Asymptotics of eigenvalues of the monodromy operator for periodic differential equations with delay (Q1902843)

The author considers linear periodic differential equations with delay, (1) \(dx (t)/dt = A(t) x(t) + B(t) x(t - \tau (t))\), where \(A\) and \(B\) are real \(n \times n\) matrices with period \(\omega\), and the delay \(\tau\) is a positive \(\omega\)-periodic function. Denote by \(U\) the monodromy operator of (1) and by \(\{\lambda_n (U)\}\) the set of eigenvalues of \(U\). In the case where this set is infinite the author proves the asymptotic estimate \(\lambda_n (U) = O(n^{- 1/\varepsilon})\), \(n \to \infty\), with an arbitrary positive number \(\varepsilon\). \{The term ``finite'' instead of ``infinite'' in the English translation of this statement is only one highlight of the sloppy translation\}.