Splitting transformation for a linear nonstationary singularly perturbed system with constant delay in the equation for the slow variable

A method of splitting with respect to rates in change of variables is developed for a linear nonstationary singularly perturbed system with constant delay in the equation for slow variables. The splitting method is based on an algebraic approach, namely, the immersion of the system with delay into a family of systems with an extended state space, and a nonlocal change of variables. The existence is proved and the asymptotics of the Lyapunov transformation is constructed, which generalizes the splitting Chang transformation to systems with delay and performs a complete splitting of a two-rate system with constant delay into two independent subsystems of lower dimensions than the original system: separately for the fast and slow variables. We prove that the split system is algebraically and asymptotically equivalent to the original system in the extended state space. The asymptotics is constructed and the action of asymptotic approximations of the splitting transform is examined.