A stability criterion for systems with delay in one critical case
The author studies the stability of equilibria for systems with delay described by the first-order equation with delay \[ \dot x(t) = ax(t) + b(t)x(t-\tau) + F\bigl(x(t),x(t-\tau)\bigr), \] where \(x\) is a scalar, \(a\) and \(b\) are constants, and \(\tau >0\). It is supposed that the analytic function \(F\) expands in a neighborhood at the point \(x_1=x_2=0\) in a series starting with terms of at least second order. The value similar to the first Lyapunov value for the equation is assumed to be equal to \(0\). The author obtains an expression for the value similar to the second Lyapunov value for the equation in terms of the coefficients of the equation. Using this expression, the author gives a criterion for the stability of the equilibrium \(x=0\) of the equation.
system of differential equations with delay
scientific article; zbMATH DE number 1285588
scientific article