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Eigenvalues and delay diﬀerential equations: periodic coeﬃcients,

impulses and rigorous numerics

Kevin E.M. Church∗

September 14, 2020†

Abstract

We develop validated numerical methods for the computation of Floquet multipliers of equilibria

and periodic solutions of delay diﬀerential equations, as well as impulsive delay diﬀerential equations.

Using our methods, one can rigorously count the number of Floquet multipliers outside a closed disc

centered at zero or the number of multipliers contained in a compact set bounded away from zero. We

consider systems with a single delay where the period is at most equal to the delay, and the latter two

are commensurate. We ﬁrst represent the monodromy operator (period map) as an operator acting

on a product of sequence spaces that represent the Chebyshev coeﬃcients of the state-space vectors.

Truncation of the number of modes yields the numerical method, and by carefully bounding the truncation

error in addition to some other technical operator norms, this leads to the method being suitable to

computer-assisted proofs of Floquet multiplier location. We demonstrate the computer-assisted proofs

on two example problems. We also test our discretization scheme in ﬂoating point arithmetic on a gamut

of randomly-generated high-dimensional examples with both periodic and constant coeﬃcients to inspect

the precision of the spectral radius estimation of the monodromy operator (i.e. stability/instability check

for periodic systems) for increasing numbers of Chebyshev modes.

Key words— impulsive delay diﬀerential equations, Floquet multipliers, Chebyshev series, rigorous

numerics, computer-assisted proofs

1 Introduction

Linearized growth and decay rates near steady states and invariant manifolds play a central role in the

analysis of dynamical systems. When these manifolds have simple descriptions such as ﬁxed points or

periodic orbits, the computation of these growth rates is equivalent to an eigenvalue problem. For delay

diﬀerential equations (or more generally, retarded functional diﬀerential equations), several authors have

proposed solutions to the “delay eigenvalue problem”. For autonomous linear equations, these include

methods based on discretization of the associated inﬁnitesimal generator [3, 4, 18, 33] and the solution

operator [15]. In the scope of equations with periodic coeﬃcients, there are several results concerning

discretization and characteristic matrices [15, 16, 30, 26, 29].

Discretization schemes can provide strong convergence properties, but these still may not be able to

provide mathematical proof concerning one or more approximate eigenvalues. For example, the spectral

accuracy of the inﬁnitesimal generator method with Chebyshev collocation [5] guarantees that each

eigenvalue of the delay diﬀerential equation (DDE) is well-approximated by some eigenvalue of the

discretized problem. Such convergence results exist also for methods based on the solution operator

[16] and they can be summarized by the statement: every eigenvalue of the DDE is the limit of some

eigenvalue of the discretized problem. However, one is often interested in a situation dual to this. That

is, one wants to know when one or more eigenvalues of the discretized problem are close (or have the

same relative location in the complex plane) as a pairing of eigenvalues of the DDE.

There has been progress recently on methods that can automatically prove (with the assistance of

a computer and interval arithmetic) results concerning the location of eigenvalues of DDEs based on

the eigenvalues of some discretization. We highlight the 2017 paper by Miyajima [23] on veriﬁed error

∗Department of Mathematics and Statistics, McGill University, Montreal, Canada.

Mailing address: Burnside Hall, 805 Sherbrooke Street West, Montreal, Quebec, Canada H3A 0B9

†Accepted date. Originally submitted April 29, 2020, revised version submitted September 11, 2020.

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K E M Church Eigenvalues and impulsive delay diﬀerential equations: rigorous numerics

bounds for particular eigenvalues, and the 2020 paper by Lessard and Mireles James [21] on validation

of generalized Morse indices. These papers both concern autonomous problems, so one motivation of the

present paper is to build on the results of Lessard and Mireles James to accomodate DDEs with periodic

coeﬃcients. Since we get a numerical discretization scheme for the monodromy operator (i.e. the period

map) for free out of our analysis, we simultaneously get an alternative to the method of Gilsinn and

Potra [16] for the computation of Floquet multipliers (the linearized stability-determining quantities) of

DDEs with periodic coeﬃcients.

