Publication: Understanding complex dynamics by means of an associated Riemann surface
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2012-08-15
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Elsevier
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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.
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© 2012 Elsevier B.V. All rights reserved.
It is a pleasure to acknowledge illuminating discussions with Carl Bender, Boris Dubrovin, Yuri Fedorov, Jean-Pierre Fran¸coise, Peter Grinevich and Fran¸cois eyvraz. The research of DGU was supported in part by MICINN-FEDER grant MTM2009-06973 and CUR-DIUE grant 2009SGR859 and he would like to thank the financial support received from the Universita di Roma “La Sapienza” under the Accordo Bilaterale with Universidad Complutense de Madrid.
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