Convergent Chaos

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.

This talk reports a collaboration with Greg Huber (KITP, UCSB), Marc
Pradas (Open University) and Alain Pumir (ENS-Lyon).