Stability Threshold for Scalar Linear Periodic Delay Differential Equations

We prove that for the linear scalar delay differential equation (x) over dot(t) = -a (t)x (t) + b(t)x(t-1) with non-negative periodic coefficients of period P > 0, the stability threshold for the trivial solution is r := integral(P)(0)(b(t)-a(t))dt = 0, assuming that b(t+1) - a (t) does not change its sign. By constructing a class of explicit examples, we show the counter-intuitive result that, in general, r = 0 is not a stability threshold.