Dynamical systems and \sigma-symmetries

TL;DR

This paper explores how -symmetries, a deformation of standard symmetries, can be applied to reduce dynamical systems of first-order ODEs, extending their utility beyond differential equations of order one.

Contribution

It demonstrates the application of -symmetries to reduce sets of first-order ODEs, addressing limitations of previous reduction methods for equations of order one.

Findings

-symmetries enable reduction of dynamical systems.

The method extends symmetry reduction techniques to first-order ODE systems.

Potential for broader application in analyzing dynamical systems.

Abstract

A deformation of the standard prolongation operation, defined on sets of vector fields in involution rather than on single ones, was recently introduced and christened "\sigma-prolongation"; correspondingly one has "\sigma-symmetries" of differential equations. These can be used to reduce the equations under study, but the general reduction procedure under \sigma-symmetries fails for equations of order one. In this note we discuss how \sigma-symmetries can be used to reduce dynamical systems, i.e. sets of first order ODEs in the form dx^a/dt = f^a (x).