Table Of ContentCHARACTERISTICMATRICESFORLINEARPERIODICDELAYDIFFERENTIAL
EQUATIONS
JAN SIEBER∗ AND ROBERT SZALAI†
Abstract.Szalaietal.(SIAMJ.onSci.Comp.28(4),2006)gaveageneralconstructionforcharacteristic
matricesforsystemsoflineardelay-differentialequationswithperiodiccoefficients. First,weshowthat
matricesconstructedinthiswaycanhaveadiscretesetofpolesinthecomplexplane,whichmaypossibly
obstructtheirusewhendeterminingthestabilityofthelinearsystem.Thenwemodifyandgeneralizethe
originalconstructionsuchthatthepolesgetpushedintoasmallneighborhoodoftheoriginofthecomplex
1 plane.
1
0 AMSsubjectclassifications.34K06,34K08,34K20
2
n Keywords.delaydifferentialequations,characteristicmatrix,stabilityofperiodicorbits
a
J 1. Introduction. Lineardelaydifferentialequations(DDEs)withcoefficientspe-
1 riodicintimecomeupnaturallywhenoneanalysesnonlinearDDEs: oneofthemost
2
common and simplest invariant sets encountered in nonlinear systems are periodic
orbits[5]. Ifonewantstofindoutifaperiodicorbitisdynamicallystableorunstable,
]
S or how its stability changes as one changes system parameters one has to linearize
D the nonlinear DDE system in the periodic orbit, which results in a linear DDE with
. time-periodiccoefficients[2,6]. ThislinearDDEistypicallywrittenintheform
h
t
a x˙(t)=L(t)x (1.1)
t
m
[ wherethecoefficients(hiddenintheoperator L(t))areperiodicin t. Withoutlossof
generalitywecanassumethatthetimedependenceof L isofperiod1(thiscorresponds
3
to rescaling time and L). Then the exponential asymptotic stability of the periodic
v
orbitinthefullnonlinearsystemisdeterminedbytheeigenvaluesofthetime-1map
2
2 T generated by the linear DDE (1.1). See the textbooks [2, 6] for a summary of
5 the fundamental results on Floquet theory and [11] for robust numerical methods
4 to calculate these eigenvalues in practical problems. For analytical purposes (and,
.
5 possibly,fornumericalpurposes)itisusefultoreducetheeigenvalueproblemforthe
0 infinite-dimensionaltime-1map T toanalgebraicproblemviaacharacteristicmatrix
0 ∆(λ)∈(cid:67)n×n. Onewouldexpectthatthismatrix∆shouldsatisfy,forexample,that
1
(i) ∆(λ) is regular if (and only if) λ ∈ (cid:67) is in the resolvent set of T (that is,
:
v λI−T isanisomorphism),
Xi (ii) λ is a root of det∆(·) of order q if and only if λ is an eigenvalue of T of
algebraicmultiplicityq(see[8]orLemma4.2in§4howonecanconstructtheJordan
r
a chainsof T from∆),and
(iii) dimker∆(λ)isthegeometricmultiplicityofλasaneigenvalueof T.
Animmediateconsequenceoftheexistenceofsuchamatrix∆wouldbetheupperlimit
n(thedimensionof∆(λ))onthegeometricmultiplicityofeigenvaluesof T.
Fortime-invariantlinearDDEs(where L doesnotdependon t in(1.1))theexis-
tenceofacharacteristicmatrixhasbeenknownandusedforalongtime. Thegeneral
theoreticalbackgroundandconstruction(extendedtomoregeneralevolutionarysys-
tems such as, for example, neutral equations and some classes of hyperbolic partial
differentialequations)isgivenbyKaashoekandVerduyn-Lunelin[8].
For time-periodic linear DDEs the textbook of Hale and Verduyn-Lunel [6] has
developedcharacteristicmatricesonlyforthecasewherealltimedelaysaremultiples
oftheperiod. Thishasbeengeneralizedtodelaysdependingrationallyontheperiodin
ordertoderiveanalyticalstabilityresultsforperiodicorbitsoftheclassicalscalardelayed
∗Dept.ofMathematics,universityofPortsmouth(UK)
†Dept.ofEngineeringMathematics,UniversityofBristol(UK)
1
2 J.SieberandR.Szalai
positivefeedbackproblem[12]. Acharacteristicfunctionintheformofacontinued
fraction expansion is given for a scalar DDE in [7]. In general, precise analytical
statementsaboutthelocationofFloquetmultipliers(orthenumberofunstableFloquet
multipliers)canbemadeonlyforDDEswithveryspecialstructure,typicallyscalarDDEs
withsingledelaysandparticularrequirementsontheright-handside,see[1,3,14,15]
forclassicalworks. AnextensionoftheseresultstocyclicDDEsusingadiscreteLyapunov
functionisgivenin[10].
