Complex dynamics studies what happens when a function is repeatedly applied to points in the complex plane:
z_{n+1}=f(z_n).
Even the simple quadratic map f(z)=z^2+c can produce fixed points, periodic cycles, attracting regions, escape to infinity, and chaotic boundaries.
For a fixed c, varying z_0 reveals the Julia set. Fixing z_0=0 and varying c produces the Mandelbrot set—the parameters for which the orbit remains bounded.
Simple rules. Infinite detail.
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