2.5 Chaos theory

In his revised and published paper, Poincaré corrected his error, explaining why the small body’s behaviour is in general unpredictable. His insight laid the foundations for the mathematical concept today known as chaos theory, which became established in the second half of the twentieth century.

For most systems, the equations resulting from Newton’s laws cannot be solved exactly to obtain the position and velocity of all parts at any given time. In these cases, a computer program must be used to ‘solve’ the equations. But these computer programs generate tiny rounding errors at each step of the calculation. This combines with Poincaré’s observation that such systems are extremely sensitive with respect to initial conditions. Ultimately, even if we knew the initial conditions exactly, our ability to predict the future behaviour of these systems – which we now call ‘chaotic systems’ – is strictly limited.

Chaos is ubiquitous, both in nature and in human behaviour: from the dripping of a tap to the currents of the oceans, from the beating of the human heart to the function of the brain, from work motivation to trading in financial markets. This is because the mathematical equations which describe physical systems, and the difficulties in solving them, have analogues in other disciplines. A particularly well-studied example is the atmosphere, which you’ll take a look at next.