Criteria for dangerous and safe boundaries of stability domains for second-order delay equations (Q2703950)

The author investigates the second-order delay differential equation of the form NEWLINE\[NEWLINE\ddot x= a_1 x+ b_1x(t-\tau)+ a_2\dot x+ b_2\dot x(t-\tau)+ f(x,\dot x, x(t-\tau),\dot x(t-\tau)),NEWLINE\]NEWLINE where \(a_1\), \(a_2\), \(b_1\) and \(b_2\) are constants and \(\tau> 0\). Under a certain assumption on the roots of the characteristic equation NEWLINE\[NEWLINEp(p- a_2- b_2 e^{-p\tau})- a_1- b_1 e^{-p\tau}= 0,NEWLINE\]NEWLINE the author obtains formulas for analogues of the first and second Lyapunov quantities. These formulas enable him to obtain a criterion for the safety of the boundary of the stability domain of the above equation.