Simple Stability Tests for Second Order Delay Differential Equations
DOI:
https://doi.org/10.14529/ctcr180215Keywords:
second order delay differential equations, exponential stability, reducation to systemsAbstract
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.
References
Berezansky L., Diblik J., and Smarda Z. Positive Solutions of Second-order Delay Differential Equations with a Damping Term. Computers and Mathematics with Applications, 2010, 60, pp. 1332– 1342. DOI: 10.1016/j.camwa.2010.06.014
Berezansky L., Braverman E., Domoshnitsky A. Stability of the Second Order Delay Differential Equations with a Damping Term. Differential Equations and Dynamical Systems, 2008, vol. 16, issue 3, pp. 185–205. DOI: 10.1007/s12591-008-0012-4
Cahlon B., Schmidt D. Stability Criteria for Certain Second-order Delay Differential Equations with Mixed Coefficients. Journal of Computational and Applied Mathematics, 2004, vol. 170, issue 1, pp. 79–102. DOI: 10.1016/j.cam.2003.12.043
Cahlon B. Unconditional Stability for Certain Delay Differential Equations. Dynamic Systems Applications, 1996, vol. 5, pp. 583–594.
Gyori I., Hartung F. Asymptotically Exponential Solutions in Nonlinear Integral and Differential Equations. Journal of Differential Equations, 2010, vol. 249, issue 6, pp. 1322–1352. DOI: 10.1016/j.jde.2010.06.017
