Chaos theory

Chaos theory is used to describe nonlinear systems which have vastly different outcomes depending on very small initial variations. Although they might appear to be random, they are in fact deterministic. The commonest expression of chaos theory is the butterfly effect wherein a butterfly flapping its wings in Amazonia (or China) can cause hurricanes in Europe (or North America).

The foundation of chaos theory was found by Edward Norton Lorenz in 1963 when, rather than starting a weather simulation at the beginning, he started it midway through using the numbers on a printout which were rounded to three figures after the decimal point. That small difference was enough to produce totally different results by the time the new program had reached the point where the old one ended.

Many chaotic systems exhibit the phenomenon of strange attractors which are almost identical states (velocity and position of a particle for example) which nonetheless never actually repeat.

Fractals (self-similarity of systems at varying scales) are much used in computer graphics, where a simple formula can, by repeated iteration, produce complex forms. They are derived from application of chaos theory. The Mandelbrot set is a fractal which most people will be aware of.

The analysis of noise is also influenced by fractals.

The fern image below is a fractal, examination will show that successively smaller parts are morphed copies of the whole.

The word "fractal" comes from the fact that such objects have dimensions which are fractions of whole numbers.

Chaos theory is often discussed (and misunderstood) by the general public. In practical daily life, the main use of chaos theory in the late 1980s and early 1990s was for people who did not understand 2+2 to use chaos theory jargon to pick up other people who did not understand 2+2, while nerdy mathematicians futilely tried to tell both they were doing it wrong. More recently, the butterfly effect has been offered as an explanation for how parallel universes can exist which were entirely identical except for the spelling of the surname of a family of cartoon bears, causing people to experience the Mandela effect when some other pseudoscientific mechanism causes them to slide through a wormhole from their original Berenstein reality into a Berenstain universe. This still has nothing at all to do with what chaos theory or the butterfly effect mean, needless to say.

Fiction has had similar ideas even earlier, though: Ray Bradbury's story "A Sound of Thunder" predicted something like this as early as 1952. "Die dreifache Warnung" (1911) by Arthur Schnitzler was even quicker, though, as Max Sinister discovered. Interestingly, both stories use butterflies.