Chaos is a state of apparent disorder and irregularity whose evolution in time, though governed by simple and exact laws, is highly sensitive to starting conditions. Small variations in these conditions produce widely different results, in such a way that the long term behaviour of chaotic systems cannot be predicted.
It is impossible to accurately predict the motion or progression through time of even a simple chaotic system - herein lies the beauty of the double pendulum. Compared to real world systems such as the weather; molecular vibrations; fluid dynamics; or solar systems and galaxies, it is an extremely simple system which can be easily modelled and understood mathematically. However, unlike a simple single pendulum, it is impossible to predict the long term behaviour of the double pendulum.
Although Chaos Theory is a well defined branch of mathematics, agreement on a definition of the phrase itself has not been finalised. However, it is generally accepted that all chaotic systems share the following features:
- Deterministic. In a mathematical sense a deterministic system is fundamentally straightforward (you can model it with a formula) and there are no random elements involved.
- Sensitive to initial conditions. Infinitesimally small changes to the initial conditions in an otherwise perfect experiment will produce widely different results. This sensitivity to initial conditions is popularly known as ‘the butterfly effect’.
- Unpredictable and unrepeatable. No two experiments produce the same results and it is impossible to predict the outcome of an experiment.
Put another way, the behaviour of a chaotic system depends so sensitively on the system's precise initial conditions that it is, in effect, unpredictable and cannot be distinguished from a random process, even though it is deterministic in a mathematical sense.
To experience chaos theory first hand, repeatedly spin your chaotic pendulum ensuring that you have all initial conditions (initial position, initial angles, initial force…) as closely matched to the previous experiment as possible. You will see within a matter of seconds a vastly different outcome.
The great French mathematician Laplace said:
If we were to know with precision the positions and speeds of all the particles in the universe then we could predict the future with certainty…
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