Dynamics and Chaos

Dynamics is the general study of how systems change over time. The Theory of Chaos is a relatively new subject in mathematics and physics. Sometimes it has been called the third revolution of the 20th century after the theory of relativity and quantum mechanics.

For the ancient Greeks, the word chaos meant disorder. It was opposed to Kosmos that was used for the universe regarded as a complex and orderly system. But nowadays, the word chaos refers to a particular “disorder” with some curious and interesting properties. A chaotic system is essentially unpredictable due to the sensitivity to initial conditions. This means that a very small change in the initial value of the position, temperature, etc. can give rise to completely different results. This concept was popularized by the meteorologist Edward Lorenz through what he called the butterfly effect. It is named after the metaphor of a butterfly flapping its wings being able to influence several weeks later a tornado far away. The image of the results also resembles a butterfly’s wing and it quickly became a symbol for scientists working on dynamical systems.

In 1961, Lorenz was trying to solve his weather equations in one of the primitive computers available at that time. One day, wanting to check some results that were taking a long time for the machine, he took a shortcut. Instead of repeating all the process he had already done, he started midway through. Then, when the results were computed, he realized that there was no resemblance with the previous results. Lorenz’s results were correct, the problem was the shortcut, it meant that he needed to input the values of his previous calculations. The preceding results had an accuracy of three digits, however this accuracy was proved ridiculous for systems as the one he was studying. In fact, errors and uncertainties quickly grow ending inexorably in chaos.

Nature is full of systems like weather. Some examples are animal populations, epidemics or stock market (Econophysics). Those systems are called ‘nonlinear’ meaning that they follow relationships that are not strictly proportional.

The double pendulum is a perfect example of chaotic system, if you want to know more about this watch our experiments!