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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
∗Department of Mathematics and Statistics, McGill University, Montreal, Canada.
Mailing address: Burnside Hall, 805 Sherbrooke Street West, Montreal, Quebec, Canada H3A 0B9
†Accepted version of manuscript published at Journal of Dynamics and Differential Equations, doi.org/10.1007/
s10884-020- 09900-0 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
the eigenvalues of some discretization. We highlight the 2017 paper by Miyajima [23] on verified error
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