Understanding complex dynamics by means of an associated Riemann surface

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 with arbitrarily large periods.

► Dense branching of solutions of dynamical systems as an obstruction to integrability.
► A Riemann surface with 1 parameter is associated to an illustrative dynamical system.
► A formula for the periods of the orbits is derived by analyzing the Riemann surface.
► Rational values of the parameter: the system is isochronous for generic initial data.
► For irrational values, the branch points are dense and aperiodic orbits are observed.