Understanding complex dynamics by means of an associated Riemann surface

TL;DR

This paper analyzes the complex dynamics of a model using Riemann surface geometry, deriving explicit formulas for orbit periods that depend on initial data and coupling ratios, revealing rich periodic and quasi-periodic behaviors.

Contribution

It introduces a geometric approach to understanding the model's dynamics via Riemann surfaces, providing explicit period formulas linked to initial conditions and coupling ratios.

Findings

Explicit period formulas depend on initial data and coupling ratios.

Rational ratios lead to all orbits being periodic and system isochronous.

Irrational ratios exhibit both periodic and quasi-periodic orbits with variable periods.

Abstract

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.