Understanding complex dynamics by means of an associated Riemann surface

Gómez-Ullate, David, Santini, Paolo, Sommacal, Matteo and Calogero, Francesco (2012) Understanding complex dynamics by means of an associated Riemann surface. Physica D: Nonlinear Phenomena, 241 (16). pp. 1291-1305. ISSN 01672789

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Official URL: http://dx.doi.org/10.1016/j.physd.2012.04.004

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 with arbitrarily large periods.

Item Type: Article
Uncontrolled Keywords: Dynamical systems, integrable systems, isochronous systems, Riemann surfaces
Subjects: G100 Mathematics
Department: Faculties > Engineering and Environment > Mathematics and Information Sciences
Depositing User: Becky Skoyles
Date Deposited: 10 Jan 2014 08:29
Last Modified: 10 Aug 2015 11:12
URI: http://nrl.northumbria.ac.uk/id/eprint/15050

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