SPECTRUM AND AMPLITUDE EQUATIONS FOR SCALAR DELAY-DIFFERENTIAL EQUATIONS WITH LARGE DELAY

. The subject of the paper is scalar delay-diﬀerential equations with large delay. Firstly, we describe the asymptotic properties of the spectrum of linear equations. Using these properties, we classify possible types of destabilization of steady states. In the limit of large delay, this classiﬁcation is similar to the one for parabolic partial diﬀerential equations. We present a derivation and error estimates for amplitude equations, which describe universally the local behavior of scalar delay-diﬀerential equations close to the destabilization threshold.

1. Introduction. Delay differential equations (DDE) with large delay appear to be a very useful tool for modeling many systems. For example, in laser devices with optical feedback [28,17] a large time delay appears due to the very fast internal time scale dynamics of the laser. As a result, the external feedback introduces a new time scale in the system, which can be modeled as the large time delay.
From the theoretical viewpoint, DDEs with large delay can be considered as singularly perturbed problems, which are challenging for analytical as well as numerical analysis. During the last decades, a number of important results have been obtained that shed some light on the properties of these systems [19,2,20,1,14,10,12]. In particular, the authors of these works studied the existence and asymptotic behavior of periodic orbits of a class of DDEs, and noticed that their properties can be related to the properties of a reduced discrete map. For example, a flip bifurcation of the map implies square waves in the corresponding singularly perturbed system of DDEs. Another important milestone in the analysis of singularly perturbed DDEs was the derivation of amplitude equations [7,15,6,30], which describe the local dynamics of DDEs at the bifurcation threshold. These derivations have been performed by formal asymptotic methods. Recently, the structure of the spectrum of linear systems with large delay was described for the cases of constant [18,31] and periodic [27] coefficients. Extensions for chaotic systems and systems with multiple delays are known as well [11,13].
In this paper, we describe in detail the asymptotic properties of the spectrum of a linear scalar DDE operator with large delay. Using these properties, we classify possible types of instabilities. Finally, for the case of vanishing quadratic nonlinearities, we present the derivation and an error estimate for the amplitude equations, which proves closeness of a solution of the amplitude equation and the corresponding solution of the original DDE with large delay. Even though the mathematical justification of amplitude equations has been established for certain classes of partial differential equations [29,16,26,24,22] or nonlinear oscillator chains [5,8], similar results for systems with time delay are still challenging because of the strong differences in the corresponding analysis for the associated linear evolution semigroups. In particular, a proof of the validity of the amplitude equations was not available previously.
We consider the following DDE with y (t) ∈ R and a smooth function g : R 2 → R, which vanishes at zero, i.e., g (0, 0) = 0. The delay τ > 0 is assumed to be large. For convenience, we introduce the small parameter ε = 1/τ . Our main goal is to describe properties of system (1) for ε → 0 in the vicinity of the equilibrium y = 0. To this end, we describe in Sec. 2 the spectral properties of the linearized system and determine possible types of destabilizing bifurcations. In Sec. 3, we show that, sufficiently close to the bifurcation point, the solutions can be described by an amplitude equation, i.e., by solutions of a partial differential equation (PDE) of the form with some periodicity conditions and real coefficients, which will be specified later [see Theorem 3.1]. To obtain this result, we assume that the function g has no quadratic nonlinearities (i.e. D 2 g (0, 0) = 0) but a nontrivial cubic term (i.e. D 3 g (0, 0) = 0). Sec. 4 contains several Lemmas concerning exponential estimates for linear and weakly non-linear DDEs with large delay and the proof of Theorem 3.1. We present an example and discuss the results and possible directions of further research in in Sec. 5.
2. Asymptotic properties of the spectrum. In this section, we consider the spectral properties of the linear DDE This equation corresponds to the linearization of (1) in y = 0 with a = ∂ 1 g (0, 0) and b = ∂ 2 g (0, 0). If not stated otherwise, we assume that b = 0 and a = 0. The spectrum of systems of linear DDEs with large delay have been studied in [18]. In the present section we focus on the scalar case y ∈ R and treat it in more detail.
