The spectrum of delay differential equations with multiple hierarchical large delays

Abstract

We prove that the spectrum of the linear delay differential equation x (t) = A 0 x(t) + A 1 x(t − τ 1 ) + . . . + Anx(t − τn) with multiple hierarchical large delays 1 τ 1 τ 2 . . . τn 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 τ 1 , τ 2 , . . . , τn. 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 τn.

A delay equation model for the Atlantic Multidecadal Oscillation

“…We observe that the delays occurring in the MoC system are of different magnitude: τ + ≪ τ − , where τ + is of order O (1) in the time scale of (4.28). For hierarchical large delays, Yanchuk and co-workers [28,29] provide a simple approximation of the spectrum for (4.28), which captures the range of possible curvatures of the curves along which the eigenvalues shown in figure 6 align. Any eigenvalue λ of (4.28) satisfies truedetfalse[−ϵλI−I+C1normale−false(α+λfalse)τ++C2normale−false(α+λfalse)τ−false]=0, with ϵ ≪ 1, τ + = O (1) and τ + ≪ τ − .…”

“…It is very sensitive with respect to small changes of τ + (since ϕ + = ωτ + / ϵ ) such that the location of the eigenvalue curves will vary strongly depending on τ + or ϵ within the range given by ϕ + ∈ [0, 2 π ]. Ruschel & Yanchuk’s [29] analysis shows in general that for hierarchically large delays the spectrum ‘fills an area’ of the complex plane under small parameter variations.…”

Absolute stability and absolute hyperbolicity in systems with discrete time-delays

“…Asymptotic spectrum for multiple hierarchically large delays and its relation to the conditions for absolute stability. Let us briefly review some concepts for the spectrum of systems with hierarchically large time-delays [72]. This spectrum can be generically divided into m + 1 parts corresponding to different timescales:…”

“…This implies that the destabilization of the system with hierarchical delays with det A m = 0 can occur only due to some B m,j spectral component, which is caused by the largest delay τ m . In a degenerate case of det A m = 0, stable parts of other spectral components may appear as well, see more details in [5,7,59,72].…”

Delay master stability of inertial oscillator networks

“…Recent work has led to a thorough analytical understanding of the spectrum in the limit of large delay, with applications in, e.g., optoelectronics [8][9][10][11]. In many cases, however, time lags may match the system's dynamical timescales and may play a critical role for stability.…”