In the case of autonmous DDEs, if the explicit location of an eigenvalue is required to high accuracy

then there are several ways this can be accomplished [7, 21, 23]. Of mention is that in this case, the

eigenvalues are precisely the zeroes λof the characteristic equation det(∆(λ)) = 0, where ∆(λ) is the

characteristic matrix of the DDE. The characteristic equation is transcedental in λ, but the result is still a

scalar (complex) zero-ﬁnding problem (∆(λ) is a d×dmatrix with dequal to the dimension of the DDE),

so a given eigenvalue can be veriﬁed to a provable level of accuracy using the radii polynomial method; see

Theorem 2.2 of [21] for a direct application of the method to the present situation. While characteristic

matrices for periodic DDEs can be constructed [26, 30] and the eigenvalues of these matrices are related to

the Floquet multipliers, a discretization step must still be performed. Even if the characteristic matrix is

explicitly available, such as when the delay is a multiple of the period [32], the computation of the matrix

generally requires at least computing the Cauchy matrix of a time-periodic ordinary diﬀerential equation

and computing weighted integrals involving this and the periodic coeﬃcients. We have been unable to ﬁnd

publications concerning error estimates between the approximate eigenvalues of discretized characteristic

matrices for periodic DDEs and the Floquet multipliers. As such, another goal of the present paper

is to devise a strategy to validate the location of speciﬁc Floquet multipliers of DDEs with periodic

coeﬃcients.

Some physical systems are characterized by smooth evolution in addition to brief bursts of activity.

This might be due to exogenous forcing or it might be an intrinsic property of the system. If these

bursts of activity incur relatively large distrubances to the state and do not occur too frequently, it can

be beneﬁcial to model them as discontinuities. One mathematical formalism for such a construction

is impulsive dynamical systems, which include impulsive diﬀerential equations [2, 25, 20], impulsive

evolution equations [13] and impulsive functional diﬀerential equations [8, 28]. One of the simplest classes

of impulsive dynamical system is one in which these discontinuities (hereafter called impulses) occur at

ﬁxed times. Even for more complicated classes of problems where impulses are triggered according to

mixed spatio-temporal relations, the case of ﬁxed (in time) impulses is still relevant since linear systems

of this type arise linearization at bounded or periodic trajectories [2]. When the sequence of impulses has

periodic structure and this is compatible with the continuous dynamics, the result is a periodic system.

The Floquet theory has been developed for linear periodic impulsive retarded functional diﬀerential

equations [10] and characteristic equations/matrices for special cases of periodic linear impulsive DDE

[17, 27] have been developed, but at present there is no rigorous numerical method to compute Floquet

multipliers for a general class of such equations. Our approach in this paper will simultaneously address

the approximation of Floquet multipliers for impulsive delay diﬀerential equations by way of a Chebyshev

spectral method, computer-assisted proof of Floquet multiplier location, and computer-assisted proof of

the count of the number of unstable Floquet multipliers (or generally, the number of Floquet multipliers

having absolute value greater than some prescribed value). Formally, the most broad class of systems

for which our numerical method applies is systems of the form

˙x=A(t)x(t) + B(t)x(t−q), t /∈pZ

∆x=C1x(t−) + C2x(t−q), t ∈pZ,

for (suﬃciently regular) real matrix-valued functions A(t) and B(t), real matrices C1and C2, and natural

numbers p(period) and q(delay). Necessary background on these systems appears in Section 2. More

generally, we can treat systems with commensurate period and delay by way of a time scaling.

With the previous paragraph in mind, we will state and prove all results in the context of periodic

impulsive delay diﬀerential equations. This class of equations includes delay diﬀerential equations with

periodic coeﬃcients by taking C1=C2= 0 in the displayed equation above, so our results can be applied

to those equations as well. Finally, although this is an extreme level of reduction, the results also apply

to ordinary and impulsive diﬀerential equations without delays. To give the reader a taste of the kinds

of results that can be proven using our validated numerics framework, we still state two theorems that

are consequences of results that appear in (and are proven in) Section 8.

