The Mandelbrot and Julia sets. Introduction.
We are amazed by turbulent
but even ordinary differential equations can have chaotic
Moreover dynamics of nonlinear maps is surprisingly complicated.
We start with a Quick Tour around the Mandelbrot and Julia sets.
The Mandelbrot set is made by iteration of the complex map (function)
zn+1 = zn2 + C for different C
and zo = 0.
Points on the parameter plane C with bounded zn
form the Mandelbrot set (the black region on these pictures, colors outside
the Mandelbrot set show, how fast zn go to infinity).
The Mandelbrot set contains small copies of the "main cardioid"
(in the white square) shown on the right picture. These small copies are
connected with the main cardioid by filaments which are formed by other
tiny cardioids. These structures are called the "Mandelbrot hair"
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Each point C in the parameter space specifies the geometric
structure of the corresponding Julia set J(C) in the dynamical
plane z. If C1 is in the Mandelbrot set, the Julia
set is connected. If C2 is not in the Mandelbrot
set, the Julia set is a Cantor dust. You see connected 'Douady's rabbit'
and 'Cantor dust' below.
You see that every J set is self-similar. But the most amazing thing
is that the M's filaments and corresponding J set are similar too
(see the right top and bottom images). And two more similar M and
J sets below.
Two more fractal images and
The Julia set trip CPU-based animation.
Next: Rotation Numbers and Internal angles of the M-bulbs
updated 19 Dec 2013