The characteristic equation of (2) reads Let us discuss the asymptotic behaviors of infinitely many roots {λ j (ε) , j ∈ Z} of (3) with respect to the small parameter ε. In [18] it is proven that, for all j, one of the following two cases occurs: Definition 2.1. The set of roots that satisfy condition (a) will be called pseudocontinuous spectrum (PCS). The roots that have asymptotics (b) will be called strongly unstable.
(a) The pseudo-continuous spectrum. The PCS satisfies Re [λ/ε] → c = const. In order to find such solutions, we use the ansatz Substituting (4) into (3) and keeping the terms of order O(ε 0 ), we obtain the equation − iω + a + be −µ e −iω/ε = 0. (5) Comparing moduli we see that µ must satisfy Given the function (6), the curve contains all solutions of the equation (5), i.e., it approximates the PCS provided the real parts are rescaled by ε (compare (4) and (7)). The positions of the solutions on the curve (7) are determined by an additional equation for ω, which is obtained by insertion of (6) into (5) arg Clearly, equation (8) has a countable number of solutions ω k whose distances are proportional to ε. Consequently, the points µ(ω k ) + iω k cover the curve (7) densely as ε → 0.
Definition 2.2. The curve (7) will be called the asymptotic continuous spectrum (ACS), since the PCS approaches it asymptotically (see Lemma 2.3). Note that the PCS is a discrete set while the ACS is a continuous curve.
(b) The strongly unstable spectrum. The strongly unstable spectrum satisfies the asymptotic relation Re [λ/ε] → ∞ as ε → 0. In this case be −λ/ε → 0 and necessarily λ → a, which is possible only if a > 0. Thus, the strongly unstable spectrum exists in the case a > 0 and satisfies λ SU → a, a > 0.
Using the implicit function theorem, it can be shown that the strongly unstable spectrum contains a single eigenvalue for a > 0 and is empty for a < 0 [18]. The presence of a strongly unstable spectrum indicates strong instability in the sense that almost all solutions escape a vicinity of the origin after a time ∼ 1/a, which is much less than the delay time τ = 1/ε. On the other hand, the unstable PCS may introduce a weak instability, since Re λ P CS ∼ ε, and the time that solutions are staying in the vicinity of the origin is of the order of the delay τ .
In the following, we assume that a < 0, since otherwise the stationary state is strongly unstable. In this case, the destabilization of the equilibrium can only be mediated by the PCS. The following lemma summarizes the main properties of the PCS. Lemma 2.3. Let b = 0 and a = 0. Then, the PCS approaches the ACS (7) as ε → 0. More specifically, for any Ω > 0 there exist ε 0 > 0 and M > 0 such that for all ε, 0 < ε < ε 0 , all roots λ of (3) which lie in the set Q (M, Ω, ε) = {z ∈ C : |Re z| ≤ M ε, |Im z| ≤ Ω} have the form where k ∈ Z and h : [−Ω, Ω] × [0, ε 0 ] → C is an analytic function, with µ(ω) given by (6) and ψ (ω) := arg iω−a b ∈ (−π, π]. Proof. We only consider the case ab < 0, the case ab > 0 can be treated analogously. First observe that an arbitrary root λ ∈ Q (M, Ω, ε) of (3) can be written as with Inserting (12) into (3) we find that g has to solve for ω = ω k . The equation (13) locally defines an implicit function g = G (ω, ε) since ∂ g M (ω, ε, g) = ε + be −g = 0 for sufficiently small ε. Because M is analytic, so is G (ω, ε). Now we show that there exists an ε 0 > 0 such that (13) exhibits exactly one solution g ∈ A (M ) for |ω| ≤ Ω and ε ≤ ε 0 . This will guarantee that there are no other solutions except those which are extensions of ε → G (ω, ε) for ε ∈ [0, ε 0 ]. Existence of a unique solution g follows from the contraction mapping principle applied to the function whose fixed points correspond to solutions of (13). Let us show that, for sufficiently large M > 0 and sufficiently small ε 0 > 0, F maps A (M ) into itself and is a contraction for arbitrary −Ω ≤ ω ≤ Ω and ε ≤ ε 0 . First, choose an arbitrary Since the real part of F (g) depends continuously on ε uniformly in |ω| ≤ Ω, its modulus is smaller than M for sufficiently small ε. Furthermore, observe that the modulus of the imaginary part of F (g) is strictly smaller than π/2 for ε < |a|/M , since then ((iω − a + εg)/b) > 0 (here we used the assumption ab < 0). Therefore, F maps A (M ) into itself for ε ≤ ε 0 and sufficiently small ε 0 > 0. Moreover, F is a contraction, since |F (g)| = O (ε 0 ) uniformly for all |ω| ≤ Ω, g ∈ A (M ) , and ε ≤ ε 0 .
