Integrable deformations of R\"ossler and Lorenz systems from Poisson-Lie groups

TL;DR

This paper introduces a method to create integrable deformations of Hamiltonian systems with Lie-Poisson symmetries using Poisson-Lie groups, applied specifically to Rössler and Lorenz systems, enabling systematic and potentially automated deformations.

Contribution

It presents a novel, algorithmic approach to deform Hamiltonian systems via Poisson-Lie groups, extending integrability to coupled and uncoupled Rössler and Lorenz systems.

Findings

Constructed integrable deformations of Rössler and Lorenz systems.

Deformations are non-polynomial, derived through exponentiation of Lie bialgebras.

Method is largely algorithmic and suitable for computer implementation.

Abstract

A method to construct integrable deformations of Hamiltonian systems of ODEs endowed with Lie-Poisson symmetries is proposed by considering Poisson-Lie groups as deformations of Lie-Poisson (co)algebras. Moreover, the underlying Lie-Poisson symmetry of the initial system of ODEs is used to construct integrable coupled systems, whose integrable deformations can be obtained through the construction of the appropriate Poisson-Lie groups that deform the initial symmetry. The approach is applied in order to construct integrable deformations of both uncoupled and coupled versions of certain integrable types of R\"ossler and Lorenz systems. It is worth stressing that such deformations are of non-polynomial type since they are obtained through an exponentiation process that gives rise to the Poisson-Lie group from its infinitesimal Lie bialgebra structure. The full deformation procedure is essentially algorithmic and can be computerized to a large extent.