Lecturer: Michael Wilkinson, Department of Mathematics and Statistics,

The Open University, Walton Hall, Milton Keynes, MK7 6AA, England

Abstract:


Chaos is widely understood as being a consequence of sensitive dependence 
upon initial conditions. Despite their overall intrinsic instability, trajectories may be very strongly
convergent in phase space over extremely long periods, as revealed by our 
investigation of a simple chaotic system (a realistic model for small bodies in a turbulent flow).
We establish that this strong convergence is a multi-facetted phenomenon, in which the 
clustering is intense, widespread and balanced by lacunarity of other regions.
Power laws, indicative of scale-free features, characterise the distribution of 
particles in the system. We use large-deviation and extreme-value statistics to 
explain the effect. Our results show that the interpretation of the 'butterfly 
effect' needs to be carefully qualified. This notion of convergent chaos, which 
implies the existence of conditions for which uncertainties are unexpectedly small, may 
also be relevant to the valuation of insurance and futures contracts.