Let us now consider again ω = ω k = 2πkε. This case corresponds to a characteristic root λ = iω k + εG (ω k , ε). As a solution to (13) the value G (ω k , 0) fulfills which implies where h : [−Ω, Ω]×[0, ε 0 ] is analytic. The higher terms of (14) are readily computed as Remark 1. It is easy to check that all solutions of the characteristic equation with |Im λ| > Ω satisfy This means that for large enough Ω, they all have negative real parts with sufficiently large magnitude. Thus, the critical spectrum is indeed described by Lemma 2.3. The real part µ(ω) of the ACS (7) is a smooth unimodal function that has maximum at µ (0) = − ln |a/b| [see Fig. 1]. Lemma 2.3 states that in rescaled coordinates (Reλ/ε,Imλ), the critical spectrum of the DDE approaches the curve of the ACS, which is independent of ε. As a result, the stability of the equilibrium is determined by the ACS and is independent of ε if it is sufficiently small.
The fact that the spectrum of the DDE with large delay asymptotically approaches a continuous spectrum, allows us to classify possible destabilization scenarios similarly to the case of spatially extended systems [31]. In the case of scalar DDEs a possible scenario is shown in the figure 1. For |b| < |a|, the PCS is stable and, if additionally a < 0, there is no strongly unstable spectrum. As a result the equilibrium is asymptotically stable for sufficiently small ε. The destabilization occurs at |a| = |b| and a < 0, where the PCS crosses the imaginary axis. After the crossing, for |b| > |a|, the PCS is unstable and there are always unstable eigenvalues provided ε is sufficiently small. The number of such unstable eigenvalues grows linearly with τ → ∞ and cover the whole curve of the ACS. The figure 1 shows also numerically computed eigenvalues for τ = 70, for comparison. As we have proved before, the rescaled real part of the ACS, µ(ω), has the form of a unimodal function with maximum at ω = 0. Hence, the leading eigenvalues (those with the maximal real parts) correspond to small ω values. Note that the strongly unstable spectrum is absent if a < 0. The following lemma provides the third-order expansion for the leading eigenvalues.
Lemma 2.4. Assume a < 0 and b = 0. Then the leading eigenvalues of the scalar DDE (2) satisfy the following estimates. For the case ab < 0: (15) and for the case ab > 0: Here k = 0, ±1, ±2, . . . and q = − ln a b = µ(0) is the maximal value of the real part of the ACS.
