x_{n+1}$

Analysis of the convergence of these matrices  sees a connection to the direction of the matrix and a special number called an $eigenvalue (\lambda)$ and its corresponding $eigenvector$



Iteration of Quadratic Functions

We have already looked at iterations of linear functions, so a natural extension would be to look at quadratic functions in the form $f(x)=ax(x-1)$.  There are some familiar questions, such as what happens when $a$ is varied with a fixed $x_0$ and vice versa.

There is a little twist, though.  Iteration around a fixed may cause nearby points to converge towards that fixed point.  That point is then called an attractor.  Nearby points could also diverge away from the fixed point.  If so, the point is then called a repeller.  Sometimes there is no pattern at all and can look quite chaotic……..

Chaos will be looked at and that is the major draw for me to consider this chapter.  Let’s have some random fun!!!

Iterations of Quadratic Functions

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?