Theorem. Let β= 0.1,ρ= 1,K= 1,d1= 0.02,d2= 0.03 in the following time-delay predator-prey

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K E M Church Eigenvalues and impulsive delay diﬀerential equations: rigorous numerics

model with impulsive harvesting

˙x=rx(t) (1 −x(t)/K)−βx(t)y(t)

˙y=ρβe−d1τx(t−τ)y(t−τ)−d2y(t), t /∈Z

∆y=−hy(t−), t ∈Z.

The equilibrium (K, 0) = (1,0) enjoys the following properties for any r > 0:

•its unstable manifold is one-dimensional if h= 0.060 and τ∈ {1,2,3};

•it is locally asymptotically stable if h= 0.075 and τ∈ {1,2,3};

•its unstable manifold is one-dimensional for all h∈[0,0.060] if τ= 1;

•with τ= 1, one of its Floquet multipliers crosses the unit circle at some h∈[0.065,0.066], and over

this entire range of hthere is exactly one eigenvalue whose absolute value is greater than 0.8.

Theorem. Consider the following two-dimensional delay equation modeled on the normal form of the

Hopf bifurcation:

˙x=βx(t)−πy(t)−x(t)(x2(t−τ) + y2(t))

˙y=πx(t) + βy(t)−y(t)(x2(t) + y2(t−τ)).

For τ∈N, there is a nontrivial (i.e. nonzero) branch of periodic solutions parameterized by β. At

parameter β=3

2and τ= 1, the unstable manifold of this periodic solution is at least two-dimensional.

1.1 Overview of the paper

The scope of this paper is fairly broad. It is therefore expected that many readers will only be interested

in the numerical method and its applications to stability. Others, however, will want to see all the

technical details concerning computer-assisted proofs. The diﬃculty is that the theory is intrinsically

tied to the numerical method and its machine implementation. To ensure a suﬃciently wide range of

researchers are able to appreciate the content, we will overview the paper here with these points in mind.

The core of the numerical method is derived in Section 3 and Section 4 with the help of the back-

ground from Section 2. The ﬁrst, Section 3, concerns an explicit representation of the monodromy

operator. In Section 4.1 we review Chebyshev expansion and convolution, and in Section 4.2 we rep-

resent the monodromy operator on a sequence space by way of correspondence with Chebyshev series.

The representation of the operator is on an inﬁnite-dimensional space, and the explicit numerical method

is obtained by projecting to a ﬁnite-dimensional subspace by mode truncation and projection. These

mechanisms are described at the beginning of Section 4.3, in Section 4.4 and in Remark 6.3.1. The last

remark involves some technical machinery, and a simpliﬁed implementation that would function only in

the case of equal period pand delay qcould be obtained by following Remark 6.1.1 from Section 6.1 in-

stead. Once these details are ironed out, implementing the method for general period and delay amounts

to simple matrix algebra and this is covered in Section 7.2, where some convergence guarantees are also

discussed. Several examples featuring the double arithmetic implementation are provided in Section 9 to

show how the implementation scales with respect to the dimension of the problem and the order of the

method.

Concerning the rigorous numerics, we prove a few necessary results concerning Floquet theory in

Section 2. Section 3 is devoted to an abstract functional-analytic representation of the monodromy

operator. In Section 4 we transition from the abstract representation to a concrete representation of the

monodromy operator in terms of Chebyshev series and establishes the basis for computer-assisted proof

of eigenvalue location. Section 5 contains several auxiliary bounds of linear maps and operators that are

needed later. Section 6 contains proofs of (computable) upper bounds for various abstract operators that

are needed for computer-assisted proofs of eigenvalue location. We discuss MATLAB implementation in

Section 7, while Section 8 is devoted to examples of computer-assisted proof.

We wrap up the paper with a conclusion in Section 10.

2 Preliminaries

In this section we will state relevant background concerning impulsive delay diﬀerential equations and

Floquet multipliers. We will state some basic assumptions that will be needed throughout the paper,

and we will prove results concerning the regularity of the eigenfunctions that will be needed later.

