Multiple basins of attraction

When we iterate a polynomial, any attractive periodic orbit must contain a critical point in its basin of attraction. Now, a quadratic has only one critical point and, as a result, can have only one bounded basin of attraction. The images below show the Julia sets of \(f(z)=z^2-1\) and \(f(z)=z^2-0.123+0.745i\), which have attractive orbits of periods two and three respectively. You can hover over those images to see the orbits.

A cubic, however, has two critical points so it can have more 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.