Lemma 2.4 implies that there are two cases, for which the location of leading eigenvalues is qualitatively different. These cases differ by the sign of the product ab. In the case when ab < 0, there is a real eigenvalue and the imaginary parts of Figure 2. Two possible types of bifurcations in the scalar DDEs. The case (a) a = −b corresponds to a transcritical or pitchfork bifurcation of the reduced map as well as for (2). In this case, there is a leading real eigenvalue. The case (b) a = b corresponds to a flip bifurcation of the reduced discrete map [eq. (19)], resp. Hopf bifurcation for (2). The leading pair of eigenvalues has imaginary parts ∼ ±iπε.' the other leading eigenvalues are given asymptotically as 2πkε. In the second case, when ab > 0, there is no real eigenvalue and the imaginary parts of the leading eigenvalues are given asymptotically as iπ(1 + 2k)ε. Both cases are illustrated in Fig. 2(a) and (b). The two above mentioned cases correspond to qualitatively different dynamics in the vicinity of equilibrium. The reasons for such difference can be understood by inspecting the properties of the corresponding reduced map. Observe that equation (1) is equivalent to the following singularly perturbed equation after the transformationst = t/τ andȳ (t) = y (tτ ). The reduced map can be obtained by substituting formally ε = 0 into (18) g (ȳ (t) ,ȳ (t − 1)) = 0.
Introducing the discrete variable z n =ȳ (t − 1), we obtain the one-dimensional discrete map z n → z n+1 determined by In general, the properties of the map (19) have much in common with the properties of DDE [20,14,10]. For example, the asymptotic stability of an equilibrium for the map (19) implies the asymptotic stability of the corresponding equilibrium for the DDE with sufficiently large delay. Indeed, denoting a = ∂ 1 g (0, 0) and b = ∂ 2 g (0, 0), the linearization of (19) around the origin is which is asymptotically stable if |b| < |a|. The bifurcation of the map occurs when as well as the bifurcation of the ACS (7) for the DDE (1), since µ(0) = 0 when (20) is fulfilled. Returning now to the two qualitatively different cases for the spectrum configuration shown in Fig. 2, we notice that they correspond to different bifurcations of the reduced map occurring at the moment |a| = |b|: 1. When a = −b, the map (19) has the multiplier 1, which may correspond to a transcritical or pitchfork bifurcation. The corresponding bifurcation of the DDE (1) is mediated by the critical eigenvalues with imaginary parts i2πkε, in particular by the leading eigenvalue λ crit (0) = 0, cf. Fig. 2(b). 2. When a = b, the map (19) undergoes a flip bifurcation. In this case, the critical eigenvalues of DDE have imaginary parts iπ(1 + 2k)ε, cf. Fig. 2(a). In particular, the most unstable pair of eigenvalues produces a Hopf bifurcation with the period 2π/πε = 2τ . The appearance of square waves with the period 2τ in this situation has been studied in [19,20,2,1].
In the both above mentioned cases, the primary destabilization bifurcation is followed by subharmonic Hopf bifurcations with frequencies ω H = 2πkε or ω H = πε (1 + 2k), respectively, as a b increases.
3. Amplitude equations. In this section we derive a PDE-approximation for the DDE In the vicinity of the stationary state y = 0, solutions of (21) can be approximated by solutions u = u(θ, x) of a corresponding parabolic PDE of the Ginzburg-Landau type, namely with appropriate periodicity conditions in x ∈ R. The equation (22) is called amplitude equation. An important feature of (22) is that it is no longer singularly perturbed. Moreover, it provides a unified description of the dynamics at the destabilization for dynamical systems that are close to a certain instability [23]. For example, Ginzburg-Landau type equations are already well established normal forms for analyzing destabilization of reaction-diffusion systems or oscillator chains [25,21,23,16,9].
In what follows, we assume that g ∈ C 4 (R 3 ). We adopt the notation g 1 := ∂ 1 g (0, 0, 0) and g 2 := ∂ 2 g (0, 0, 0) for the derivatives with respect to the instantaneous and the delayed state variable respectively. As follows from the linear theory for large τ given in Sec. 2, the system admits two types of bifurcations for the ground state y = 0, which are governed by the pseudo-continuous spectrum (PCS) and occur when the conditions g 1 = g 2 in the flip case or g 1 = −g 2 in the pitchfork/transcritical case are satisfied.
