Simple Stability Tests for Second Order Delay Differential Equations

Abstract

For linear and nonlinear delay differential equations of the second order with damping terms exponential stability and global asymptotic stability conditions are obtained. The results are based on the new sufficient stability conditions for systems of linear equations. The results are illustrated with numerical examples, and a list of relevant problems for future research is presented.

We proposed a substitution which exploits the parameters of the original model. By using that 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. There is a natural extension of this approach to stability analysis of high-order models. For the nonlinear second order non-autonomous 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 Kaldor–Kalecki business cycle model. Several numerical examples illustrate the application of the stability tests. We suggest that a similar technique can be developed for higher order linear delay equations, with or without non-delay terms.

AMS Subject Classification: 34K20, 34K25.