Mandelbrot Magic

Perturbation

To 'perturb' means to slightly alter the normal functioning of a system in some way. Perturbation can shed light on the inner functioning of systems. As an illustration, consider a pendulum, which is in a state of stable equilibrium. Perturb it by poking it slightly, and it will return to its original position. By contrast, imagine a rigid pendulum balanced upsidedown. This is in a state of unstable equilibrium, so that when you perturb it with a slight push, it will fall over.

In this section we will see alter the normal functioning of the Mandelbrot Set by making small, deliberate changes to the starting conditions, and seeing what happens after hundreds of iterations. As we commonly find when studying fractals, small changes can result in some big differences!

When we generate the Mandelbrot Set, we take the equation Z_n+1 = Z_n² + C and ask what happens to Z for all the different startng points C in the complex plane.
Normally, we start with Z = 0, so the iterations look like:

Z₁     =     Z₀² + C     =     C

Z₂     =     Z₁² + C     =     C² + C

Z₃     =     Z₂² + C     =     (C² + C)² + C

Now we are going to explore what happens if we start with Z₀ not equal to 0.

Remember that Z is a complex number, so it has two coordinates, a real and an imaginary component. The applet below has a slider beneath it to adjust the real component of Z and a slider on the right to adjust the imaginary component.

Play with the applet a bit to explore what happens when you change the value of Z₀. You can click to zoom into the image after you've adjusted the value of Z₀ to see up close how the Mandelbrot Set changes.

It can be very informative when studying a complex dynamic system such as this one to see how it responds to small changes in the starting conditions. In this case, we are "perturbing" the normal state of the system by giving Z₀ a little nudge in one direction or another.

Observe how sensitive the Mandelbrot Set is to perturbations of the value of Z₀. The normal Mandelbrot Set is a single connected object. However, as soon as we start to perturb Z₀, the Set breaks apart and little disconnected islands form.

Try perturbing one slider or the other, and notice the different effects of real vs imaginary perturbations. In one case, the resulting Mandelbrot Set preserves its symmetry, but in the other case it does not.

Questions:
A perturbation of which component of Z preseves the symmetry of the Mandelbrot Set? [ ]
What happens to the little embedded replicas of the Mandelbrot Set when you perturb Z₀? [______]

We have now seen several examples of Sensitivity to Initial Conditions, in which small changes to the starting conditions can cause big changes in the outcome.

In the next section, we'll see a different phenomenon, Robustness, which is a kind of stability. It means that the equation can be changed significantly, and yet it will still create a map in the complex plane that contains perfect replicas of the Mandelbrot Set.