Properties of solutions to one class of nonlinear systems of differential equations with a parameter

A system of nonlinear ordinary differential equations of large dimension with a parameter is considered. We investigate asymptotic properties of solutions to the system in dependence on the growth of the number of the equations or parameter. We prove that, for sufficiently large number of differential equations, the last component of the solution to the Cauchy problem is an approximate solution to an initial problem for one delay differential equation. For a fixed number of equations and a sufficiently large parameter, the solution to the Cauchy problem for the system is an approximate solution to the Cauchy problem for a simpler system.