The complex squaring function provides a great introduction to
complex dynamics. We define \(f(z)=z^2\) and iterate it on the complex
plane. In spite of the simplicity of the function, most of the hallmarks
of complex dynamics and chaos are right there in front of us. There is a
stable domain that comes in two separate pieces and the boundary between
those pieces forms a chaotic set. Even on the chaotic set, there is
order within the chaos.

One common feature of chaotic dynamics is missing. Generally, the chaotic
set is quite complicated and crinkly - fractal
in fact. For \(z^2\), the chaotic set is a perfect circle. Nonetheless, this
is part of what makes the example relatively easy to understand.

Stability

The first demo below shows the unit circle in the complex plane. Hover your
mouse over the demo to choose an initial seed \(z_0\) and view its
orbit under the complex squaring function.

If that number is inside the unit
circle, then it's absolute value is less than one. Thus if we square it, it
should move closer to the origin, as will the subsequent iterates. The interior
of the unit circle displays a stability in the sense that all points inside it
have a similar fate.

If the initial seed is outside the unit
circle, then it's absolute value is greater than one. Now if we square it, it
should move farther away from the origin and its orbit will diverge to \(\infty\).
We again have stability but with a different fate.

Chaos

While the previous demos was cool, it doesn't really show what happens right
on the unit circle. If you square a number with absolute value one, you definitely
produce another number with absolute value one. You can try to click right on the
unit circle in the first demo, though, yet it
just doesn't seem to work. Clearly, the behavior there is a bit more difficult.

The demos below has two buttons. You can click the "start with random point"
button to choose a random point right on the unit circle and then watch as the
firt 1000 iterates are generated. The code ensures that the orbit
stays
on the unit circle. Nonetheless, the orbit jumps around quite chaotically never
repeating itself. It seems quite chaotic.

But, if we connect the dots in the order that
they appeared, a hidden pattern emerges.