Logistic Equation

This picture is a Mandelbrot fractal given by the logistic equation

zn+1 = p(1 - zn) zn ,

with the initial value z0= (0.1, 0) over the rectangular area defined by xmin = -2.5, xmax = 4.5, ymin = -2.5, ymax = 2.5. When each zn assumes a real value, the equation is used by ecologists to model population growth of certain species. If the "critical point" z0= (0.5, 0) is used for the initial value, the "figure eight" portion becomes smoother and symmetric about its intersection point as you can see in the Symmetric Logistic Equation . It is my personal preference to use the asymmetric picture.

Birth of Chaos: The figure below is a replica of the above image, and there the green area is given by the convergence scheme and the black background by the divergence scheme. The blue area consists of the parameters corresponding to the sequences that eventually become periodic of period greater than 1. For example, the red bumps in the original picture above consists of the "periodic points" of period 2.

Be aware, however, that the small computer-generated picture is too coarse to show what is happening on the boundaries between different regions. Consider, for example, the real line (middle horizontal line):

The important points regarding the real logistic equation are labeled in the figure, where v and w are approximately 3.44 and 3.57, respectively. These points are related to the behaviors of the real logistic sequences zn as follows:

  • If 0 < p < 1 then zn converges to 0;
  • If 1 < p < 3 or p = 1 then zn converges to 1 - p-1;
  • If 3 < p < v or p = 3 then zn eventually becomes periodic with period 2;
  • If v < p < w then zn eventually becomes periodic with period of the form 2k, and as p increases, the period increases in an orderly fashion like 22, 23, 24 , etc. without bound; thus, from p = 1 to about 3.57, the behavior of zn shows the phenomenon of so-called bifurcation;
  • If w < p < 4, (it was discovered that) zn becomes quite unpredictable or chaotic.

  • The "invisible" interval between w and 4 is part of the boundary of the black region interacting with other regions in a complex fashion.

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