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K E M Church Eigenvalues and impulsive delay diﬀerential equations: rigorous numerics

2.1 Impulsive delay diﬀerential equations

In this paper, a nonlinear periodic impulsive delay diﬀerential equation will consist of of a delay diﬀerential

equation together with a discrete-time update rule:

˙x=f(t, x(t), x(t−τ)), t 6=tk(1)

∆x=g(k, x(t−), x(t−τ)), t =tk.(2)

The periodicity comes from the assumption that there exists some n > 0 and T > 0 such that tk+n=

tk+Tfor all integers k, while also g(k+n, ·,·) = g(k, ·,·) and f(t+T , ·,·) = f(t, ·,·). Equation (2) should

be interpreted as

x(tk)−x(t−

k) = g(k, x(t−

k), x(tk−τ)),(3)

where x(t−) denotes the limit from the left at time t. Systems with multiple and time-varying delays

can also be considered. More generally, impulsive retarded functional diﬀerential equations can be

considered without periodicity assumptions. We refer the reader to [1, 2, 25, 28] for background on

impulsive diﬀerential equations. In future we will refer to (1)–(2) simply as a nonlinear IDDE, and will

drop the reference to periodicity. In what follows we will assume that fis C1in its second and third

variables and continuous from the right and bounded in its ﬁrst, while gis C1in its second and third

variables.

The following deﬁnitions are adapted from [8]. A solution (unconditional on any initial data) x:R→

Rdof (1)–(2) is a function that is continuous from the right, possesses limits on the left, and satisﬁes both

equations (1) and (2), with the derivative being interpreted as a right-hand derivative. These solutions

are continuous from the right at times tkwith ﬁnite limits on the left. Any natural phase space for

(1)–(2) with a vector space structure must therefore contain functions that have many discontinuities.

To see why, observe that at each time tk, the solution in Rdwill have a discontinuity, so the solution

history θ7→ xt(θ) for xt: [−τ, 0] →Rdand deﬁned by xt(θ) = x(t+θ) will have a discontinuity at the

lagged argument θwhenever t+θ=tk. This observation leads naturally to the choice of phase space

RCR([−τ , 0],Rd) = {φ: [−τ, 0] →Rd:φis continuous from the right and has limits on the left}.

These are the right-continuous regulated functions. We will often write it simply as RCR when there is

no ambiguity. When equipped with the supremum norm ||φ|| = supθ∈[−τ,0] |φ(θ)|for |·|some norm on

Rd,RCR becomes a Banach space. One can then deﬁne a solution satisfying the initial condition xs=φ

for some (s, φ)∈R× RCR to be a function x: [s−τ, s +β)→Rdsuch that x|(s,s+β)satisﬁes (1)–(2)

and xs=φ. Under the conditions described above, such a solution is guaranteed to exist and be unique

for any (s, φ)∈R× RCR, and deﬁned on a maximal interval of existence. Following this, an IDDE

generates a (nonlinear) two-parameter semigroup on RCR through the solution map in the usual way.

It is typical for nonlinear IDDE to possess no ﬁxed points, and these will typically not be robust

under perturbations of the vector ﬁeld fand jump map g. This can be seen by observing that the

problem of ﬁnding zeroes of the map F:Rd→Rd×Rddeﬁned by F(x)=(f(x), g(x)) is generally

overdetermined. However, a generic periodic solution (of period j T for j∈N) is robust under small

T-periodic perturbations (this is a fairly direct consequence of Theorem 5.1.1 from [10], the Floquet

theory and the implicit function theorem) of fand g, so in this regard periodic solutions are the simplest

structurally stable invariant sets one can study in IDDE.

2.2 Monodromy operator and Floquet multipliers

The local stability of a periodic solution γis determined by the trivial solution y= 0 of the linearization

˙y=D2f(t, γ(t), γ (t−τ))y(t) + D3f(t, γ(t), γ(t−τ))y(t−τ), t 6=tk(4)

∆y=D2g(k, γ(t−), γ(t−τ))y(t−) + D3g(k, γ (t−), γ(t−τ))y(t−τ), t =tk.(5)

If γhas period jT for some j∈N, then (4)–(5) has period ˜

T=jT .

Let t7→ y(t, s, φ) denote the unique solution of (4)–(5) satisfying the initial condition ys(·, s, φ) = φ.