Szalai et al. give a general construction for a characteristic matrix ∆ for time-
periodicDDEsin[13]thatlendsitselfeasilytorobustnumericalcomputation. Since
oneexpectsthat∆(λ)musthaveanessentialsingularityatλ=0(theeigenvaluesof
the compact time-1 map T accumulate at 0) the constructed matrix is rather of the
form ∆(µ) where µ = λ−1. We show (in §3) that the characteristic matrix ∆(µ) as
constructedbySzalaietal.typicallyhaspolesinthecomplexplane,makingitunusable
whenever eigenvalues of T of interest coincide with these poles. However, the set
of these poles is discrete and accumulates only at ∞. Moreover, in §4 we provide a
modification∆ (µ)oftheoriginalmatrix∆(µ),whichallowsustopushthepolesof
k
∆ totheoutsideofanygivenballofradiusR≥1inthecomplexplanebyincreasing
k
the dimension of ∆ . This modification reduces the eigenvalue problem for T to a
k
root-findingproblemfordet∆ (µ)foralleigenvaluesof T withmoduluslargerthan
k
1/R,whichisusefulbecauseoneistypicallyinterestedinthelargesteigenvaluesof T.
Tokeepthenotationaloverheadlimitedwedevelopourargumentsforthecaseofa
DDE with a single delayed argument and a constant delay τ<1. In Appendix B we
showthatthegeneralizationtoarbitrarydelays(distributedandreachingarbitrarilyfar
intothepast)isstraightforward.
One immediate application for the characteristic matrix came up in [16] where
YanchukandPerlikowskiconsiderperiodicorbitsofnonlinearDDEswithasinglefixed
delay τ but change the delay to τ+kP (where k is an integer and P is the period
of the periodic orbit), and then study the stability of the periodic orbit (which does
not change its shape) in the limit k → ∞. The limit is, of course, a singular limit if
one considers the time-P map T but the characteristic matrix ∆ has a well-defined
k
regularlimit. Thispermittedtheauthorstodrawconclusionsaboutthelinearstability
properties of periodic orbits for sufficiently large delays from the properties of the
so-calledpseudo-continuousspectrum[16]. As[16]reliesontheoriginalconstruction
from [13], which may have poles near the unit circle, our note closes a gap in the
argumentof[16].
2. Construction of characteristic matrix for a single fixed delay. Consider a
periodiclineardifferentialequationofdimensionnwithasingledelayτandcontinuous
periodiccoefficientmatricesA(t)andB(t)(bothofsizen×n):
x˙(t)=A(t)x(t)+B(t)x(t−τ) (2.1)
whereweassumethattheperiodofthetimedependenceofAandB equals1without
lossofgenerality(thiscorrespondstorescalingtime,τ,AandB). Wealsoassumefor
simplicityofnotationthatτ<1. Letusdenotethemonodromyoperator(alsocalled
thetime-1map)for(2.1)by T :C([−1,0];(cid:67)n)(cid:55)→C([−1,0];(cid:67)n). C([−1,0];(cid:67)n)isthe
spaceofcontinuousfunctionsontheinterval[−1,0]withvaluesin(cid:67)n. Thatis, T maps
an initial condition x in C([−1,0];(cid:67)n) to the solution of (2.1) after time 1 starting
from the initial value x(0) (the head point) and using the history segment x(·). For
x ∈C([−1,0];(cid:67)n)wedefine Tx preciselyasthesolution y ∈C([−1,0];(cid:67)n)of
(cid:168)y(t−τ) if t∈[τ−1,0]
˙y(t)=A(t)y(t)+B(t)
x(1+t−τ) if t∈[−1,τ−1) (2.2)
y(−1)=x(0).
CharacteristicmatricesforperiodicDDEs 3
Thecomplexnumberλisaneigenvalueof T (aFloquetmultiplierfor(2.1))ifonecan
findanon-zero x suchthat Tx =λx. Weknowthat T iscompact[6]forτ<1(and
that,forarbitraryτ,wecanfindapower Tm thatiscompact). Hence,anynon-zeroλ
iseitherintheresolventsetof T (thatis,λI−T isregular)oritisaneigenvalueof T
withafinitealgebraicmultiplicity.
Ifweintroduceµ=1/λwemaywritetheeigenvalueproblemfor T asfollows:
(cid:168)x(t−τ) if t∈[τ−1,0],
x˙(t)=A(t)x(t)+B(t) (2.3)
µx(1+t−τ) if t∈[−1,τ−1),
x(−1)=µx(0) (2.4)
where x ∈ C([−1,0];(cid:67)n) is continuous. Reference [13] constructs a characteristic
matrix∆(µ)for T basedonthefollowinghypothesis.
Hypothesis2.1(Szalai’06)
Theinitial-valueproblemontheinterval[−1,0]
(cid:168)x(t−τ) if t∈[τ−1,0],
x˙(t)=A(t)x(t)+B(t) (2.5)
µx(1+t−τ) if t∈[−1,τ−1),
x(−1)=v (2.6)
hasauniquesolution x ∈C([−1,0];(cid:67)n)forallµ∈(cid:67)and v∈(cid:67)n.
