# The InterFACE Project

### The Lorenz Equations

In the early 1960s, Edward Lorenz, working at the Massachussetts Institute of Technology, came across and investigated some of the first real-world chaos work ever performed.

Lorenz was attempting to create mathematical models of weather patterns, but he soon discovered chaotic behavior in his systems.

Today, three of the twelve equations which Lorenz was using - the so-called Lorenz equations - are some of the standard equations used for demonstrating chaotic systems.

Now let's look at the equations. Lorenz used twelve equations in all, but the three most important are:

• dx/dt = a * y - a * x
• dy/dt = r * x - y - x * z
• dz/dt = x * y + b * z

These are all differential equations - they describe the rate of change of x, y, and z respectively.

These equations attempt to model convection processes in the atmosphere - how air is warmed, then rises, is cooled, and falls again. Each of the variables, x, y and z, has a specific meaning in this system:

• x - This variable is proportional to the speed of motion of the air due to convection.
• y - This is a measure of the temperature difference between the warm, rising air and the cool, falling air.
• z - This is a measure of the vertical temperature difference as you move through the system from top to bottom.

There are also three constants in the equation set, which have a large impact on the system:

• a - This is proportional to the Prandtl number, which is based upon the nature of the air involved. This usually takes a value of 10.
• b - This represents the size of the area represented by the modelling - it was originally set to 8/3 (2.666...).
• r - This is the system's Rayleigh number - a parameter which dictates at which point convection will start in the system. It is vital for changing from steady, stable convection to chaotic convection.

Like the equations in the previous section, the Lorenz equations demonstrate a sensitivity to initial conditions.

If two graphs are plotted to show the graph of the x variable after several thousand iterations, but with different initial conditions, they will appear very different. This is a sign of chaotic behavior.

It is perhaps fortunate that Lorenz discovered his chaotic consequences while trying to model a real-world system, as the mathematical novelties suddenly took on a real significance. In his paper to the American Journal of the Atmospheric Sciences, entitled "Deterministic Nonperiodic Flow", Lorenz speculated that accurate long-term weather forecasting would be impossible, as the amazing sensitivity of the system to initial conditions would make it impossible to collect sufficiently-accurate data for use. Errors could not be expected to "cancel out" - instead, they would cause massive changes to the system in question.

This has given rise to the so-called Butterfly Effect. This uses the image of a butterfly flapping its wings in the Amazon Jungle, and causing thunderstorms over New York. The reason for this is that the fluttering of the butterfly's wings causes air currents, which cause minute changes to the initial starting conditions of the weather system - and, as the Lorenz equations demonstrate, this small change can have a major effect.