3.3 - Painlevé analysis and normal forms

Summary

Abstract

This work explores various general properties of systems of nonlinear ordinary differential equations obtained by using nonlinear transformations: further reductions of Poincaré normal forms for autonomous and non-autonomous systems, Lotka-Volterra universal standard format and generalization of Painlevé singularity analysis for detecting integrable systems. These methods are progressively implemented in the computer algebra program NODES.

Introduction

The main purpose of this work is to present some general results for systems of nonlinear ODE's obtained through quasi-monomial transformations. These transformations (discovered independently by a mathematician: A. Br'uno [1]; engineers: M. Peschel and W. Mende [2]; and physicists: the authors of the present article [3, 4, 5]), provide a powerful algebraic computational scheme for the analysis of nonlinear ODE's.

Essentially, three types of results are obtained in this framework:

I. Direct decoupling and/or integrability conditions under quasi-monomial transformations (QMT) with explicit closed-form construction of the reduced ODE systems and first integrals.

II. Reduction through QM-transformation to the Lotka-Volterra standard format.

III. Extension of the Painlevé test for integrability.

- Results of type I are based on a general matrix representation of systems of ODEs with polynomial nonlinearities closely associated to the QM transformations. Decoupling and integrability conditions arise from singularities of the matrices involved in this representation [5]. Such a singularity is generic in Poincaré-Dulac normal forms. As a consequence, the dimensions of these systems are reduced after a well-defined QM transformation.