Notethattheproblem(2.5)–(2.6)differsfromtheboundary-valueproblem(2.3)–(2.4)
becausewehavereplacedtheboundarycondition x(−1)=µx(0)byaninitialcondition
x(−1)=v. Theformulationofthehypothesisin[13]is(onlyapparently)weaker: it
claimstheexistenceonlyforµ∈(cid:67)forwhich1/µiseitherintheresolventsetof T or
aneigenvalueof T ofalgebraicmultiplicityone. IfHypothesis2.1istruethenonecan
definethecharacteristicmatrix∆(µ)via
∆(µ)v=v−µx(0) (2.7)
where x(0) is the value of the unique solution of (2.5)–(2.6) at the end of the time
interval[−1,0].
Integralequationfor(2.5)–(2.6). Forsystem(2.5)–(2.6)onehastoclarifywhatit
meansfor x ∈C([−1,0];(cid:67)n)tobeasolution. Wecall x ∈C([−1,0];(cid:67)n)asolutionof
(2.5)–(2.6)if x satisfiestheintegralequation
x(t)=v+M (µ)[x](t) where (2.8)
1
(cid:90) t (cid:168)x(s−τ) ifs∈[τ−1,0],
(cid:2)M (µ)x(cid:3)(t)= A(s)x(s)+B(s) ds (2.9)
1 µx(1+s−τ) ifs∈[−1,τ−1)
−1
pointwise for all times t on the interval [−1,0]. The linear operator M (µ) maps
1
C([−1,0];(cid:67)n)backintoC([−1,0];(cid:67)n)andiscompact. (M willbegeneralizedlater
1
to M fork>1.)
k
Lemma2.2
Forµsufficientlycloseto0theinitial-valueproblem(2.5)–(2.6)hasaunique
solutionforall v.
Proof:. For(2.8)tohaveauniquesolutionitisenoughtoshowthat I−M (µ)is
1
invertible. Sincethetime-1map T iswelldefinedandlinearin x =0(thatis, Tx =0
for x =0)wehavethat(2.2)hasonlythetrivialsolutionfor x =0. Thisinturnimplies
thatthecorrespondingintegralequation
y =M (0)y (2.10)
1
4 J.SieberandR.Szalai
hasonlythetrivialsolutioninC([−1,0];(cid:67)n). Thus,thekernelof I−M (0)istrivial,
1
and,since I−M (0)isacompactperturbationoftheidentity, I−M (0)isinvertible.
1 1
Moreover, the real-valued function µ(cid:55)→(cid:107)M (µ)−M (0)(cid:107) (using the operator norm
1 1
induced by the maximum norm of C([−1,0];(cid:67)n)) is continuous in µ because M
1
dependscontinuouslyonµ(seedefinitionof M in(2.9)). Thisimpliesthat I−M (µ)
1 1
isalsoinvertibleforsmallµ,whichinturnmeansthat(2.8)–(2.9)hasauniquesolution
forsmallµ. (cid:131)
The following lemma states that the function on the complex domain µ(cid:55)→[I−
M (µ)]−1, which has operators in (cid:76)(C([−1,0];(cid:67)n);C([−1,0];(cid:67)n)) as its values, is
1
welldefinedandanalyticeverywhereinthecomplexplaneexcept,possibly,inadiscrete
setofpoles.
Lemma2.3
Theoperator-valuedcomplexfunctionµ(cid:55)→[I−M (µ)]−1 canhaveatmosta
1
finitenumberofpolesinanyboundedsubsetof(cid:67). Allpolesof[I−M (µ)]−1(if
1
thereareany)areoffinitemultiplicity. Ifµisnotapolethenµ(cid:55)→[I−M (µ)]−1
1
isanalyticinµ.
Proof:. Wecansplittheoperator M (µ)intoasumoftwooperators:
1
M (µ)=M (0)+µL where (2.11)
1 1
(cid:90) t (cid:168)0 ifs∈[τ−1,0],
[Lx](t)= B(s) ds. (2.12)
x(1+s−τ) ifs∈[−1,τ−1)
−1
Theoperator I−M (0)isinvertible(andacompactperturbationoftheidentity)and
1
L:C([−1,0];(cid:67)n)(cid:55)→C([−1,0];(cid:67)n)iscompact. Thus,wecanrewrite I−M (µ)as
1
I−M (µ) =(cid:0)I−M (0)(cid:1)(cid:148)I−µ(I−M (0))−1L(cid:151) and,thus,
1 1 1
[I−M (µ)]−1=λ·(cid:148)λI−(I−M (0))−1L(cid:151)−1(I−M (0))−1
1 1 1
(keeping in mind that µ=1/λ). The function λ(cid:55)→[λI−(I−M (0))−1L]−1 on the
1
right-handsideisthestandardresolventofthecompactoperator(I−M (0))−1L,which
1
hasonlyfinitelymanypolesoffinitemultiplicityoutsideofanyneighborhoodofthe
complexoriginaccordingto[9],andisanalyticeverywhereelse. (cid:131)
Lemma2.3carriesovertothecharacteristicmatrix∆(µ)definedin(2.7): since
x =[I−M (µ)]−1v (ifweextend v∈(cid:67)n toaconstantfunction),∆(µ)v=v−µx(0)is
1
ananalyticfunctionwithatmostadiscretesetofpolesoffinitemultiplicity(possibly
accumulatingat∞).
