## 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).
You can find a discussion about the origin of this equation and several
of its properties in pp. 69-80 of the book
Notice the part where the value of
For the first two steps the values are close but then diverge widely.
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
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
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.x[n]
As the value of by doing your own
simulations and also convince yourself that the system is not random.
(For r under 3 r settles
into a steady value as shown in Figure 4. For x
between 3 and 3.5 r oscillates between two values
as shown in Figure 5. For x around 3.5 r
oscillates between 4 values and as x increases
the oscillations become less regular.)r
Version of October 12, 2009 |