Each (eventually compact) linear operator Mt:RCR → RCR deﬁned by

Mtφ=yt+˜

T(·, t, φ) (6)

is called a monodromy operator. The spectrum of each monodromy operator is identical, so by convention

we will usually refer to M:= M0as the monodromy operator. γis locally asymptotically stable if all

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K E M Church Eigenvalues and impulsive delay diﬀerential equations: rigorous numerics

eigenvalues of Mare in the open ball B1(0) = {z∈C:|z|<1}. When some eigenvalues have unit

modulus but all others have modulus less than one, the stability of γdepends on some more algebraic

properties (eg. whether the restriction of Mto the direct sum of generalized eigenspaces associated to the

eigenvalues of unit modulus has a diagonalizable representation) and dynamical properties (i.e. the ﬂow

on its centre manifold). If any eigenvalue has modulus greater than one, γis unstable. The eigenvalues

of Mare called Floquet multipliers.

The computation of Floquet multipliers is important for several reasons. Apart from verifying stability

or instability of a periodic solution, the number of Floquet multipliers outside of the disc B1(0) together

with the dimension of their generalized eigenspaces dictates the dimension of the unstable manifold of

γ. The classiﬁcation of a bifurcation in a parameter-dependent system depends on the structure of

the centre ﬁbre bundle in additional to higher-order normal form data, all of which depends on the

Floquet multipliers on the unit circle and their associated eigenfunctions. For these reasons, we will now

dispense with the explicit dependence on γand consider the more general linear periodic impulsive delay

diﬀerential equation

˙y=A(t)y(t) + B(t)y(t−τ), t 6=tk(7)

∆y=C1(k)y(t−) + C2(k)y(t−τ), t =tk.(8)

We will refer to such a system as a linear periodic IDDE (impulsive delay diﬀerential equation). The

periodicity here means that there exists T > 0 and n > 0 such that A(t+T) = A(t), B(t+T) = B(t),

C1(k+n) = C1(k), C2(k+n) = C2(k), and tk+n=tk+T.

2.3 Eigenfunctions are densely nonsmooth

Our approach to rigorous computation of Floquet multipliers will be based in part on a suitable rep-

resentation of the eigenfunctions of the monodromy operator as inﬁnite series with good convergence

properties. Unfortunately, the problem of computing the Floquet multipliers for the fully general peri-

odic system (7)–(8) seems not very amenable to this approach, even if we assume Aand Bare analytic.

The following proposition and its associated proof should demonstrate the main problem.

Proposition 2.3.1. Let φbe an eigenfunction of the monodromy operator Massociated to the scalar

impulsive delay diﬀerential equation

˙y=αy(t−τ), t 6=k∈Z,

∆y=βy(t−), t =k∈Z,

for α6= 0,β6=−1and τ∈R\Qirrational. Let (a, b)⊂[−τ, 0]. There exists q > 0such that φ|(a,b)is

at most qtimes diﬀerentiable.

Proof. t7→ y(t; 0, φ) is a solution that can be represented in the form etλ p(t) for pperiodic with period

one and diﬀerentiable from the right. Substituting this ansatz into the IDDE and simplifying, it follows

that pis a 1-periodic solution of

˙p=−λp(t) + αe−λτ p(t−τ), t 6=k

∆p=βp(t−), t =k.

We ﬁrst prove that pis discontinuous at each integer k∈Z. Suppose not, then we must have p(k) = 0

since β6=−1. pis therefore a 1-periodic solution of the DDE

˙p=−λp(t) + αe−λτ p(t−τ)

with p(0) = 0. The phase space of the above DDE decomposes C=C([−τ, 0],R) as

C=R⊕S⊕N⊕U,

where S,Nand Uare the stable, centre and unstable subspaces respectively, while Rgenerates those

solutions that have superexponential decay: ψ∈Rif and only if

lim

t→∞ p(t, 0, ψ)ert = 0,∀r > 0.

