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.

Mandelbrot

$$ 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.