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Fractal Chaos Explorer — Mandelbrot, Julia & Bifurcation, Interactive

Explore chaotic and fractal systems interactively: progressive escape-time renders of the Mandelbrot and Julia sets, the logistic map's period-doubling bifurcation tree that is one of the clearest visual routes into chaos theory, and chaos-game fractal builders where simple random rules produce structured self-similar patterns.

How to use Fractal Chaos Explorer

  1. Pick a mode — escape-time map, bifurcation tree, or chaos game.
  2. Pan and zoom into the complex plane or parameter space.
  3. Adjust iteration depth and colouring.
  4. Watch structure emerge from the underlying simple rule.

Frequently asked questions

What is the bifurcation diagram actually showing?
It plots the long-term behaviour of the logistic population-growth equation as its growth-rate parameter increases — a single stable value splits into two, then four, then chaos, in the same period-doubling route to chaos found across many nonlinear systems.
Why does zooming into the Mandelbrot set keep revealing more detail?
The boundary is a fractal — infinitely detailed at every zoom level, with smaller copies of the whole set appearing recursively no matter how far in you go.
What is a "chaos game" fractal?
A surprisingly simple rule — repeatedly jump partway toward a randomly chosen vertex of a polygon — that produces a highly structured fractal (like the Sierpinski triangle) purely from random input.

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