Dynamical systems and σ-symmetries

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 "σ-prolongation"; correspondingly one has "σ-symmetries" of differential equations. These can be used to reduce the equations under study, but the general reduction procedure under σ-symmetries fails for equations of order one. In this note we discuss how σ-symmetries can be used to reduce dynamical systems, i.e. sets of first order ODEs in the form dx^a/dt = f^a (x).

References in corpus (2)

• A generalization of λ-symmetry reduction for systems of ODEs: σ-symmetries
• Reduction of systems of first-order differential equations via Lambda-symmetries

Cited by in corpus (7)

• ${\mathcal{C}}^{\infty }$-symmetries of distributions and integrability
• Dealing with Rational Second Order Ordinary Differential Equations where both Darboux and Lie Find It Difficult: The $S$-function Method
• Integration of differential equations by ${\mathcal{C}}^{\infty }$-structures
• Symmetry and Lie-Frobenius reduction of differential equations
• ${\mathcal{C}}^{\infty }$-structures in the integration of involutive distributions
• Generalized notions of symmetry of ODE's and reduction procedures
• On the Geometry of Twisted Symmetries: Gauging and Coverings