Introduction to Chaos

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The word 'chaos' is frequently colloquially used to describe a system which exhibits a high state of disorder. In fact, chaos has a much more specific meaning and a reasonably short history.

Although Poincaré had touched upon the idea of chaos in the 1800s, the subject remained largely neglected in favour of other studies of nonlinear behaviour until Dr. Edward Lorenz tried to use high-speed computers to model atmospheric convection currents in 1961.

Lorenz used simplified versions of the Navier-Stokes equations - a set of equations that describe fluid motion - in order to model the convection currents in the atmosphere to try to gain an insight into the weather which was notoriously unpredictable. However, when Lorenz obtained the long-term solutions to his equations, he found that they never settled to any kind of equilibrium. Instead, irregular, aperiodic oscillations persisted.

Additionally, he found that slight differences in the initial values resulted in behaviours that rapidly diverged to totally different profiles.

Although from these results it appeared that these systems were inherently unpredictable, Lorenz found some structure when he plotted his results in three-dimensions and found a butterfly-shaped set of points. A 2-D plot can also show this pattern and the figure on the right indicates what he saw. The 3-D and 2-D plots are examples of 'strange attractors' and are an indication that the boundary of the phase space (where an output variable might be located) could be reasonably well defined but predicting exactly where the output might lie is very difficult.

Thus, chaos is defined by Strogatz as 'aperiodic long-term behaviour in a deterministic system that exhibits sensitive dependence on initial conditions'. In a deterministic system, in theory, as long as the initial conditions are 'known', i.e. accurately measured, the future state of the system can always be calculated. In reality, the knowledge of the future state is extremely difficult to obtain due to the sensitivity to initial conditions resulting in chaotic (aperiodic) behaviour that renders long-term predictions impossible. We can see from an example that even if the measuring system accuracy could be improved by a huge amount, the effort expended will always be in vain.

We can see the effect that the intial conditions have as a simulation proceeds by looking at the figure on the right. Two of the three initial values used were identical and the third value differed by 1 part in a million between runs. Even with this very small difference,the two systems deviate after just over 30 seconds.

Chaotic systems can be found in a multitude of disciplines including engineering, meteorology, economics, geology and microbiology; however the waterwheel experiment that has been built provides an example of chaotic motion.