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Dynamical systems on infinitely sheeted Riemann surfaces

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Fedorov, Yuri N. and Gómez-Ullate Otaiza, David (2007) Dynamical systems on infinitely sheeted Riemann surfaces. Physica D: nonlinear phenomena, 227 (2). pp. 120-134. ISSN 0167-2789

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Official URL: http://dx.doi.org/10.1016/j.physd.2007.02.001




Abstract

This paper is part of a program that aims to understand the connection between the emergence of chaotic behaviour in dynamical systems in relation with the multi-valuedness of the solutions as functions of complex time tau. In this work we consider a family of systems whose solutions can be expressed as the inversion of a single hyperelliptic integral. The associated Riemann surface R -> C = {tau} is known to be an infinitely sheeted covering of the complex time plane, ramified at an infinite set of points whose projection in the tau-plane is dense. The main novelty of this paper is that the geometrical structure of these infinitely sheeted Riemann surfaces is described in great detail, which allows us to study global properties of the flow such as asymptotic behaviour of the solutions, periodic orbits and their stability or sensitive dependence on initial conditions. The results are then compared with a numerical integration of the equations of motion. Following the recent approach of Calogero, the real time trajectories of the system are given by paths on R that are projected to a circle on the complex plane tau. Due to the branching of R, the solutions may have different periods or may be aperiodic.


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© 2007 Elsevier B.V. All rights reserved.
We thank L. Gavrilov, P. Santini, and V. Enolski for discussions and valuable remarks. Our research was partially supported by the Spanish Ministry of Science and Technology under grant BFM 2003-09504-C02-02.

Uncontrolled Keywords:Analytic structure; Periodic-solutions; Evolution-equations; Integrable systems; Painleve property; Motions
Subjects:Sciences > Physics > Physics-Mathematical models
Sciences > Physics > Mathematical physics
ID Code:30880
Deposited On:15 Jun 2015 10:14
Last Modified:10 Dec 2018 15:09

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