This project is the second in a series of projects beginning with my project 'y = (x + 6)/2 – A Mathematical Sink Hole' that can be seen at https://scratch.mit.edu/projects/12537308/ In the above project, x=y=6 is the single-point attractor. This project, as you can see from the defining equation y=x^2 + c, is a quadratic equation. To explore the onset of 'chaos' in this simple system, use these parameters. c = -0.5 zoom = 100, # of attractors = c = -1.1 zoom = 100, # of attractors = c = -1.3 zoom = 100, # of attractors = c = -1.38 zoom = 100, # of attractors = c = -1.395 zoom = 100, # of attractors = c = -1.755 zoom = 100, # of attractors = c = -2.00, zoom = 50, # of attractors = I used a 'seed' value of x=0.7. You can experiment to determine if the seed makes a difference. If you've done the above, you've seen what is called 'period doubling' as the number of attractors keep splitting until the system, even though it is deterministic, appears to be random at c=-2. This is called 'deterministic chaos'.
On page 6 of 'Chaos and Fractals - The Mathematics Behind the Computer Graphics' by Robert Devaney et. al, the authors say 'At this point we urge the reader to write a computer program which computes the orbits of various functions. You will see a vast array of different dynamical behavior if you compute the orbits of various x-values for T(x) = x^2 + c.' This project accepts the challenge. I've plotted both the points on the parabola and, since y=x in iteration, the y=x points on the y=x line.