The characteristic matrix ∆(µ) (where it is well defined) has the properties one
expects: for a complex number µ in the domain of definition of ∆, 1/µ is a Floquet
multiplieroftheDDE(2.1)ifandonlyif∆(µ)issingular(det∆(µ)=0). Anyvectorvin
itskernelisthevalueofaneigenfunctionfor1/µattime t=−1. Thefulleigenfunction
can be obtained as the solution of the initial-value problem (2.5)–(2.6). If ∆(µ) is
regular then 1/µ is in the resolvent set of the eigenvalue problem (2.3)–(2.4). The
Jordanchainsassociatedtoeigenvalues1/µcanbeobtainedbyfollowingthegeneral
theorydescribedin[8]. Wegiveaprecisestatement(seeLemma4.2inSection4)about
theJordanchainstructureafterdiscussingHypothesis2.1.
Theconstructioncanbeextendedinastraightforwardmannertomultiplefixed
discrete delays and distributed delays. The only modification is that higher powers
ofµhavetobeincludedfordelayslargerthan1. Forexample,ifτ∈(1,2]thenthe
initial-valueproblem(2.5)–(2.6)ismodifiedto
(cid:168)µx(1+t−τ) if t∈[τ−2,0],
x˙(t)=A(t)x(t)+B(t)
µ2x(2+t−τ) if t∈[−1,τ−2),
x(−1)=v.
CharacteristicmatricesforperiodicDDEs 5
Thus, alsoforarbitrarydelays∆(µ)alwayshasatworstfinitelymanypolesoffinite
multiplicity in any bounded subset of (cid:67). Appendix B discusses the case where the
functionalaccessingthehistoryof x isaLebesgue-Stieltjesintegral.
3. Thepolesofthecharacteristicmatrix. Hypothesis2.1isnottrueinitsstated
form(andneitherintheformstatedin[13]). Letusanalysethesimplescalarexample
(cid:18) 1(cid:19)
x˙(t)=x t− (3.1)
2
ontheinterval[−1,0]. Problem(2.5)–(2.6)restatedforthisexampleisequivalentto
x˙ (t)=µx (t), x (0)=v
1 2 1
(3.2)
x˙ (t)=x (t), x (0)=x (1/2)
2 1 2 1
where x and x areinC([0,1/2];(cid:67)),and x (t)=x(t−1)and x (t)=x(t−1/2)for
1 2 1 2
t intheinterval[0,1/2]. Thetime-1/2mapfortheIVPoftheODE(3.2)withinitial
conditions x (0)=v, x (0)=w is
1 2
(cid:112) (cid:112) (cid:112)
(cid:20)x (1/2)(cid:21) cosh(cid:128)1 µ(cid:138) µsinh(cid:128)1 µ(cid:138)(cid:150)v(cid:153)
x12(1/2) =sinh(cid:128)1(cid:112)2µ(cid:138)/(cid:112)µ cosh(cid:128)1(cid:112)2µ(cid:138) w
2 2
(cid:112)
where µreferstotheprincipalbranchofthecomplexsquareroot(thesetofsolutions
is the same for both branches as all entries in the matrix are even). The continuity
condition x (0)=x (1/2)in(3.2)issatisfiedifandonlyif
2 1
(cid:20) (cid:112) (cid:18)1(cid:112) (cid:19)(cid:21) (cid:18)1(cid:112) (cid:19)
1− µsinh µ w+cosh µ v=0,
2 2
whichhasauniquesolutionforarbitrary v ifandonlyif
(cid:112) (cid:18)1(cid:112) (cid:19)
0(cid:54)=1− µsinh µ .
2
Thus, we see that for example (3.1) the initial-value problem (2.5)–(2.6) cannot be
solvedfornon-zero v wheneverµisarootofthefunction
(cid:112) (cid:18)1(cid:112) (cid:19)
f(µ)=1− µsinh µ .
