{'fraction_non_alphanumeric': 0.05366684003971819, 'fraction_numerical': 0.02212870584897631, 'mean_word_length': 4.124787981584686, 'pattern_counts': {'":': 0, '<': 6, '<?xml version=': 0, '>': 6, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 6, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Absolute stability and absolute hyperbolicity in systems with discrete time-delays

An equilibrium of a delay differential equation (DDE) is absolutely stable, if it is locally asymptotically stable for all delays. We present criteria for absolute stability of DDEs with discrete time-delays. In the case of a single delay, the absolute stability is shown to be equivalent to asymptotic stability for sufficiently large delays. Similarly, for multiple delays, the absolute stability is equivalent to asymptotic stability for hierarchically large delays. Additionally, we give necessary and sufficient conditions for a linear DDE to be hyperbolic for all delays. The latter conditions are crucial for determining whether a system can have stabilizing or destabilizing bifurcations by varying time delays.

It is a basic fact that the equilibria of a DDE do not change under variations of the delay time. In general, their stability properties may change under such variations. Indeed, in many cases increasing delay is known to induce additional instabilities. However, there is also the case, called absolute stability, where the stability of an equilibrium remains unchanged for all possible non-negative delay times. Considering linear DDEs with discrete delays where I is the identity matrix. We present a new criterion for the absolute stability of Eq. (1.1), i.e., a necessary and sufficient condition on the matrices A k such that all roots λ of the quasipolynomial (1.2) have negative real parts for arbitrary non-negative delays τ k . Our Theorems 2 and 3 generalize known results [12,32,35,[41][42][43][44][45][46][47][48][49][50][51][52][53][54][55] and have three main advantages: • simple to check (conditions on compact sets); • they give necessary and sufficient conditions; • geometric interpretation using certain limiting spectral sets.
Moreover, the absolute stability appears to be equivalent to the asymptotic stability for hierarchically large delays 1 ≪ τ 1 ≪ · · · ≪ τ m , which, for the case m = 1, is the asymptotic stability for a single large delay.
Additionally, we provide a criterion for system (1.1) to be hyperbolic for all time delays, i.e., the condition for the absence of the roots of the characteristic polynomial λ with zero real parts. In particular, this means that under the obtained conditions one cannot change the stability of the equilibrium in (1.1); it remains either asymptotically stable or unstable for all delays.
Let us first give a brief overview of the known results on the absolute stability. One of the first conditions is due to Pontryagin [32]. This criterium involves the verification of certain properties of the characteristic equation evaluated along the whole imaginary axis ∆(iy) as well as some additional implicit conditions. The Potryagin conditions have been used in many applications [35,56].
In [44], Brauer gave sufficient conditions for the absolute stability of the characteristic equation which is a polynomial of the first order in e −λτ . Comparing it with (1.2), this corresponds to a single delay and a rank one matrix A 1 . On the other hand, equation (1.3) can appear in some cases with distributed delays, which we do not consider here. The Brauer's conditions have been applied in, e.g. [12,43]. Cooke and van der Driessche also considered Eq. (1.3) as well as a generalization to multiple delays in [43]; they provided sufficient conditions for the absolute stability. Chin Yuang-Shun [41] gave criterion for the case of one delay. This criterion requires Q(iy) = 0 for all y ∈ R and all τ 1 ≥ 0, which includes the time-delay as a parameter. Instead, a practically employable criterion for absolute stability in the case of a single delay should be delay-independent and given by an at most one-parameter condition. In section 4, we provide such a criterion and explain its geometrical meaning. The Pontryagin type conditions, in contrast, are hard to check, and in the case of multiple delays they are very laborious.
Several other sufficient conditions are given in [42,51], for the case of two delays in [45], neutral equations in [57], and some special types of equations in [46][47][48][49][50]55]. In [54,58], a strong delay-independent stability is used to give sufficient conditions for the absolute stability, which is called there weak delay-independent stability. Applications to control problems are considered in [52,53].

