Understanding complex dynamics by means of an associated Riemann surface
Matteo Sommacal (Northumbria University Newcastle)
Tuesday 13th May, 2014 15:00-16:00 Maths 326
We provide an example of how the complex dynamics of a recently introduced model can be understood via a detailed analysis of its associated Riemann surface. Thanks to this geometric description an explicit formula for the period of the orbits can be derived, which is shown to depend on the initial data and the continued fraction expansion of a simple ratio of the coupling constants of the problem. For rational values of this ratio and generic values of the initial data, all orbits are periodic and the system is isochronous. For irrational values of the ratio, there exist periodic and quasi-periodic orbits for different initial data. Moreover, the dependence of the period on the initial data shows a rich behaviour and initial data can always be found with arbitrarily large periods.
This toy model is meant to provide a prototype of a general mechanism explaining, in a deterministic context, the transition from regular to irregular motions as travel on Riemann surfaces.