Based on our analysis of the linear equation in Sec.2, it is not difficult to check that if the system is ε 2 -close to the bifurcation, then the number of unstable eigenvalues stays asymptotically bounded with ε → 0. Indeed, the arclength of the unstable part of the ACS (7) is proportional to ε if µ(0) ∼ ε 2 . Since two neighboring eigenvalues of the PCS have a mutual distance ∼ ε while deviating from the ACS maximally to order O(ε 2 ) the number of unstable eigenvalues is bounded [see Fig. 1 and Lemma 2.4]. This is the reason why we assume the perturbation parameter in the right hand side of (21) to be proportional to ε 2 . If the perturbation were of order O (ε), the number of unstable eigenvalues would grow with ε → 0 and we expect some singularly perturbed problem in the limit. For convenience, let us rewrite Eq. (21) by explicitly separating linear parts as where κ = a b = −sgn (b). This means, κ = 1 stands for the flip case and κ = −1 for the pitchfork/transcritical case. The function f ∈ C 4 (R 3 ) represents the nonlinearities of g, i.e., f 1 = f 2 = 0. Since we consider the case of cubic nonlinearities, we have f 11 = f 12 = f 22 = 0. Here again, the corresponding derivatives are denoted by indices of the function f , e.g. f 12 = ∂ 2 ∂ 1 f (0, 0, 0) and so on.
be a solution of system (22) with initial data A 0 that satisfies A 0 (x) = −κA 0 (x + 1) , Let y A be the formal approximation Then, for each d 0 > 0, there exist C > 0 and ε 0 > 0 such that the following statement holds for all ε ∈ [0, ε 0 ]. If y = y (t; ϕ 0 ) is a solution of the DDE (23) with an initial value y (s; ϕ) = ϕ 0 (s) , s ∈ [0, τ ], which satisfies then y (t, ϕ 0 ) is defined for t ∈ I ε = 0, T 0 /ε 3 and The proof of the theorem will be given in the following section.

4.
Estimates for evolution operators and proof of Theorem 3.1. In this section we derive estimates for evolution operators of linear, nonautonomous DDEs (Lemmas 4.1 and 4.2) and of a weakly, nonlinearly perturbed DDE (Lemma 4.3) that enables us to prove Theorem 3.1 at the end of this section. Although the Lemmas 4.1 and 4.2 are not directly used in the proof of the theorem, they provide properties of the evolution operators that are useful in various physical applications, such as stability of synchronization in delay-coupled systems [11]. Moreover, they explain the origin of the additional smallness factor ε in the exponential growth rates that plays a crucial role for the proof of the amplitude equations estimate in Theorem 3.1.
We remark that Lemma 4.1 is also valid for systems of DDE's. The proof is literally the same. The following Lemma gives an exponential estimate for the DDE with a nonautonomous perturbation Lemma 4.2. Let a < 0 and |a| = |b|. Assume also that a 1 (t) and b 1 (t) are continuous and uniformly bounded, i.e., there exists a constant M > 0 such that |a 1 (t)| ≤ M and |b 1 (t)| ≤ M for all 0 ≤ t ≤ T . Then there exists an ε 0 > 0 such that, for 0 < 1/τ = ε ≤ ε 0 , the solution of the initial value problem for (32) with an initial condition ϕ(s), s ∈ [−τ, 0], satisfies the following estimate |b|−ε n 0 M . The proof of Lemma 4.2 will be given together with the proof of Lemma 4.3.

Remark 2.
If the nonautonomous perturbations in (32) are absent and a = −b, then all solutions are bounded. This follows from the fact that d dt y 2 (t) ≤ 0 at timepoints t with |y (t)| ≥ |y (s)| for all earlier times s ≤ t. For the case a = b < 0, the zero solution is asymptotically stable, since all eigenvalues have negative real parts.
