The
image on the right in the graphic below is the iconic image of what is called

*deterministic chaos*. To understand the mathematical construction of that image (called a map) requires beginning with the graph of the quadratic equation y = x^{2}+ c, as is shown in the image on the left (c = -1.3).**To simplify the process of how you get from the parabola to the map, I’ve written a series of four Scratch projects. The two concepts needed to understand the transition are the concept of**

*iteration*and the concept of a mathematical

*attractor*.

The
first project looks at a linear equation and computes a single attractor of x =
6.

**y = (x + 6)/2 – A Mathematical Sink Hole**can be seen at the following link.

The
next project plots the attractors of the quadratic equation y = x

^{2}+ c by plotting on the parabola and on the y = x line (the definition of iteration line).**x**

^{2}

**+ c Plots**can be seen at this link.

This
project drops plotting the parabola, stores the iterations in a list, and lets
you scan the list for patterns in the attractors.

**The Attractors of y = x**

^{2}

**+ c**is at this link.

The
last project in the series plots just the x values of each iteration for
successive values of c beginning with c = -0.5 until c < -2.

**Map of y =x**

^{2}+ c as a Function of c.
A free pdf file containing more information about the mathematics and coding in each project is available on request at grandadscience@gmail.com.