Chaos!

We will now explore the transition from order to chaos in these simple dynamic systems with the help of a new tool, the Bifurcation Diagram.

Instead of using the logistic function, as we did in the last section, which describes population dynamics, we'll use the closely related quadratic function, which generates the Mandelbrot Set. The basic dynamics of bifurcation and period doubling apply in both functions.

Bifurcation (branching)

First, let's examine the dynamics of this system using the same tool of graphical iteration that we used with the Logistic Function. In this case, the Quadratic Equation is simply:

Z_n+1 = Z_n² + C

Explore the behavior of the function for the range of C from 0 to -2. Look for areas where the function behaves periodically. Try to count the periodicities at various values of C. You can do this by counting the number of times the graphical iteration touches the parabola in the left panel, or by seeing how many points occur in the right panel before the same pattern repeats.

Let's note some key periodicities.
At C = 0, the function has a period of 1, i.e. it has one fixed point solution.
At C = -1.0, the function has bifurcated to period 2, that is, it oscillates between two fixed point solutions.
At C = -1.3, the function has bifurcated to period 4
At C = -1.3875, the function has bifurcated to period 8.

This is known as the Period Doubling Cascade, and it leads directly into Chaos.

Let's explore a different way of looking at this now, using what's called the Bifurcation Map.

This map shows the value of C on the X-axis plotted against the orbit behavior (periodicity, shown as the number of branches) on the Y-axis.

Click to zoom in, <Ctrl>-click to zoom out.
Applet courtesy of Yevgent Demidov and Andrei Buium.

Click on this map to explore its behavior. Notice the examples of self-similarity in the little copies of the entire shape, repeating at multiple scales.

Zoom all the way out to the beginning. The simple section on the right (SOON) from 0 to .75 is a single branch that corresponds to the range of the Quadratic Function where it has a simple, period-1 behavior. Let's keep lowering C, which means going from right to left on this map. By the time C = -1, the equation has bifurcated into period two behavior, and you can see two branches in the bifurcation map above. Each of these two branches quickly bifurcates again, into four arms, representing period 4 behavior. The bifurcations get closer together and smaller, until at around -1.4 the whole map appears to turn messy. This signifies aperiodic behavior, or possibly periodic behavior where the period is so high it's unrecognizable. Perhaps there's a pattern to the orbits of the equation for those values, but you might have to iterate the equation millions of times before the pattern repeats itself and we can appreciate its periodicity.

As the value of C continues further left into the disorderly areas, something unexpected happens. There are small windows of order amidst the chaos. Zoom into one of these, the window around C = -1,76.

Questions:
Use the graphical iteration applet at the top to determine the periodicity at:C = -1.76: [ ]
Find this value of C in the bifurcation map, using the coordinates below it.

This is the largest of the many small windows of order in the disorderly range of the bifurcation map. Keep exploring this beautiful and infinitely complex structure for a while.

You can see a static, hi-res version of the very similar bifurcation map for the Logistic Function here.

The Mandelbrot Set

We've been exploring the Quadratic Function here in one dimension only, using real numbers. Rememeber that we generated the Mandelbrot Set by iterating this same equation using 2-Dimensional numbers in the complex plane. So it should not be a surprise that there's a strong connection between the bifurcation diagram and the Mandelbrot Set. To explore this relationship, let's look at the behavior of the Mandelbrot Set along the real (X) axis, and compare it with the bifurcation map.

The X-axis of the Mandelbrot Set corresponds to the Bifurcation Diagram above. Click in the top panel to explore the relationship between the two panels.

The range from C = 0 to -.75 is the period 1 area of the bifurcation map, and is also the main body of the Mandelbrot Set.
The period 2 section of the bifurcation map corresponds to the main bulb (the 'head') of the Mandelbrot Set.
Each additional bifurcation in the Map corresponds to the next bulb on top of the head moving leftward on the X-axis.
At C=-1.76, the window in the bifurcation map corresponds to the biggest of the small replicas on the spike of the Mandelbrot set.
In fact, every small window in the bifurcation map represents a tiny replica in the Mandelbrot Set!

These ideas of period doubling leading to chaos are not confined to the pure math of the Quadratic Function. As we saw with the logistic Function, period doubling occurs in ecological population modeling. It has also been observed in many other natural systems, including the aperiodic rhythms of the heart, where some disorder or chaos is healthy, while too regular a heartbeat is actually dangerous and can signal an impending heart attack! Period doubling also occurs in the transition of liquids from smooth flow into turbulence, the behavior of electronic circuits, in lasers, and in the behavior of complex chemical reactions.

In the next section, we will see how chaos manifests itself in the behavior of a single brain cell, or neuron, being stimulated electrically.