Chaos Theory is a mathematical discipline that studies chaos in dynamical systems. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, repetition, self-similarity, and fractals, amongst other features. The underlying principle behind chaos theory is the Butterfly Effect, which describes how a small change in the initial state of a deterministic nonlinear system can result in large differences after much time.
What is the Butterfly Effect?
In 1961, Lorenz was using a computer to simulate weather patterns using various initial conditions. A particular sequence of data intrigued him, and he wanted to observe the end of the simulation again. He then entered data corresponding to the middle of a particular simulation. But, to his surprise, he saw drastically different behaviour this time! Why, you may ask? As it turns out, the computer he used worked with 6-digit precision but the data he entered rounded variables to 3 digit precision. The difference was so tiny that the scientific consensus at the time would have been that this would have no visible effect. However, Lorenz discovered that this small change in initial conditions produced drastic changes in long-term system behaviour!
With this observation, Lorenz laid down the foundation for a whole new field of mathematics: chaos theory. The butterfly effect's name comes from the paper Lorenz submitted on the matter in 1972, titled Predictability: Does the Flap of a Butterfly's Wings in Brazil set of a Tornado in Texas?
What is the Lorenz Attractor?
The system of differential equations in the first slide and the corresponding plot is what is named the Lorenz attractor. Attractors in dynamical system theory are simply states toward which a system tends to evolve, for a wide variety of starting conditions of the system. The plot in the first slide is then known as the attractor corresponding to the Lorenz system of differential equations.
(1) dx/dt = σ(y -x)
(2) dy/dt = x(ρ – z) - y
(3) dz/dt = xy - βz
This system was used by Lorenz to model the weather, specifically atmospheric convection. In particular, the equations describe the change of three quantities with respect to time: x is proportional to the rate of convection, y to the horizontal temperature variation, and z to the vertical temperature variation. The constants σ, ρ, β are system parameters proportional to various atmospheric measurements. This system also arises in the study of lasers, DC motors, electric circuits, chemical reactions, and osmosis!
The plot seen in the first slide is simply the pattern that is often seen for a relatively large interval of system parameters σ, ρ, β. It is generated by tracing the (x, y, z) coordinates for each t.
The Butterfly Effect has become a widely popular term in modern culture.
Legacy
Lorenz is remembered as a pioneer and a genius in many ways. In 2011, the MIT Lorenz Centre was named after him. The Lorenz Centre is a climate think tank devoted to fundamental scientific inquiry founded in honour of Lorenz's fundamental contributions to the field of mathematics and atmospheric science.
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