Explicit Construction of First Integrals by Singularity Analysis in Nonlinear Dynamical Systems

TL;DR

This paper introduces a new explicit algorithm that constructs first integrals of nonlinear ODEs using Painleve-Laurent series and singularity analysis, enhancing the toolkit for identifying integrable systems.

Contribution

The paper presents a novel algorithm that explicitly constructs first integrals from singularity analysis, bridging a gap between Painleve methods and integral calculation in nonlinear systems.

Findings

Successfully constructs first integrals for Hamiltonian systems with polynomial potentials

Applies the method to generalized Volterra systems with effective results

Demonstrates the algorithm's ability to identify integrability in complex nonlinear systems

Abstract

The Painleve and weak Painleve conjectures have been used widely to identify new integrable nonlinear dynamical systems. The calculation of the integrals relies though on methods quite independent from Painlev\'e analysis. This paper proposes a new explicit algorithm to build the first integrals of a given set of nonlinear ordinary differential equations by exploiting the information provided by the Painleve - Laurent series representing the solution in the neighbourhood of a movable singularity. The algorithm is based on known theorems from the theory of singularity analysis. Examples are given of the explicit construction of the first integrals in nonlinear Hamiltonian dynamical systems with a polynomial potential, and in generalized Volterra systems.