**Chaos**

*"Chaos breeds life, when order breeds habit." *

- Henry Adams

## In the late 1700s, the eminent work initiated by the English mathematician and physicist Sir Isaac Newton was carried forward by the French mathematician and physicist Pierre-Simon Laplace, also known as the “French Newton.” Laplace firmly believed in** determinism**, the idea that the past completely determines the future, that given enough information about a system, it is possible to make completely accurate predictions about the future state of that system.

In 1814, Laplace published the first formal articulation of determinism, called **Laplace’s Demon** (1):

*“We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.”*

By the 1800s, determinism was all the rage in mathematics and physics. There were certain “special cases” that did not fit with the idea of determinism, but these misfits were pushed aside.

Over one century later, on a winter day in 1961, the mathematician and meteorologist Edward Lorenz studied one such special case - **the weather**. Lorenz used a model based on 12 equations to run simulations that predicted weather patterns on his rather archaic digital computer. To save time, he entered data from the mid-point of his earlier simulations so that he could examine the later weather predictions of his model without having to wait through the entire simulation. When Lorenz entered the data, he rounded it off to three digits whereas the data from the earlier simulations contained six digits, a difference he assumed to be inconsequential since temperature is rarely measured to within one part in a thousand. However, when Lorenz compared the new simulations with the earlier ones, he saw this:

## The two simulations matched each other closely at the start, but after several model “months” they **wildly diverged**, not even remotely resembling each other in the longer term. Upon completing an extensive analysis, Lorenz realized that his assumed inconsequential round-off error was in fact highly consequential, that given enough simulation time even the tiniest round-off errors produced massive changes in the entire system.

Fascinated, Lorenz pursued these observations with an even simpler model, consisting of only three basic equations, that described **rolling fluid convection** in a box full of gas heated from below. These were the three equations:

**Dx/dt = P(y-x)**

Dy/dt = Rx-y-xz

Dz/dt = xy-By

Dy/dt = Rx-y-xz

Dz/dt = xy-By