2
Thisfunction f isagloballydefinedrealanalyticfunctionofµ(theexpressiononthe
(cid:112)
right-handsideisevenin µ). Ithasaninfinitenumberofcomplexrootsaccumulating
atinfinity. Oneoftherootsisreal: f(0)=1,and f(µ)→−∞forµ→+∞andµ∈(cid:82)
(let’scallitµ : µ ≈1.8535). Consequently,example(3.1)providesacounterexample
0 0
toHypothesis2.1. Ifoneextendsexample(3.1)byadecoupledequationthathasits
Floquetmultiplierat1/µ (forexample, ˙y =−[logµ ]y)thenitalsocontradictsthe
0 0
hypothesisasstatedin[13].
AswehavediscussedinSection2,onefindsthatµ=1/λisapoleof∆(µ)ifλis
anon-zeroeigenvalueoftheoperator[I−M (0)]−1L where M and L aredefinedin
1 1
(2.9) and (2.12). Thus, whenever [I−M (0)]−1L has a non-zero spectral radius we
1
mustexpectthat∆(µ)haspoles.
4. The extended characteristic matrix. The construction of ∆(µ) suffers from
theproblemthatthepolesof∆(µ)maycoincidewiththeinverseofFloquetmultipliers
of(2.1)ofinterest. Asimpleextensionoftheconstructionof∆(µ)permitsonetopush
thepolestotheoutsideofacircleofanydesiredradiusR. Thenthecharacteristicmatrix
canbeusedtofindallFloquetmultipliersoutsideoftheballofradius1/R. Weexplain
6 J.SieberandR.Szalai
theextensionindetailforasingledelayτ<1(thestraightforwardgeneralizationto
thegeneralcaseisrelegatedtoAppendixB).
The idea is based on the observation that the linear part of the right-hand side
in the integral equation formulation (2.8)–(2.9) of initial-value problem (2.5)–(2.6)
hasanormlessthan1iftheintervalissufficientlyshort(thus,makingthefixedpoint
problem(2.8)–(2.9)uniquelysolvable).
Similartoamultipleshootingapproach, wepartitiontheinterval[−1,0]into k
intervalsofsize1/k:
(cid:20) i i+1(cid:19)
Ji =(cid:2)ti,ti+1(cid:1)= −1+ k,−1+ k fori=0,...,k−1. (4.1)
Thenwewriteasystemofkcoupledinitialvalueproblemsforavectorofkinitialvalues
(v0,...,vk−1)T ∈(cid:67)nk:
(cid:168)x(t−τ) if t∈[τ−1,0],
x˙(t)=A(t)x(t)+B(t) (4.2)
µx(1+t−τ) if t∈[−1,τ−1),
x(t )=v fori=0,...,k−1 (4.3)
i i
where t∈[−1,0]. System(4.2)–(4.3)correspondstoacoupledsystemofkdifferential
equationsforthekfunctions x| inthetimeintervalsJ. Moreprecisely, x isdefinedas
J i
i
asolutionofthefixedpointproblem
t
(cid:90) (cid:168)x(s−τ) ifs∈[τ−1,0],
x(t)=v(t)+ A(s)x(s)+B(s) dswhere (4.4)
s µx(1+s−τ) ifs∈[−1,τ−1)
a (t)
k
v(t)=v if t∈J, (4.5)
s i i
a (t)=t if t∈J. (4.6)
k i i
We point out that a solution x of (4.2)–(4.3) is not necessarily in C([−1,0];(cid:67)n)
x(t )
2
−
v2=x(t2)+
v0=x((cid:0)1)+ v1=x(t1)+
x(0)(cid:0)
x(t1)(cid:0)
1=t t t 0=t
0 1 2 3
−
J J J
0 1 2
Fig.4.1: Illustrationofsolution x ∈C formultiple-shootingproblem(4.2)–(4.3)with k=3
k
sub-intervals. Theright-sidedlimits x(ti)+areequaltothevariablesvi. Thegapsbetweenthe
left-sided limits x(ti)− and x(ti)+ are zero if [v0,...,vk] is in the kernel of the characteristic
matrix∆ (µ).Theequationsfor x| onthesub-intervalsarecoupledduetothetermst−τor
1+t−τkin(4.2). Ji
becauseitwilltypicallyhavediscontinuitiesattherestartingtimes t fori=1...k−1
i
CharacteristicmatricesforperiodicDDEs 7
as illustrated in Fig. 4.1 . Thus, we should define the space in which we look for
solutionsasthespaceofpiecewisecontinuousfunctions:
C ={x :[−1,0](cid:55)→(cid:67)n: x continuousoneach(half-open)subintervalJ (4.7)
k i
andlimt(cid:37)t x(t)existsforalli=1...k.}
i
C ={x ∈C :x(t )=0foralli=0...k−1}. (4.8)
k,0 k i
Wecall x ∈C asolutionof(4.4)–(4.6)ifitsatisfies(4.4)forevery t ∈[−1,0]. Both
k
spaces, C and C are equipped with the usual maximum norm. For example, the
k k,0
piecewiseconstantfunctionv asdefinedin(4.5)isanelementofC . Severaloperations
s k
areusefulwhendealingwithfunctionsinC tokeepthenotationcompact. Anyelement
k
x ∈C haswell-definedone-sidedlimitsattheboundaries t ,whichwedenotebythe
k i
subscripts+and−:
x(ti)−=tl(cid:37)imt x(t) fori=1...k,
i
x(ti)+=tl(cid:38)imt x(t)=x(ti) fori=0...k−1.
