Chaos Theory

Lorenz Attractor Turbulence from an aeroplane wing

Chaos

Chaos theory describes the behaviour of deterministic dynamical systems, particularly those which are affected by the “butterfly effect” – critical dependence on initial conditions. This sensitivity manifests itself as an exponential growth of perturbations in the initial conditions, the behaviour of chaotic systems appears to be random.

This is not the case however, as the systems are deterministic, and so its future dynamics are defined by its initial conditions.

Where is Chaos found?Chaos is found in many areas of science, including:• Weather systems• Quantum mechanics• Chemical reactions• Population growth• Lasers• Fluid dynamics• Plate tectonics• Market behaviour in economicsWhere systems seemingly produce random behaviour.

The discovery of ChaosChaos, and “random” behaviour in deterministic systems,

was discovered by many different mathematicians and scientists; however, all of these people put the results down to faulty equipment, or miscalculations. The first person to accept and document Chaos was French mathematician, physician and philosopher Henri Poincaré in 1890. He did this whilst studying the orbits of various planets and satellites, noticing that there can be apparently random non-periodic orbits. After Poincaré, many people picked up on the idea of Chaos theory, although it remained a controversial topic, with many people rejecting the idea.

Famous instances of Chaos

One of the most famous, and earliest chaotic systems was “Hadamard’s Billiards” (1898). In this dynamical system, Hadamard considered the motion of a free particle on a frictionless surface of constant negative curvature, genus two.

Hadamard was able to show that all trajectories (infinitesimally close to one another), move away from each other dramatically – they have a positive Lyapunov exponent.

Many people consider this to be the first-ever examination of chaotic dynamical systems. Both Albert Einstein and Ernst Mach were greatly influenced and inspired by this experiment.

Hadamard’s Billiards

In 1961, Edward Lorenz was working on weather prediction; he created his own weather simulation machine, into which he would enter certain figures, for other figures to be returned. This would include temperature, moisture etc.

During one study he wanted to seem the same sequence again, so he started the system by entering the figures from half-way through his last attempt. However, the results he received differed from those in the previous test. Although the difference was minimal, Lorenz could see that it was increasing, and so over time would grow out of control.

Lorenz had discovered that the system was dependant on its initial conditions, and that changes in these conditions would have a large affect on the outcome. Lorenz therefore argued that it was impossible to predict the weather for beyond a week at most.

Lorenz Attractor