Nonlocal Symmetries, Telescopic Vector Fields and $\lambda$-Symmetries of Ordinary Differential Equations

Abstract

This paper studies relationships between the order reductions of ordinary differential equations derived by the existence of $\lambda$-symmetries, telescopic vector fields and some nonlocal symmetries obtained by embedding the equation in an auxiliary system. The results let us connect such nonlocal symmetries with approaches that had been previously introduced: the exponential vector fields and the $\lambda$-coverings method. The $\lambda$-symmetry approach let us characterize the nonlocal symmetries that are useful to reduce the order and provides an alternative method of computation that involves less unknowns. The notion of equivalent $\lambda$-symmetries is used to decide whether or not reductions associated to two nonlocal symmetries are strictly different.

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Cited by in corpus (8)

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• Jacobi multipliers, non-local symmetries and nonlinear oscillators
• Dealing with Rational Second Order Ordinary Differential Equations where both Darboux and Lie Find It Difficult: The $S$-function Method
• On the symmetries of a nonlinear non-polynomial oscillator
• Interplay of symmetries and other integrability quantifiers in finite dimensional integrable nonlinear dynamical systems
• Generalized symmetries, first integrals, and exact solutions of chains of differential equations
• Symmetry and Lie-Frobenius reduction of differential equations
• On the Geometry of Twisted Symmetries: Gauging and Coverings