THE SPECTRUM OF DELAY DIFFERENTIAL EQUATIONS WITH MULTIPLE HIERARCHICAL LARGE DELAYS

We prove that the spectrum of the linear delay differential equation x'(t) = A(0)x(t) + A(1)x(t - tau(1)) -+ ... +A(n)x(t - tau(n)) with multiple hierarchical large delays 1 << tau(1) << tau(2) << ... << tau(n) splits into two distinct parts: the strong spectrum and the pseudo-continuous spectrum. As the delays tend to infinity, the strong spectrum converges to specific eigenvalues of A(0), the so-called asymptotic strong spectrum. Eigenvalues in the pseudo-continuous spectrum however, converge to the imaginary axis. We show that after rescaling, the pseudo-continuous spectrum exhibits a hierarchical structure corresponding to the time-scales tau(1), tau(2), ..., tau(n), Each level of this hierarchy is approximated by spectral manifolds that can be easily computed. The set of spectral manifolds comprises the so-called asymptotic continuous spectrum. It is shown that the position of the asymptotic strong spectrum and asymptotic continuous spectrum with respect to the imaginary axis completely determines stability. In particular, a generic destabilization is mediated by the crossing of an n-dimensional spectral manifold corresponding to the timescale tau(n).