General criterion for absolute stability
First, we introduce some notation and definitions. Our notation is that of Ref. [37]. Given a bounded linear operator A, its spectrum is denoted by σ(A) and its spectral radius is denoted by ρ(A).
Given a finite family of operators A k : C n → C n for k = {0, 1, . . . , m} of Eq. (1.1), we consider feedback phases Φ = (ϕ 1 , . . . , ϕ m ) ∈ T m and Our key object is the phase dependent spectrum σ(S(Φ)) ⊂ C, which will contain key information about the stability of the system. As follows from the general DDE theory [37], in case of absolute stability, all solutions of the initial value problem for DDE (1.1) are exponentially asymptotically stable, i.e. x(t; ϕ) → 0 exponentially fast with t → ∞ for any initial function The following theorem provides a general criterion for the absolute stability in the case of multiple discrete delays. Let us discuss the meaning of the above conditions. Condition (A1.1) [instantaneous stability] means that the corresponding instantaneous ODE systemẋ = A 0 x must be exponentially stable. Condition (A1.2) [nonsingular S(0)] is equivalent to the requirement that the characteristic quasipolynomial (1.2) does not possess a zero root. We will later show that, taking into account (A1.1) [instantaneous stability] and (A1.3) [no resonance], the condition (A1.2) can be replaced by the requirement that S(0) is Hurwitz. Hence, (A1.2) [nonsingular S(0)] contributes to the exponential stability of the ODE systemẋ = S(0)x obtained from (1.1) for zero delays.
Condition (A1.3) [no resonance] means that the spectrum of the m-parametric set of matrices S(Φ) cannot cross the imaginary axis apart from the origin. We will show later that, taking into account (A1.1) [instantaneous stability], the condition (A1.3) is equivalent of having S(Φ) "almost Hurwitz", i.e., ℜσ (S(Φ)) < 0 except that the possible zero eigenvalue. We will also show that σ (S(Φ)) can be in a certain sense related to the asymptotic spectrum in delay systems with hierarchically long delays. Moreover, purely imaginary eigenvalues iω of σ(S(Φ)), which we call resonances, appear as characteristic roots of (1.2) at an infinite sequence of resonant delay times.
Moreover, purely imaginary values iω ∈ σ(S(Φ)) correspond to certain "resonances" and the appearance of critical characteristic roots for countable number of delays.
The conditions for absolute stability can be equivalently formulated as follows. The proof will be given in Sec. 6. Combining the asymptotic spectral theory from [59,60] for the case of one delay with Theorem 2, we can show that the absolute stability is determined by the stability at large delays. In particular, we obtain the following Corollary 4. System (1.1) with one delay is absolutely stable if and only if it is asymptotically exponentially stable for all sufficiently large delays, i.e. there exists τ L such that ℜ(λ) < 0 for all characteristic roots and all τ > τ L .
In fact, Corollary 4 is a consequence of the following more general statement for the case of multiple delays.
The stability for one large delay has a useful interpretation from the point of view of a singular map. By rescaling the time t = T /ε with ε = 1/τ , we obtain By neglecting formally the left-hand side, we obtain the singular map , This hints that the stability of the system can be obtained at a formal level by a discrete dynamical system. There are many publications devoted to relations between the DDE (2.2) and the singular map (2.3), see [37,[61][62][63][64][65][66][67][68][69][70]. In fact, in order to obtain equivalent stability conditions, one should consider an extended singular map We will provide a discussion about this form in Sec. 4.5. Using this dynamical system we can conclude absolute stability as shown in the following • A 0 is Hurwitz; • the discrete dynamical system (2.4) is asymptotically exponentially stable for ω = 0; • for ω = 0, the discrete dynamical system (2.4) possesses multipliers µ with |µ| ≤ 1 and µ = 1, i.e., it is either asymptotically exponentially stable or neutral with µ = e iϕ , ϕ = 2πk.
Organisation of the manuscript. We provide examples of the application of Theorem 2 to scalar DDE with multiple delays in Sec. 3 and give a geometric interpretation of the obtained criterion for one delay in a system of DDE's in Sec. 4 emphasising the role of asymptotic spectrum for large delays. We consider the case of multiple hierarchical delays in Sec. 5. We offer proofs of Theorems 2 and 3 in Sec 6. Finally, we provide conclusions and some open problems in Sec. 7.

