The Mandelbrot and Julia sets. Introduction.

We are amazed by turbulent fluid dynamics but even ordinary differential equations can have chaotic strange attractors. 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) z_n+1 = z_n² + C for different C and z_o = 0. Points on the parameter plane C with bounded z_n form the Mandelbrot set (the black region on these pictures, colors outside the Mandelbrot set show, how fast z_n 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" or filaments.

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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.

Contents Next: Rotation Numbers and Internal angles of the M-bulbs
updated 19 Dec 2013