SCIENCE

CHAOS THEORY

Introduced by Prof. C Budd, University of Bath, and Prof. J Keating, University of Bristol, on 26 November 1999

Professor Budd opened the session with reference to some common examples of chaotic behaviour in nature, including air turbulence and changes in the weather. The important distinction between random and chaotic behaviour was drawn. For example, in turbulence, system particles at any instant all conform to classic physical laws, whereas if the motion were truly random, they would not. The difficulty in predicting future behaviour in such a system lies in defining the precise state of the system at any instant. The result is that motions diverge rapidly from predicted values based on classical mechanics and the assumed initial state of the system. In this sense chaotic behaviour could be considered to be a deterministic disorder or complex behaviour.

The lecture then moved on to a detailed presentation of how seemingly simple mathematical models can show extremely complex behaviour. The example chosen was that of population growth in a closed system. In the Malthusian scenario, population growth is expressed as ever-increasing and based on a linear growth function. In the real world, population growth is restrained when the population becomes excessive because other influences come into play, such as disease, famine etc.. In order to take into account such effects a simple modification has been proposed (May 1971) in the form of the logistic difference equation. This equation displays non-linear behaviour as the population growth rate is increased. At low rates of growth the population settles to a steady state value year after year; as the rate increases a bifurcation occurs and the steady state population oscillates between two values every other year. Further increase in the growth rate eventually leads to unpredictable yearly population levels which appear to be random.

C.Budd

Professor Keating opened his lecture by describing the wave like nature of matter and the notion that it is actually impossible to precisely measure the exact position and momentum of a particle at any instant (Heisenberg's Uncertainty Principle). By implication therefore it is impossible to precisely establish initial conditions. This effectively rules out the possibility of precisely defining `initial states' as required by classical mechanics and the methods of Laplace , Lagrange and Hamilton etc.. The lecture then explored further the difficulties of combining quantum mechanics with classical physical theories.

J. Keating