We cosider a certain class of systems of differential equations of “neutral” type containing linear delay γ(t) = (1-μ)t , which increases indefinitely when t → ∞. For a small positive μ the asymptotic behavior of these systems is investigated. Based on the studied properties of such systems, a method of stabilization of first approximation systems containing also a constant small delay is proposed. Some methods of research of stability of these systems are given. The stability of some neutral-type systems has been studied by transition from these systems to counting systems with a delay without neutral terms. In further research, the methods of a small parameter with a derivative as well methods of studying difference systems are also used. The methods can be used to study the process of vertical oscillations of a current collector when interacting with a contact wire in the case of an elastic support at the point of fixation of the contact wire. When considering the problem of stability of systems, a Banach space is introduced, in which some properties of shift operators are investigated. Since differential equations with a deviating argument are integrated in a closed form only in exceptional cases, numerical methods are used to integrate them. Graphs of the corresponding example illustrating the effect of the value μ on the asymptotic properties of neutral-type systems with linear delay are constructed. These graphs of the numerical solution of the system under consideration are obtained using the Matlab application software package. They show the asymptotic stability and instability of this system, both with and without neutral terms.