i
Wealsodefinethefourmaps
S:(cid:67)nk(cid:55)→Ck S[v0...vk−1]T(t)=vi if t∈[ti,ti+1)fori=0...k−1,
Γ+:Ck(cid:55)→(cid:67)nk Γ+[x(·)]=(cid:2)x(−1)+,x(t1)+,...,x(tk−1)+(cid:3)T,
Γ−(µ):Ck(cid:55)→(cid:67)nk Γ−(µ)[x(·)]=(cid:2)µx(0)−,x(t1)−,...,x(tk−1)−(cid:3)T,
M (µ):C (cid:55)→C where
k k k,0
t
(cid:90) (cid:168)x(s−τ) ifs∈[τ−1,0],
(cid:2)M (µ)x(cid:3)(t)= A(s)x(s)+B(s) ds
k µx(1+s−τ) ifs∈[−1,τ−1)
a (t)
k
(4.9)
(thepiecewiseconstantfunctionak wasdefinedin(4.6)asak(t)=ti if t∈[ti,ti+1)).
The map S takes a tuple of vectors v0,...vk−1 and maps it to a piecewise constant
function, assigning x(t )= v and then extending with a constant to the subinterval
i i
Ji. For example, the function vs in (4.4)–(4.5) is equal to Sv. The map Γ+ takes the
right-sidelimitsofapiecewisecontinuousfunction(thus,Γ+S istheidentityin(cid:67)nk).
ThemapΓ−(µ)takestheleft-sidelimitsattheinteriorboundaries ti (i≥1),andinits
firstcomponentittakestheleft-sidelimitat t =0(theendoftheinterval)multiplied
k
byµ. Themap M (µ)isthegeneralizationof M definedin(2.9): inthedefinitionof
k 1
M thelowerboundaryintheintegralisnot−1buta (t)suchthatwecanestimateits
k k
norm:
(cid:107)Mk(µ)(cid:107)∞≤ 1k(cid:20)t∈m[−a1x,0](cid:107)A(t)(cid:107)∞+|µ|tm∈[a0,x1](cid:107)B(t)(cid:107)∞(cid:21)=: C∗(k|µ|). (4.10)
AccordingtotheTheoremofArzelá-Ascoli(see[4],page266)theoperatorM (µ)isalso
k
compactbecauseitmapsanyboundedsetintoasetwithuniformlyboundedLipschitz
constants. Usingtheabovenotationwecanwritethedifferentialequation(4.2)–(4.3)
(or,rather,thecorrespondingintegralequation(4.4))asafixedpointproblemin C .
k
Foragiven v=[v0,...,vk−1]∈(cid:67)nk andµ∈(cid:67)wefindafunction x ∈Ck suchthat
x =Sv+M (µ)x. (4.11)
k
Duetothenormestimate(4.10)on M (µ)wecanstatethefollowingfactaboutthe
k
existenceofafixedpoint x of(4.11)foragiven v andµ.
8 J.SieberandR.Szalai
Lemma4.1
LetR≥1begiven. Ifwechoose k>C∗(R)thenthesystem(4.2)–(4.3)hasa
unique solution x ∈ Ck for all initial values (v0...vk−1)T and all µ satisfying
|µ|<R.
Proof:. Thedifferentialequation(4.2)–(4.3)isequivalenttofixedpointproblem
(4.11). The norm estimate (4.10) implies that I −M (µ) is invertible such that the
k
solution x of(4.11)isgivenby[I−M (µ)]−1Sv. (cid:131)
k
Lemma4.1impliesthatforµsatisfying C∗(|µ|)<k theoperator I−Mk(µ)isan
isomorphismonC .
k
4.1. Extended characteristic matrix. We can use Lemma 4.1 to define the ex-
tended characteristic matrix ∆k(µ) for µ satisfying C∗(|µ|) < k. This is a matrix in
(cid:67)(nk)×(nk) andisdefinedas
v v0−µx(0)
∆k(µ)v=∆(µ) ...0 = v1−x...(t1)− (4.12)
vk−1 vk−1−x(tk−1)−
where x =(cid:2)I−M (µ)(cid:3)−1Sv∈C istheuniquesolutionofthecoupledsystemofinitial-
k k
valueproblems,(4.2)–(4.3),withinitialcondition v=(v0,...,vk−1)T ∈(cid:67)nk. Thefirst
rowintherightsideofdefinition(4.12)correspondstotheboundarycondition(2.4).