Scalar DDEs
In the case of scalar DDEs the absolute stability conditions can be significantly simplified.  Proof. We verify that the conditions of Theorem 3 are equivalent to (3.2)-(3.3). In order to simplify the condition (A2.2) [almost Hurwitz S(Φ)] for the scalar case, we observe that the maximum of the real part of a 0 + m k=1 a k e iϕ k is achieved at ϕ k = − arg a k , k = 1, . . . , m, and it equals For ℑ(a 0 ) = 0, this isolated maximum has nonzero imaginary part and must be negative accordingly to (A2.2). Therefore, we obtain (3.2) with strict inequality as an equivalent to (A2.2).
Finally, we observe that Numerical examples with scalar DDEs will be presented in Secs. 4.4 and 5.3.

The case of one delay, geometric interpretation
Since the case of one discrete delay appears most often in applications, we discuss it here in more detail. In particular, we give a geometric interpretation using the asymptotic spectrum for large delay.
4.1. Auxiliary results. The following technical Lemmas will be needed.

4.2.
Absolute stability conditions in terms of extended singular maps (2.4). The following lemma shows that condition (A2.2) [almost Hurwitz S(Φ)] can be recast in terms of a spectral radius criterion.
Lemma 10. Assume A 0 is Hurwitz. Then the following statements hold: Proof. (I) follows from the equivalent expressions Then (I) implies that the matrix A 0 + A 1 e iϕ possesses no purely imaginary eigenvalues. Since A 0 is Hurwitz, Lemma 9 implies that A 0 + A 1 e iϕ is also Hurwitz.
To prove the converse, assume A 0 + A 1 e iϕ is Hurwitz and let us show that the condition ρ (iω − A 0 ) −1 A 1 < 1 holds for all ω. It clearly holds for sufficiently large ω. If, it fails for some ω, then, there must exist (III) This statement follows from the continuity of eigenvalues as functions of ω and statements (I) and (II).
With Lemma 10 we obtain that for systems with one delay the criteria for absolute stability from Theorems 2 and 3 can be equivalently reformulated as follows.
Lemma 11. System (1.1) with a single delay is absolutely stable if and only if the following conditions are satisfied: Lemma 11 implies immediately the statement of Corollary 6.

4.3.
Absolute stability and asymptotic spectrum. In view of Corollary 4, the stability for large delays and the absolute stability are equivalent. In this section, we discuss this relation in more details. The spectrum of DDEs can be well approximated in the limit τ → ∞. More specifically, the spectrum of DDEs with one large discrete delay can be generically divided into two parts [59,60,69]: (i) The strongly unstable part S su , which is approximated by the unstable spectrum of A 0 , i.e. σ(A 0 ) with ℜσ (A 0 ) > 0, and (ii) the pseudo-continuous spectrum S pc , which is approximated by the curves in the complex plane. The functions γ j (ω) are given by where Y j (ω), j = 1, rankA 1 , are the roots of the spectral polynomial In particular, the functions γ j (ω) are continuous except for the isolated points ω s where lim ω→ωs γ j (ω) = ±∞. The points ω s where lim ω→ωs γ j (ω) = +∞ are determined by the condition iω s ∈ σ(A 0 ). Clearly, if such a point exists, it leads to an instability for large delays.
We are now ready to provide an interpretation of the conditions of Lemma 11 in terms of the asymptotic spectrum. Condition (A), i.e. ℜ (σ(A 0 )) < 0, is also the same as (A1.1) [instantaneous stability] in Theorems 2 and 3. It guarantees that, first, the strongly unstable spectrum is absent, and, second, the asymptotic continuous spectrum possess no singularities, see which means that all roots Y j (ω), j = 1, rankA 1 , of the spectral polynomial (4.9) satisfy |Y j (ω)| > 1 for ω = 0, implying γ j (ω) < 0 for all ω = 0.
It is easy to see that the conditions of the Corollary 13 imply the stability of the asymptotic spectrum. The asymptotic continuous spectrum is allowed to touch the imaginary axis at the origin, and this is the case when a 0 + |a 1 | = 0, however, the additional condition a 0 + a 2 = 0 forbids the appearance of the trivial eigenvalue.

