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By Mark Gilbert
“Chaos” is one of maths’ more emotive words. In non-technical language, chaotic is equated with a complete lack of order, randomness and even despair. On the other hand, as a mathematical concept, its definition is far more nuanced, and the consequences far more interesting.
Chaos is not random, but occurs in ‘deterministic’ systems, where the current state completely determines the system’s behaviour at any point in the future. For many systems, such as a swinging pendulum, or a single planet orbiting a star, knowing even the approximate state now is enough to predict its approximate behaviour forever. These systems were so open to the methods of classical mechanics, that in the late 1950s, the American mathematician, Steven Smale optimistically conjectured that all ‘dynamical systems’ would eventually settle to either steady equilibria or oscillation.
Exceptions were soon found, and throughout the 1960s, scientists such as Edward Lorenz discovered more and more deterministic systems where the behaviour didn’t settle down to ‘nice’ behaviour, where they were still predictable in theory, but unpredictable in practice. They called this ‘Chaos’.
Despite his error, Smale later succeeded in capturing the essence of this ‘Chaos’. He used geometric maps which involved folding and stretching squares. However, this essence of chaos can be captured with a simpler transformation called the `tent map’.
This map takes a value between 0 and 1, say x, and maps it to either 2x or 2-2x, whichever is smaller. The graph of this function is an isosceles triangle, and the function is called the ‘tent map’.
Exploring the properties of the tent map, we find that it meets the three mathematical criteria for chaos:
(1) sensitivity to initial conditions: the tiniest change in the starting point completely changes the way the system behaves after the same map has been applied many times, for example, the initial values of the trajectories plotted below differed by just 0.0001
(2) topological mixing: a fancy way of saying that for any numbers a<b, the image of the set of numbers less than b, but greater than a, under sufficiently many transformations, will always be the whole set of numbers between 0 and 1
(3) density of periodic orbits (repeating sequences): this is a fascinating example of the strange forms of order which define mathematical chaos. Any point between 0 and 1 is arbitrarily close to a point which, unlike the non-repeating trajectories of some of its neighbours, cycles through the same points as the map is applied again and again. For the tent map, there are infinitely many periodic points, which are generated by taking two integers k,n and calculated by 2n/(2^k+1).
This order in chaos is surprising, and in more complicated systems, is related to even more complex objects such as fractals and strange attractors, but it has proved incredibly useful in illuminating the behaviour of the strange systems now found in disciplines as diverse as economics, ecology and engineering.