One day in the winter of 1961 Edward Lorenz, a meteorologist at the Massachusetts Institute of Technology, was running simple weather simulations through a program he had devised for his cumbersome vacuum-tube computer. He wanted to repeat and extend one particular run, but rather than start from scratch – always a time-consuming business – he started half-way through, taking the initial values from a print-out of the previous run. He broke off for a coffee, leaving the computer running, and came back expecting to find that the second half of the original simulation had been duplicated. Instead, to his astonishment, he discovered that what was supposed to be a re-run bore little resemblance to the previous version.
Lorenz’s first thought was that a tube had blown, but then the truth began to dawn on him. It wasn’t a matter of a computer malfunction. The figures he had typed in for the second run had been rounded by the computer – on the print-out but not in its memory – from six decimal places to three. He had assumed that such a tiny discrepancy – about one part in a thousand – would make no difference to something on the scale of a meteorological projection. But it had. A minute difference in the initial conditions had caused a major difference in outcome.
Scientific models It is in response to the huge complexity of nature – climate being a prime example – that scientists devise models. Models are simplified approximations of the real world that allow regularities to be discerned and described mathematically (i.e. by mathematical equations). It is assumed that such models behave deterministically: that a future state of the model can be derived fully, in principle at least, by applying appropriate equations to data describing the current state. This procedure can be ‘iterated’ – repeated again and again using the output of one run as the input for the next – to move the forecast further and further into the future.
It was this kind of method that Lorenz was following in running his simulation program in winter 1961. The fact that, after just a few iterations, the program had produced two wildly different outcomes from practically identical input data cast doubt over the whole methodology. His model had apparently behaved unpredictably and produced random results: it had exhibited – in a terminology that did not yet exist – chaotic behaviour.
Seagulls and butterflies So why did Lorenz’s climate simulation behave chaotically?
The mathematical equations used in weather forecasting describe the atmospheric motions of the relevant variables, such as temperature, humidity, wind speed and wind direction. An important characteristic of these variables is that they are interdependent: for instance, the level of humidity is affected by temperature, but the temperature is itself affected by humidity. In mathematical terms, this means that these variables are in effect functions of themselves, so the relations between them have to be described by so-called ‘nonlinear’ equations. Put simply, these are equations that cannot be represented by straight lines on graphs.
It turns out that one of the properties of nonlinear equations is that they exhibit the kind of sensitivity to initial conditions that had caused Lorenz such a shock in 1961. He went on to demonstrate that sensitivity of this kind was not merely a consequence of complexity by showing that it also occurred in a much simpler model (of convection) that could be described by just three nonlinear equations. In 1963 Lorenz recorded a colleague’s remark that, if his ideas were correct, ‘one flap of a seagull’s wings would be enough to alter the course of the weather forever’. By 1972 the beast behind the requisite atmospheric disturbance had shrunk, as was reflected in the title of Lorenz’s paper of that year: ‘Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?’ The ‘butterfly effect’ had been born.
Making sense of disorder The butterfly effect has been warmly embraced by popular culture, but its true implications are often not properly understood. Usually it is used as a loose metaphor for the way momentous events can be triggered by apparently trivial ones, but its significance goes far beyond this. The flap of a butterfly’s wings is only the cause of a tornado in the weak sense that the tornado might not have happened if the flap had not occurred first. But there are millions of other butterflies, and billions of other factors, all of which may be no less relevant in bringing about the tornado. One implication is the staggering sensitivity of the system to tiny events within it. Another implication, following from the first, is the practical impossibility of identifying the causes of any event in the system. Indeed, given that infinitesimally small events may bring about gross effects and that such tiny events may be beyond our powers of detection in principle, it may follow that the system, though fully deterministic, is entirely unpredictable.
‘Today even our clocks are not made of clockwork – so why should our world be?’
Ian Stewart, British mathematician, 1989
Lorenz drew the conclusion that long-term weather forecasting may be impossible in principle, but the implications of chaos run much deeper. The network of innumerable and interconnected factors that together determine global climate is by no means exceptional. Indeed, the great majority of physical and biological systems are of this kind, so scientific attempts to explain them mathematically are bound to involve nonlinearity; and hence they have the potential to show chaotic behaviour. From its origins in meteorology, chaos theory has spread to a wide variety of disciplines whose only link is that they deal with apparent disorder: turbulence in fluid dynamics; species fluctuation in population dynamics; disease cycles in epidemiology; heart fibrillations in human physiology; planetary and stellar motion in astronomy; traffic flow in urban engineering. In a more philosophical sense, the capacity of chaos to reveal the mesmeric and (some would say) beautiful order underlying the apparent disorder of nature gives us new hope of understanding and coping with the highly suggestive randomness of the universe.
the condensed idea
The butterfly effect