Multiple delays
5.1. Equivalence of absolute stability and asymptotic stability for hierarchically large delays. In this section we show that the criterium for the absolute stability for arbitrary positive delays is equivalent to the stability for hierarchically large time-delays, i.e., the asymptotic stability for 1 ≪ τ 1 ≪ · · · ≪ τ m . Such an equivalence is a generalization of Corollary 4 for one large delay. Interestingly, due to the symmetry of the conditions for the absolute stability with respect to the numbering of the delays, the order τ k in this case does not play any role.
For the proof, we will need several Lemmas.
Proof. The proof follows from the consecutive application Lemmas 9 and 14 to the matrices A k e iφ k , r = 0, ..., m, where M r−1 , r = 1, .., m, plays the role of A and A r plays the role of B. Note that in this way Case I of Lemma 9 transfers the Hurwitz property to the next level r, while Case II of Lemma 9 provides a resonance, which is then by Lemma 14 transfers to the next level. Case III of Lemma 9 detects a zero eigenvalue, which is transferred by Case II of Lemma 14. By considering all possible logical chains, wee see that I-III are the only possibilities that can be realized. I: Case I of Lemma 9 for all r = 1, . . . , m. In this case, all matrices are Hurwitz for all Φ. II: Case I of Lemma 9, followed by Case II of Lemma 9, possibly followed by Case I of Lemma 14. Here, we have iω ∈ σ (M r ) for some r ≤ m, ω = 0. Then the sequential application of Lemma 14 m − r times leads to iω ∈ σ(S(Φ)) withω = 0 and someΦ. II: Case I of Lemma 9, followed by Case III of Lemma 9, followed by Case I of Lemma 14. Here, the matrix M r contains zero eigenvalue for some Φ and otherwise it is Hurwitz for all other Φ. At some further application of Lemma 14 on some level r 1 > r, there appears a resonance ω = 0 such that iω ∈ σ A 0 + r1 k=1 A k e iφ k , r < r 1 ≤ m. Further application of Lemma 14 m − r 1 times leads to the statement II of this Lemma. III: Case I of Lemma 9, followed by Case III of Lemma 9, followed by Case II of Lemma 14. Similarly to the previous case, some matrix M r contains zero eigenvalue and otherwise it is Hurwitz for all other Φ. At some further applications of Lemma 14, only case II of Lemma 14 is realized. We must only show that T 0 = T m . Indeed, assuming opposite, we have 0 ∈ S(Φ) for all Φ, which implies 0 ∈ A 0 and contradicts the assumption of A 0 Hurwitz. II: Case I of Lemma 9, followed by Case III of Lemma 9, followed by Case II of Lemma 14, followed by Case I of Lemma 14. This logical chain is similar to the previous one, with only difference that the case I of Lemma 14 is realized at some later iteration.
Proof. The proof follows from the sequential application of Lemma 14 in a similar way as above.
Proof. The fact that Eq. (5.2) holds for time-delays (5.3) can be checked by substitution. Let us show that time delays can be chosen to be hierarchical, i.e., satisfy the condition (2.1) with arbitrary small ε > 0. We denote which is a small parameter for sufficiently large n 1 . We assume, in particular, that n 1 ≫ ω 0 . Such a definition of ε implies equality (2.1) for k = 1.
We remark that Lemma (17) generalizes some of the statements shown for one delay in [71]. 1. First, we show that (A1.1) [instantaneous stability] is necessary. Assume the opposite, i.e., the condition (A1.1) of Theorem 2 does not hold. Then either 1a: Consider the case iω 0 ∈ σ (A 0 ). Then, Lemma 16 implies that one of the two cases can occur: 1aa: iω ∈ σ(S(Φ)) with someω = 0. In such a case, Lemma 17 implies that iω is a solution of the characteristic equation for hierarchically large time delays (2.1). We obtain the contradiction to the absolute stability and, hence, (A1.1) holds.