Consequently, any periodic solution must be generated by an element of the centre subspace and is

therefore of the form

p(t) = c1eiωt +c2e−iωt

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K E M Church Eigenvalues and impulsive delay diﬀerential equations: rigorous numerics

for some ω > 0. Since pmust be 1-periodic, we get ω= 2π. The condition p(0) = 0 implies p(t) =

csin(2πt) for a constant c, with c6= 0 because φis an eigenfunction. Substituting this ansatz into the

delay diﬀerential equation, performing some algebraic simpliﬁcations and recalling that α6= 0, we can

derive the equation

eλτ (2π+λ)

α−cos(2πτ )sin(2πt) + sin(2πτ ) cos(2πt) = 0.

From the linear independence of sine and cosine, τmust be rational, which contradicts our assumption

of τbeing irrational. We conclude that pmust be discontinuous at the integers.

Next, we prove that if pis inﬁnitely many-times diﬀerentiable at some t∈R, then t−kτ /∈Zfor

all integers k≥0. We prove this by strong induction on k. We already know that pis discontinuous

on the integers, so we must have t /∈Zfor pto be diﬀerentiable at t. Suppose now that pis inﬁnitely

many-times diﬀerentiable and that t−jτ /∈Zfor j= 0,...,k for some k≥0. We have

p(k+1)(t) = −λp(k)(t) + αe−λτ p(k)(t−τ).(9)

If pis indeed inﬁnitely many-times diﬀerentiable at t, then the right-hand side must be continuous at

t. A straightforward inductive proof shows that each of p(k)and p(k)(t−τ) can be written as a ﬁnite

linear combination of the terms p(t−jτ ) for j= 0,...,k+ 1, and in particular the right-hand side of (9)

written in terms of these has a nonzero coeﬃcient on p(t−(k+ 1)τ). By the strong induction hypothesis,

t−jτ /∈Zfor j= 0,...,k. If the right-hand side of (9) is indeed continuous at t, then we must also have

t−(k+ 1)τ /∈Z. This completes the inductive proof, and it follows that t−jτ /∈Zfor all j≥0.

Finally, let (a, b)⊂[−τ , 0]. The restriction φ|(a,b)is C∞if and only if p|(a,b)is also C∞. Consider

the sequences

pn=nτ, cn= [pn]1,

where [x]1=x− bxc.cnis dense in [0,1], from which it follows that there exist integers k, j ≥0 such

that pk∈(a+j, b +j). Deﬁne c=kτ −j∈(a, b). From previous analysis, pis not inﬁnitely many-times

diﬀerentiable at csince c−kτ =−j∈Z.

When the delay and the period are non-commensurate, it should be expected that the eigenfunctions

of a linear periodic IDDE will have a lower order of smoothness in any subinterval of their domain. In

particular, the set of points where an eigenfunction fails to be inﬁnitely many times diﬀerentiable is dense

in in its domain.

2.4 Regularity of the eigenfunctions under commensurate delay and

period

With the observations of the previous section in mind, we will need to make the following additional

assumption on the structure of (7)–(8) in order to have a chance of eigenfunctions with appropriate series

representations. The following deﬁnition will be used fairly often.

Deﬁnition 2.4.1. Let Xbe a complex Banach space and let (ak, bk)be a (ﬁnite, inﬁnite or bi-inﬁnite)

sequence of nondegenerate open intervals with bk=ak+1 . A function f:I⊂R→X(possibly real-valued)

for an interval Iis piecewise-analytic with respect to (ak, bk)if:

•fis continuous from the right and locally bounded,

•for each k, there exists an open neighbourhood Ukof [ak, bk]in Csuch that f|(ak,bk)∩I=˜

fk|(ak,bk)∩I

for some analytic function ˜

fk:Uk→X.

Given such a function f, the sequence ˜

fkwill denote the analytic extensions from (ak, bk).

Assumption 0. The following conditions are satisﬁed.

A0.1 The sequence tksatisﬁes tk+1 =tk+Tand, additionally, C1(k)and C2(k)are constant.

A0.2 There exist p, q ∈Nsuch that qT −pτ = 0.

A0.3 The T-periodic functions t7→ A(t)and t7→ B(t)are piecewise-analytic with respect to the open

intervals Ik:= t0+kT

p, t0+(k+ 1)T

pfor k∈Z.

