Symmetry, singularities and integrability in complex dynamics II: Rescaling and time-translation in two-dimensional systems

2000/11/02 by Peter Leach, P. G. L. Leach, Spiros Cotsakis +5
Mathematics · Physics and Astronomy · #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Numerical methods for differential equations #Quantum chaos and dynamical systems #math-ph #math.MP #nlin.SI

paper · pdf · doi:10.48550/arxiv.nlin/0011002

25 pp, latex. To appear in J.Math.Anal.Appl

arxiv created 2000/11/02 · openalex publication_date 2000/11/02 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The explicit integrability of second order ordinary differential equations invariant under time-translation and rescaling is investigated. Quadratic systems generated from the linearisable version of this class of equations are analysed to determine therelationship between the Painlevé and singularity properties of the different systems. The transformation contains a parameter and for critical values, intimately related to the possession of the Painlevé Property in the parent second order equation, one finds a difference from the generic behaviour. This study is a prelude to a full discussion of the class of transformations which preserve the Painlevé Property in the construction of quadratic systems from scalar nonlinear odes invariant under time translation and rescaling.

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