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 = x2 + 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.
https://scratch.mit.edu/projects/12537308/
The
next project plots the attractors of the quadratic equation y = x2 +
c by plotting on the parabola and on the y = x line (the definition of
iteration line).
x2
+ c Plots can be seen at this link.
https://scratch.mit.edu/projects/63919408/
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 = x2 + c is at this link.
https://scratch.mit.edu/projects/62164438/
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 =x2
+ c as a Function of c.
https://scratch.mit.edu/projects/65695456/
A free pdf file containing more information about the mathematics and coding in each project is available on request at grandadscience@gmail.com.
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