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Journal of Dynamical Systems and Geometric Theories

(Since 2002)
Open Access ·Peer-reviewed·ISSN (Online): 2169-0057·ISSN (Print): 1726-037X

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Journal of Dynamical Systems and Geometric Theories
Open Access Original Articles

A Connection Between Geometry and Dynamical Systems

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pp. 25–44Vol. 3Issue 1January 2005DOI: 10.1080/1726037X.2005.10698486XML
Received:
21 Jan 2004
Published Online:
03 Jun 2013
Article type:
Original Articles
Language:
EN
Article no.:
1726037X.2005.10698486
Pages:
25–44

Abstract

This paper deals with a geometric and systematic approach to the integration of a nonlinear dynamical system: the anisotropic harmonic oscillator in a radial quartic potential. We study this system from a different angle: a) We show, using a Lax-type representation of the Hamilton’s equations of motion, that the system is linearized in the jacobian variety of a smooth genus 2 hyperelliptic curve. b) We find via Kowalewski-Painlevé analysis the principal balances of the hamiltonian vector field defined by the hamiltonian and we show that the system is algebraic complete integrable. c) We also describe an explicit embedding of the abelian variety which completes the generic invariant surface, into projective space. d) We give a direct proof that the abelian variety obtained in this paper is dual to Prym variety and can also be seen as a double unramified cover of the jacobian variety of an hyperelliptic curve of genus 2. e) We show that at some special values of the parameters λ1 and λ2, we can describe elliptic solutions which are associated with two-gap elliptic solitons of the Korteweg-de Vries equation.

Keywords

Lax representationHamiltonianIntegrable systemsCurvesAbelian varietiesJacobian varietiesPrym varieties

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