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A NEW form of chaos, which does not derive its unpredictability from the so-called ‘butterfly’ effect, has been discovered by a graduate student in New York. This new ‘complex chaos’ is so unpredictable that it is almost impossible to answer any question about its behaviour.

Until now, all known forms of chaos have resulted from the butterfly effect. For example, the world’s weather, which is chaotic, is unpredictable because we can never measure its initial state precisely. Even the tiniest fluctuation which we do not account for – perhaps the beat of a butterfly’s wings – might mean the difference between a drought and a deluge on the other side of the world.

Complex chaos requires no butterflies. Even if we measured the state of the weather so precisely that we accounted for every butterfly’s wing, we would still be unable to predict the weather’s behaviour.

Christopher Moore of Cornell University is the graduate student who has demonstrated that some chaotic systems are not just random but also highly complex. He has devised an innovative proof, which mixes topology with Alan Turing’s theory of computing (Physical Review Letters, vol 64, p 2350).

‘Usually, if you know the initial conditions accurately you have no problem,’ adds Moore. ‘But in the case of complex chaos, even if you do know the initial conditions exactly, you cannot predict anything.’

Moore has shown that all the standard tools for understanding chaotic behaviour fail when faced with complex chaos. ‘There are no short cuts,’ he says.

Moore’s proof depends on the impasse in the so-called halting problem for computers: it is impossible to devise an algorithm which can examine any computer program and decide whether a computer that runs it will either complete the program or get stuck in a never-ending loop.

To generate complex chaos, Moore creates a transformation of a square (see Diagram). He shows that the effect on any point of this transformation is equivalent to a single operation carried out by a theoretical computer called a Turing machine. He then shows that predicting the future behaviour of the point is the same as predicting whether the Turing machine will halt. Because that is undecidable, predicting the point’s behaviour is impossible. ‘The ‘undecidable’ properties of the Turing machine carry over to my transformation,’ says Moore.

Moore’s vital innovation is the type of transformation he uses to create chaos. Moore’s transformation divides a square into eight segments and transforms each segment separately.

Usually, the boundaries between segments would violate one of mathematics most cherished rules – that such transformations should be ‘continuous’, so that points that begin close together stay close together. But Moore removes the discontinuities by removing an infinitely thin net of points, leaving behind what mathematicians call a Cantor set.

When Moore repeats the transformation of the square, he generates a scrambled mess. Although it resembles other forms of chaos superficially, it is far less predictable.

Mathematicians have carried out work similar to Moore’s with mathematical objects known as cellular automata. But Moore’s work goes several steps further. It applies to simple systems which can arise in the real world. ‘We thought the unpredictability of chaos rested on the inaccuracy of our knowledge of the initial conditions,’ says Moore. ‘But it does not.’

‘It’s a fantastic paper,’ says Robert MacKay, a mathematician at the University of Warwick.

According to MacKay, Moore’s work is significant not only for what it tells us about the nature of chaos but in opening new avenues of research. ‘The systems that mathematicians have been looking at are just a few of the simplest forms. Moore has uncovered new universes of possibilities.’