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 behavior and initial data can always be found such
the period is arbitrarily high.Comment: 25 pages, 14 figures, typed in AMS-LaTe
