Simple Stability Tests for Second Order Delay Differential Equations

2018. Т. 18, No 2. С. 149–155 149


Introduction
In the present paper, a specially designed substitution transforms linear second order equations into a system, to which we apply some known exponential stability results.
In Lemma 1 does not assumed that ( ) ≥ 0 but in Lemma 2 the constant in right-hand side of the inequality (4) is better. So these lemmas are independent and we use both of them. 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 nonautonomous 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.
Proof. By substitution to equation (5) By Lemma 1 the following condition implies exponential stability of system (7).
The second case is proved similarly. Now with an additional assumption ( ) ≥ 0 we can improve Theorem 1. Theorem 2. Suppose ( ) ≥ 0, = 1, … , and the following conditions hold: The proof is based on Lemma 2 and is similar to the proof of Theorem 1.   (5) is exponentially stable.
Consider here the following equation (13) Then equation (12) is exponentially stable.
To prove the theorem we need in the following lemma.
We will assume that the initial value problem has a unique global solution on [ , ∞) for all nonlinear equations considered in this section. Suppose at least one of the following conditions holds:

Remarks and Open Problems
The technique of reduction of a high-order linear differential equation to a system by the substitution ( ) = 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 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 nondelay terms.
Solution of the following problems will complement the results of the present paper: 1. In all stability conditions obtained, we used lower and upper bounds of the coefficients and the delays. It is interesting to obtain stability conditions in an integral form.
2. Apply the technique used in the paper to examine delay differential equations of higher order. Also, the substitution used in this chapter was based on the existence of a non-delay term, it would be interesting to adjust the method for equations which have several delayed terms only.
3. Establish necessary stability conditions for the equations considered in this chapter by reduction to a system of delay differential equations. 4. For the sunflower equation and its modifications establish set of conditions to guarantee boundedness of all solutions. 5. Apply the technique used in the paper to examine delay differential equations of higher order.