Theotherk−1rows, vi−x(ti)−,guaranteefor v∈ker∆(µ)thattherightandtheleft
limitof x attherestartingtimes t agree,makingthegapsshowninFig.4.1zero. Thus,
i
x =[I−M (µ)]Sv isnotonlyinC butinC([−1,0];(cid:67)n)if∆ (µ)v=0. Whatismore,
k k k
if∆ (µ)v=0thentheright-handsideof(4.2)forthesolution x =[I−M (µ)]−1Sv
k k
is continuous such that x is continuously differentiable and satisfies for every t the
differentialequation(2.3)–(2.4),whichdefinestheeigenvalueproblemfor T.
LetuslistseveralusefulfactsaboutC ,C ,∆ (µ), T andthequantitiesdefinedin
k k,0 k
(4.9).
1. Wecanexpress∆ (µ)usingthemapsdefinedin(4.9):
k
∆k(µ)=I−Γ−(µ)(cid:2)I−Mk(µ)(cid:3)−1S. (4.13)
2. C isisomorphicto(cid:67)nk×C :
k k,0
(v,ϕ)∈(cid:67)nk×C (cid:55)→Sv+ϕ∈C
k,0 k
(4.14)
ψ∈Ck (cid:55)→(Γ+ψ,ψ−SΓ+ψ)∈(cid:67)nk×Ck,0.
3. Thedifferentialequation
(cid:168)x(t−τ) if t∈[τ−1,0],
x˙(t)=A(t)x+B(t)
0 if t∈[−1,τ−1),
(4.15)
x(−1)+=0
x(ti)+=x(ti)− fori=1,...,k−1
has only the trivial solution (it is identical to (2.5)–(2.6) for µ = 0). Its integral
formulationis x =SΓ−(0)x+Mk(0)x. Thus,theoperator I−SΓ−(0)−Mk(0):Ck(cid:55)→
C([−1,0];(cid:67)n)(whichisofFredholmindex0)isinvertible.
4. Thetime-1map T ofthelinearDDE x˙(t)=A(t)x(t)+B(t)x(t−τ),defined
in (2.2) for initial history segments in x ∈ C([−1,0];(cid:67)n), can be easily extended to
initialhistorysegmentsinC . Infact,theextensionof T toC canbedefinedby
k k
T =(cid:2)I−SΓ−(0)−Mk(0)(cid:3)−1(cid:2)S(Γ−(1)−Γ−(0))+Mk(1)−Mk(0)(cid:3),
CharacteristicmatricesforperiodicDDEs 9
whichmapsC intoC([−1,0];(cid:67)n),suchthat
k
I−µT =(cid:2)I−SΓ−(0)−Mk(0)(cid:3)−1(cid:2)I−SΓ−(µ)−Mk(µ)(cid:3), (4.16)
which maps C into C (see Appendix A for a detailed decomposition of expression
k k
(4.16)). Sincetheextensionof T mapsintoC([−1,0];(cid:67)n)(theoriginaldomainofthe
non-extended T)itsspectrumisidenticaltothespectrumoftheoriginal T.
Lemma4.1andtheabovefactspermitustoapplythegeneraltheorydevelopedin
[8],relatingtheextendedcharacteristicmatrix∆ (µ)∈(cid:67)(nk)×(nk) tospectralproperties
k
ofthetime-1map T: eigenvalues,eigenvectors,andthelengthoftheirJordanchains.
Lemma4.2(Jordanchains)
Assumethat0<|µ∗|<Rand k>C∗(R)where C∗(R)isdefinedby(4.10). If
∆k(µ∗)isregularthen1/µ∗ isintheresolventsetofthemonodromyoperator
T. If∆k(µ∗)isnotregularthenλ∗=1/µ∗ isaneigenvalueof T. TheJordan
chainstructureforλ∗ asaneigenvalueof T isalsodeterminedby∆k:
1. thedimensionl0 ofker∆k(µ∗)isthegeometricmultiplicityofλ∗,
2. the order p of the pole µ∗ of ∆k(µ)−1 is the length of the longest
Jordanchainassociatedtoλ∗,
3. theorderofµ∗ asarootofdet∆k(µ)isthealgebraicmultiplicityof
λ∗,
4. the Jordan chains of length less than or equal to p can be found as
solutions(y0,...,yp−1)(where yp−1 isnon-zero)ofthelinearsystem
0=∆(0)y
k 0
0=∆(1)y +∆(0)y
k 0 k 1
∆(2)
0= k y +∆(1)y +∆(0)y
2 0 k 1 k 2 (4.17)
.
.
.
∆(p−1)
0= (p−k 1)!y0+...+∆(k1)yp−2+∆(k0)yp−1
where∆(kj) isthe jthderivativeof∆k(µ)inµ∗ and∆(k0)=∆k(µ∗).
