KTH/Nordita/SU seminar in Theoretical Physics

Chaos and multivaluedness: travelling on Riemann surfaces

by David Gómez-Ullate (Universidad Complutense, Madrid)




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.

* 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.