chapter  A4
30 Pages

Lie point symmetries of delay ordinary differential equations

WithVladimir A. Dorodnitsyn, Roman Kozlov, Sergey V. Meleshko, Pavel Winternitz

Lie point symmetries of delay ordinary differential equations (DODEs) accompanied by an equation for the delay parameter (delay relation) are considered. A subset of such systems (delay ordinary differential systems or DODSs) which consists of linear DODEs and solution independent delay relations have infinite-dimensional symmetry algebras, as do nonlinear ones that are linearizable by an invertible transformation of variables. Moreover, the symmetry algebras of these linear or linearizable DODSs of order N contain a subalgebra of dimension dim L = 2N realized by linearly connected vector fields. Genuinely nonlinear DODSs of order N have symmetry algebras of dimension n, 0 ≤ n ≤ 2N + 2. It is shown how exact analytical solutions of invariant DODSs can be obtained using symmetry reduction. In particular we present invariant solutions of a DODS originating in a study of traffic flow.