ABSTRACT
The technique of reduction of a second order linear differential equation to a system of two differential equations of the first order by the substitution x' = y is quite common. However, this substitution does not depend on the parameters of the original equation, and therefore does not offer new insight from a qualitative analysis point of view. Instead, we proposed a substitution which exploits the parameters of the original model. By using this approach, a broad class of the second order non-autonomous linear equations with delays was examined and explicit easily-verifiable sufficient stability conditions were obtained. We widely use many transformations of a given equation. In particular, by the Mean Value Theorems one can transform one class of equations to another one. For example, a solution of a given integro-differential equation is also a solution of an explicitly constructed delay differential equation. By this method a known result, obtained for delay differential equations implies a similar result for the given integro-differential equation. For the nonlinear second order nonautonomous equations with delays we applied the linearization technique and the results obtained for linear models. Our stability tests are applicable to some milling models and to a non-autonomous KaldorKalecki business cycle model. Several numerical examples illustrate the application of the stability tests.