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K E M Church Eigenvalues and impulsive delay diﬀerential equations: rigorous numerics

We refer to the above as Assumption 0 because we will shortly perform a change of variables that

will make its characterization a bit nicer. Under this assumption, we make the change of variables

t−t0=τ

qs

for sa new rescaled time. Deﬁning z(s) = y(t(s)) and observing that τ

q=T

p, we get the IDDE

dz

ds =T

pAt0+T

psz(s) + T

pBt0+T

psz(s−q), s /∈pZ

∆z=C1z(s−) + C2z(s−q), s ∈pZ.

Observe that the matrix-valued functions s7→ ˜

A(s) = T

pA(t0+T

ps) and s7→ ˜

B(s) = T

pB(t0+T

ps) are

now periodic functions with period pand are analytic on the intervals (k, k + 1) for k∈Z, while the

restriction to each such interval is the itself the restriction of some analytic function. Dropping the tildes

and relabeling the variables, we can therefore consider without loss of generality IDDEs of the form

˙y=A(t)y(t) + B(t)y(t−q), t /∈pZ(10)

∆y=C1y(t−) + C2y(t−q), t ∈pZ,(11)

with p, q ∈Nand p-periodic functions Aand Bthat are piecewise-analytic with respect to the open

intervals (k , k + 1) for k∈Z.

Remark 2.4.1. If the linear system (7)–(8) is actually the linearization (4)–(5) at a periodic solution

from a nonlinear periodic IDDE such as (1)–(2), the piecewise-analytic condition of A0.3 will be satisﬁed

provided fis analytic, γis piecewise-analytic with respect to the intervals Ik, and both A0.1 and A0.2 are

satisﬁed. The piecewise-analyticity of periodic solutions γw.r.t. these intervals is a technical detail, but

it can be proven with similar arguments as those used in proving Proposition 2.4.1. For our examples,

however, we will be taking γto be a constant solution or an explicitly computable periodic solution, so

this detail is unimportant.

The following result will be of fundamental importance once we move onto the representation of

eigenfunctions. It states that the eigenfunctions of (10)–(11) with Aand Bpiecewise-analytic with

respect to (k, k + 1) are also piecewise-analytic with respect to the same intervals.

Proposition 2.4.1. Suppose the matrix-valued functions Aand Bin (10)–(11) are p-periodic and

piecewise-analytic with respect to (k, k + 1) for k∈Z. Every eigenfunction of the monodromy opera-

tor Mwith nonzero eigenvalue is piecewise-analytic with respect to the same intervals.

Proof. From the Floquet theory of IDDE [11], φis an eigenfunction of Mwith eigenvalue µ6= 0 if and

only if there exists a p-periodic function z:R→Cdand λ∈Csuch that y(t, 0, φ) = eλt z(t) is a solution

of (10)–(11), and µ=eλp . In particular, φ(θ) = eλθ z(θ). It follows that zis a complex-valued p-periodic

solution of the IDDE

˙z= (A(t)−λI)z(t) + B(t)e−λq z(t−q), t /∈pZ

∆z=C1z(t−) + C2e−λqz(t−q), t ∈pZ.

Deﬁne the translates zi(t) = z(t−i) for i= 0,1,...,q−1 and let [j]pdenote the remainder of jmodulo

p– that is, [j]p∈[0,...,p−1) is the unique integer such that j= [j]p+rp for some r∈Z. It suﬃces to

prove that the ziare piecewise-analytic with respect to (k, k + 1), and then to establish the analogous

analytic continuation. By construction, (z1,...,zq−1) is a p-periodic solution of

˙zi= (A(t−i)−λI)zi(t) + B(t−i)e−λq z[i−q]p(t), t /∈i+pZ

∆zi=C1zi(t−) + C2e−λqz[i−q]p(t), t ∈i+pZ.

Since each of t7→ A(t−i) and t7→ B(t−i) is piecewise-analytic with respect to (k, k + 1) for k∈Z, each

ziis also piecewise-analytic w.r.t these intervals since it can be identiﬁed