Lie group classification of second-order delay ordinary differential equations
This article is part of a research program the aim of which is to turn Lie group theory into an efficient tool for classifying, transforming and solving a further class of equations, namely Delay Ordinary Differential Systems (DODSs). Such a system consists of two equations involving one independent variable x and one dependent variable y. As opposed to ODEs the variable x figures in more than one point (we consider the case of two points, x and x − ). The dependent variable y and its derivatives figure in both x and x − . This is not a discretization since both x and y remain continuous. Two previous articles were devoted to first-order DODSs, here we concentrate on a large class of second-order ones. We show that within this class the symmetry algebra can be of dimension n with 0 ≤ n ≤ 6 for genuinely nonlinear DODSs and n = ∞ for linear or linearizable ones. The symmetry algebras can be used to perform symmetry reduction and obtain exact particular (invariant) solutions.