5.2.
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 τ k = ν k ε −k from [72]. This spectrum can be generically divided into m + 1 parts corresponding to different timescales: (i) The strongly unstable part S su , which is approximated by the unstable spec- (ii) The asymptotic continuous spectrum on different timescales can be described by the following sets where k = 1, . . . , m. The functions Y k,j (ω, ϕ 1 , . . . , ϕ k−1 ) are the j-th roots of the spectral polynomial where the index j numbers the roots. The sets B k,j correspond to the eigenvalues with the real parts converging to zero as ε k . For m = 1, the the sets B 1,j contain the asymptotic continuous spectrum of systems with one large delay τ 1 .
In the non-degenerate case of det A m = 0, the asymptotic spectrum has the form S su k=1,...,m−1 j=1,...,rankA k That is, for all spectral components that correspond to the convergence of real parts as O(ε k ) with k = 1, . . . , m − 1, only the unstable part is included. The stable part of the asymptotic continuous spectrum can contain only B m,j , which has the slowest convergence ε m of the real parts to zero. 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].
Taking into account different part of the asymptotic spectra, we can interpret the role of the conditions of Theorems 2 and 3 for the spectrum of systems with hierarchical time delays. Condition (A1.1) [instantaneous stability] guarantees the absence of the strongly unstable spectrum. Condition (A1.   Lemma 18. Let A ∈ C n×n be not Hurwitz and 0 ∈ σ(A). Then, for any B ∈ C n×n , there exists ϕ ∈ T such that the matrix A + Be iϕ is not Hurwitz and λ ∈ σ(A + Be iϕ ) with λ = 0 and ℜ(λ) ≥ 0.
Hence, for all positive delays, the roots cannot cross the imaginary axis and the asymptotic stability holds, i.e. the conditions (A1.2) and (A2.2) imply the absolute stability.
The following steps (iv)-(vi) prove that (A1. A k e −λ/ε = 0. Let U (λ 0 ) ⊂ C be a sufficiently small open neighborhood of λ 0 such that it does not contain other eigenvalues of A 0 , and ℜ(U (λ 0 )) > 0. Then, the holomorphic function P m (λ) converges uniformly to det [−λI + A 0 ] for ε → 0 on U (λ 0 ). According to the Hurwitz theorem, the characteristic equation (6.3) has an unstable root in λ ∈ U (λ 0 ) for all sufficiently small ε. This contradicts to the absolute stability assumption and, hence, (A1.1) is a necessary condition. If iω 0 ∈ σ (A 0 ), Lemma 16 implies that one of the two cases can occur: 1. iω ∈ σ S(Φ) with someω = 0. In such a case, Lemma 17 implies that iω is a solution of the characteristic equation for countable number of delays (5.3). We obtain the contradiction to the absolute stability and, hence, (A1.1) is necessary.

Conclusions
The obtained conditions for absolute stability determine a class of linear DDEs, which are asymptotically exponentially stable, independently on time-delays. Such class of systems can be useful for applications, where the robustness against timedelays is important. For nonlinear systems, these conditions exclude the possibility of any bifurcations at the corresponding equilibrium.
Bifurcations induced by varying time delay are also excluded in the case of absolute hyperbolicity. Linear systems that do not belong to one of these two classes have resonances, i.e. purely imaginary eigenvalues, which occur for countably many resonant delay times in each delayed argument, and are necessarily unstable for large delays. Note that such systems may or may not become stable for certain ranges of small delays. Even systems with strong instabilities for large delay may become stable for small delay, but only if they have unstable asymptotic continuous spectrum. This counter-intuitive conclusion follows from absolute hyperbolicity, which we showed for strongly unstable systems with stable asymptotic continuous spectrum.