mark

Let $f(z)=z^2-1$. Use our rational function demo to plot the basins of attraction of $f$ and $f\circ f$. Explain the results that you see.

An archived instance of discourse for discussion in undergraduate Complex Dynamics.

mark

RedCrayon

$f$ has an orbit of period two between $-1$ and $0$. This is because $-1$ and $0$ are super-attractive fixed points of $f\circ f$. As far as I can tell, everything in the finite basin falls into that orbit under iteration, hence the entire basin is a single color.

For $f\circ f$, the finite basin of $f$ is split into two regions by the super-attractive fixed points $-1$ and $0$. Some regions converge to $-1$ under iteration and some converge to $0$, hence these two regions are colored differently.

mark

@RedCrayon That looks great!