This equation will change how you see the world (the logistic map)

Created: 2020-08-04T15:50:43.266106

chaos theory.

Logistic map

$$ X_{n+1} = rx_n * (1-x_n)

Creates a parabola; the smaller a population the larger the result but the larger the population the smaller the result.
Initial population does not matter very much. The parabola will pull a population back to some equilibrium which is controlled by the growth rate.
r controls the growth rate and thus the equilibrium.
When growth is too high energy will oscillate instead of stay smoothly controlled.
A growth of 3.57 creates chaos and was used for pseudo-random
chaos is not maintained at all values of r; order is apparent with certain windows that create chaos.

Period Doubling Bifurcation: when stability is lost and the population oscillates between two equilibriums forever.
Bifurcation diagram: shows where equilibriums split apart.

$$ Z_{n+1} = Z_{n^2} + c

Choose some complex number c; if the algorithm goes off the rails (it grows or shrinks continuously) then it is not part of the mandelbrot set.
Some numbers such as -1 will oscillate and will not escape some initial starting bounds. Those numbers are part of the mandelbrot set.
The mandelbrot diagram is a large bifurcation diagram that has been collapsed.