Consider the following, weakly and nonlinearly perturbed equation: where τ = 1/ε. The following result gives useful a priori bounds, which will enable us to prove Theorem 3.1.
Step (ii). Next, we prove the existence of C 0 and ε 0 > 0 such that the inequality H(C 0 , ε 0 ) < C 0 holds. Substituting c 1 and c 2 in (35) we obtain
Equipped with the previous result, we can prove Theorem 3.1: Proof. For the sake of brevity we write y (t) = y (t; ϕ) and y A (t) = y A (t, ε). Consider the rescaled error Differentiating (36), taking into account that y (t) satisfies (23) and y A (t) satisfies (29), we obtain the equation for the error Since the nonlinearity f contains terms at least cubic in y(t) and y(t − τ ), we have Defining the functions a 2 , b 2 and N via we see that a 2 (t) and b 2 (t) are bounded for t ∈ I ε uniformly in ε. Moreover, the nonlinear function N satisfies the assumption (ii) of Lemma 4.3. The equation for the error becomes with initial condition Applying Lemma 4.3 for the case n = 2, g 1 = −r, g 2 = N , we obtain that for each T 0 > 0 and d 0 > 0 that there exist ε 0 , C > 0 such that for all ε ∈ [0, ε 0 ] the following statement holds: Thus, y (t) = y A (t) + ε 2 R (t) exists for t ∈ I ε and |y (t) − y A (t)| = ε 2 |R (t)| ≤ Cε 2 , for all t ∈ I ε .
This proves the theorem.

The corresponding amplitude equation is
with the antiperiodicity condition A 0 (x) = −A 0 (x + 1) .
For the following initial condition A 0 (x) = − cos (πx) , the solution of (38)-(39) is shown in Fig 3(b) as a gray-scaled plot. The corresponding solution for the DDE (37) in the coordinates x = εt + 1 a ε 2 t, θ = ε 3 t is shown in Fig 3(a). We note that both solutions show the same behavior as predicted by the theory. For more numerical examples, we refer to the papers [6,30].  Figure 3. Chart (a) shows evolution for DDE (37) in the rescaled coordinates x = ε + ε 2 /a t, θ = ε 3 t. Chart (b) shows the evolution of the amplitude equation (38)-(39). The magnitude of the solution is indicated by the different shades of gray.
Let us mention some possible generalization of our results. Suppose that the DDE has a small and slow nonautonomous perturbation of the form ε 3 h(εt + ε 2 t/a, ε 3 t), i.e., y (t) = g(y(t), y(t − τ )) + ε 3 h(εt + ε 2 t/a, ε 3 t), with h (x, θ) = −κh (x + 1, θ). Then the same procedure is applicable for (40) and, as a result, the nonautonomous and nonhomogeneous partial differential equation appears as the amplitude equation. If h is independent of θ, then the dynamics of (41) is that of a traveling gradient flow. Indeed, definingũ (x, θ) = u (θ, x + βθ), we find ∂ũ ∂θ = α ∂ 2ũ ∂x 2 + pũ + γũ 3 + h (x) , which is the L 2 -gradient flow for the energy functional Hence, for γ < 0 the global attractor for (42) contains spatially (anti-)periodic traveling waves u (x, θ) = U (x + βθ) and heteroclinic connections between them, cf. [4]. Of course, considering a forcing h (x, θ) with nontrivial θ-dependence opens up the possibility of a much richer dynamical behavior and to implement suitable control mechanisms to stabilize desirable patterns. The generalization of the presented results to the case of systems of DDEs is still open. In particular, Lemma 4.3 depends essentially on the scalar nature of the problem. Systems possess some interesting features that are not present in scalar DDEs, for example, Turing destabilization [30]. Another interesting extension concerns DDEs with multiple delays. Some preliminary results on the spectrum of such systems are obtained in [13].