Chaos and multivaluedness: travelling on Riemann surfaces

KTH/Nordita/SU seminar in Theoretical Physics [before December 2013]

Wednesday 21 November 2007
from 11:00
to 12:00 at
FA31

Speaker :

David Gómez-Ullate (Universidad Complutense, Madrid)

Abstract :

Our work is part of a program whose aim is to understand the emergence
of chaotic behaviour in dynamical systems in relation with the
multi-valuedness of the solutions as functions of complex time \tau. In
this talk we consider a family of systems whose solutions can be
expressed as the inversion of a single hyperelliptic integral. The
associated Riemann surfaces are known to be infinitely sheeted coverings
of the complex time plane, ramified at an infinite set of points whose
projection in the \tau-plane is dense. The main novelty of this work is
that the geometrical structure of these infinitely sheeted Riemann
surfaces is described in great detail, which allows to study global
properties of the flow such as asymptotic behaviour of the solutions,
periodic orbits and their stability or sensitive dependence on initial
conditions. The results are then compared with a numerical integration
of the equations of motion. Following the recent approach of Calogero,
the real time trajectories of the system are given by paths on the
Riemann surface that are projected to a circle on the complex \tau-plane.
The mechanism leading to the emergence of chaotic behaviour will be
discussed in this context, emphasizing the similarities and differences
with respect to classical indicators of chaotic phenomology.

References:
* Yu. Fedorov and D. Gómez-Ullate, Dynamical systems on infinitely
sheeted Riemann surfaces, Physica D 227 (2007), no. 2, 120--134.
* F. Calogero, D. Gómez-Ullate, P. M. Santini and M. Sommacal, The
transition from regular to irregular motions, explained as travel
on Riemann surfaces, J. Phys. A 38 (2005), no. 41, 8873--8896.
* P. Grinevich, P. M.Santini, Newtonian dynamics in the plane
corresponding to
straight and cyclic motions on the hyperelliptic curve $\mu^2=\nu^n-1, n\in{\Bbb Z}$:
ergodicity, isochrony, periodicity and fractals, Physica D 232,
(2007) 22--32.