Illustrating Chaos (draft)

A chaotic system is a dynamic where slight variations in the initial conditions result in much larger variations in the output. Weather is a prime example of a chaotic system. For example, whether it rains today or not it depend on a minute difference in atmospheric temperature a few days earlier. This why long term weather prediction is impossible. See the Wikipedia article on Lorenz equations for details on a system of equations used in weather prediction and the peculiar behavior of this equations.

However one does not need to go to something as complicated as equations for weather prediction to find chaotic behavior. Consider the following simple equation (where '*' stands for multiplication).

x[n+1] = r*x[n]*(1-x[n])

You can find a discussion about the origin of this equation and several of its properties in pp. 69-80 of the book Chaos - Making a New Science by James Gleick, (New York, Viking, 1987). Here we focus only on a numerical simulation of the equation. It turns out that the equation exhibits chaotic behavior for values of r close to 4 as shown in the Figure 1

Figure 1

Notice the part where the value of x stays for several periods at a low value and then increases suddenly. However the seemingly irregular behavior is not the defining feature of chaotic systems. Such systems are characterized by the fact that small changes in the initial state produce quite different results. Figure 2 below superimposes the runs for starting values 0.59, 0.60, and 0.61. For clarity it uses a larger horizontal scale than Figure 1.

Figure 2

For the first two steps the values are close but then diverge widely.
The run of Figure 2 in the same scale as Figure 1
A longer run for a different starting value

Chaotic behavior is different than random. If you start at 0.6 you get exactly the same graph each time. Random behavior would be different each time.

Many natural systems exhibit behavior that could be called chaotic. Figure 3 below shows average temperature variations over thousands of years (each point is the average over approximately a century) from the Vostok ice core. The original source of data is the paper
    Petit et al. (1999), "Climate and Atmospheric History of the Past 420,000 years from the Vostok
    Ice Core, Antarctica", Nature, 399, pp.429-436.
I am grateful to Professor David Black of the School of Marine and Atmospheric Sciences of Stony Brook University for providing me with an Excel file of the data.

Figure 3: The leftmost point is about 420,000 years ago and the rightmost point is at the current time. The horizontal line corresponds to the current average temperature.

Are the data of Figure 3 showing chaotic behavior?

Before jumping to any philosophical implications of the nature of chaotic systems it is worth noting that the above equation yields much simpler results for smaller values of r. x[n] may settle quickly into a constant value as shown in the first graph below or it may go into sustained oscillations as shown in the second graph.

Figure 4

Figure 5

As the value of r increases the oscillations become more complex until we reach the chaotic behavior. You can experiment with different values of r by doing your own simulations and also convince yourself that the system is not random. (For r under 3 x settles into a steady value as shown in Figure 4. For r between 3 and 3.5 x oscillates between two values as shown in Figure 5. For r around 3.5 x oscillates between 4 values and as r increases the oscillations become less regular.)

Download Simulation Program (It runs only on Microsoft Windows XP or later)

Version of October 12, 2009