Generalized notions of symmetry of ODEs and reduction procedures (Q2922232)

The focus of this article is the extension of the notion of a \(\lambda\)-symmetry along with its application to the reduction of the order of systems of ordinary differential equations and dynamical systems. These symmetries are also referred to as `combined' \textit{joint-\(\lambda\)-symmetries}, where a \(\sigma\)-prolongation is a deformed prolongation and a system of \(n\) ordinary differential equations can be defined as \(\sigma\)-symmetric if under this prolongation at the point \(\epsilon = 0\) we obtain \(0\).NEWLINENEWLINEThe notion of reducing a system is extended towards doing so for a dynamical system. Thus, it is shown that first-order ordinary differential equations can be suitably reduced if they admit a \(\sigma\)-symmetry. Considering the case of \textit{orbital} symmetries it is shown that a different form of reduction is found for dynamical systems. The authors discuss how dynamical systems can be transformed into higher order ordinary differential equations and how these symmetry properties of the dynamical system can be transferred into reduction properties of the relevant ordinary differential equation. More simply, if the dynamical system admits some symmetry then it can be said that the dynamical system can be reduced in terms of suitable symmetry-adapted variables, transferring the reduction to the higher order ordinary differential equation. Examples are provided illustrating theses ideas.NEWLINENEWLINEThe presence of \(\sigma\)-symmetries admits geometrical interpretations and algebraic aspects of interest. As such the results obtained are of significance.