Notes prepared by
Professor Marcus du Sautoy, University of Oxford

Dynamical systems, chaos and strange attractors

Dynamical systems analyse the evolution over time of mathematical and natural phenomena. For example the swinging of a pendulum or the weather can be translated into a set of numbers whose behaviour over time is then described mathematically by a system of equations.

Attractors are places where a dynamical system likes to tend towards as the system evolves. Attractors are important in any analysis as they reflect the ultimate state of the system. For example a pendulum with friction swings back and forward tending towards its stable state with the pendulum coming to a stop.

If we introduce three magnets which can attract the head of the pendulum then the magnets become new points of attraction for the system. But unlike the simple swinging pendulum, we now have a chaotic motion. The essence of chaos is sensitivity to initial conditions. If I change very slightly the initial point from which I release the pendulum, it can end up heading to a completely different magnet. This is the heart of a chaotic dynamical system.

If one plots the starting positions where the pendulum gets attracted to the blue magnet, the picture looks extremely complicated – something we call a fractal because of its very complex structure.

The actual points of attraction in the case of the example with the magnets and the pendulum are still very simple. But some dynamical systems seemed to exhibit very strange points of attraction with very complex geometrical structures. Computer analysis of one particular system controlled by the Hénon map seemed to demonstrate an attractor with an infinite amount of fine structure. Successive magnifications seemed to reveal an ever increasing amount of detail. The trouble is that computer graphics might make us think this thing looks fractal but pinning down that this thing really exists requires mathematical proof. Numerical calculations require approximations and the whole point of chaos is that it is very sensitive to small errors. In 1991 Carleson with his colleague Michael Benedicks finally gave a rigorous mathematical proof of the existence of a strange attractor conferming that the attractor for the Hénon map is as strange and chaotic as the graphics seem to imply.

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