Interactive basins of attraction

in complex dynamics

Here's one of the key ideas in complex dynamics: when we iterate a nice (say, differentiable) function \(f:\mathbb C \to \mathbb C\), certain attractive domains naturally appear. The images below allow you to choose an initial point by just hovering your mouse so that you can see this in action. The first two images show the Julia sets of \[f(z)=z^2-1 \text{ and } \,f(z)=z^2-0.123+0.745i,\] which have attractive orbits of periods two and three respectively.

As it turn out, an attractive orbit must always attract a critical point. Since those first two are quadratic functions, they have only one critical point and can, therefore, have only one attractive orbit. A cubic, however, has two critical points so it can have two attractive orbits. The image below, for example, shows the Julia set of \[f(z) = (-0.090706 + 0.27145i) + (2.41154-0.133695)z^2 - z^3.\] For this function, one critical point is attracted to an orbit of period two and the other is attracted to an orbit of period three. As a result, we see features of both quadratic Julia sets above.

The more complicated the function, the more complicated the potential behavior. The image below shows the basins of attraction for the rational function \[f(x) = \frac{z^5+0.01}{z^3}.\] There are four different colors - one for each basin of attraction, including infinity.