System(4.17)isequivalenttotherequirement
∆k(µ)(cid:148)y0+(µ−µ∗)y1+...+(µ−µ∗)p−1yp−1(cid:151)=O(cid:0)(µ−µ∗)p(cid:1)
forallµ≈µ∗ (thisis,infact,thegeneraldefinitionofaJordanchainforanonlinear
eigenvalueproblem). Thevectors y intheJordanchainsfoundassolutionsof(4.17)
j
are still vectors in (cid:67)nk. We give the precise relation between the vectors y and the
j
generalizedeigenvectorsof I−µT inEquation(4.21)intheproofbelow.
Proof:. Theproofextendsandmodifiestheconstructionusedby[13],andmakes
thegeneraltheoryof[8]applicabletotime-periodicdelayequations.
First, we give a brief summary of the relevant statement of [8]. Let G : Ω (cid:55)→
(cid:76)(X ;Y )andH:Ω(cid:55)→(cid:76)(X ;Y )betwofunctionswhichareholomorphicontheopen
1 1 2 2
subsetΩofthecomplexplane(cid:67),andmapintothespaceofboundedlinearoperators
fromoneBanachspaceX intoanotherBanachspaceY . ThefunctionsG andH are
1,2 1,2
calledequivalentifthereexisttwofunctions E:Ω(cid:55)→(cid:76)(X ;X )and F :Ω(cid:55)→(cid:76)(Y ;Y )
2 1 1 2
(alsoholomorphiconΩ)whosevaluesareisomorphismssuchthat
H(µ)=F(µ)G(µ)E(µ)forallµ∈Ω.
10 J.SieberandR.Szalai
If two operators G and H are equivalent then every Jordan chain (u0,...,up−1) of
the eigenvalue problem H(µ)u = 0 for µ∗ corresponds to exactly one Jordan chain
(v0,...,vp−1)oftheeigenvalueproblemG(µ)v=0atµ∗ viatheisomorphism
v =E u
0 0 0
v =E u +E u
1 0 1 1 0
. (4.18)
.
.
vp−1=E0up−1+...+Ep−1u0,
wheretheoperators Ej aretheexpansioncoefficientsoftheisomorphism E inµ=µ∗:
E(µ)=E0+(µ−µ∗)E1+...+(µ−µ∗)p−1Ep−1+O((µ−µ∗)p). (4.19)
We construct the equivalence initially only for the case of a single delay τ<1. The
necessarymodificationsforarbitrarydelayscanbefoundinAppendixB(theunderlying
ideaisidenticalbutthereismorenotationaloverhead). Also,werestrictourselveshere
tomerelystatingwhattheoperatorsH, F,G and E are,relegatingthedetailedchecks
andcalculationstoAppendixA.
The subset of permissible µ, Ω⊂(cid:67), is the ball of radius R around 0, where the
initial-valueproblem(4.2)–(4.3)hasauniquesolution. Inourequivalencetheoperator
functionsG andH are
G(µ)=I−µT : X =C (cid:55)→ Y =X ,
1 k 1 1
(cid:18)∆ (µ) 0(cid:19)
H(µ)= k : X =(cid:67)nk×C (cid:55)→ Y =X .
0 I 2 k,0 2 2
Note that the spaces X and X are isomorphic via relation (4.14). The eigenvalue
1 2
problemG(µ)v=0istheeigenvalueproblemfor T (reformulatedforµ=λ−1). The
eigenvalueproblemH(µ)[v,ϕ]=0isequivalentto∆ (µ)v=0,ϕ=0. Thus,establish-
k
ingequivalencebetweenG andH provesLemma4.2. Wedefinetheisomorphisms E
and F as
E(µ)[v,ϕ]=(cid:2)I−M (µ)(cid:3)−1(cid:2)Sv+ϕ(cid:3)
k
(cid:20)F (µ)ψ(cid:21)
F(µ)ψ= 1
F (µ)ψ
2
=(cid:150)(cid:2)Γ+−Γ−(0)(cid:3)ψ+Γ−(µ)(cid:2)I−Mk(µ)(cid:3)−1(cid:2)I−SΓ+−Mk(0)(cid:3)ψ(cid:153)
(cid:2)I−SΓ+−Mk(0)(cid:3)ψ
(4.20)
See(4.13)and(4.16)forexpressionsgiving∆ (µ)and I−µT,defining G and H in
k
termsofS,Γ± and Mk. Theinverseof I−Mk(µ)isguaranteedtoexistbyLemma4.1.
Theonlychoicewemakeisthedefinitionof E,whichisclearlyanisomorphismfromX
2
toX (because(v,ϕ)(cid:55)→Sv+ϕisanisomorphismfromX toX ). Thedefinitionof F is
1 2 1
thenuniquelydeterminedifwewanttoachievetheequivalence F(µ)G(µ)E(µ)=H(µ),
andfollowsfromastraightforwardbuttechnicalcalculation,giveninAppendixA.The
concreteformofEandF allowsustospecifythepreciserelationbetweenJordanchains
of∆k(µ)y =0and(I−µT)x =0: (x0,...,xp−1)∈C([−1,0];(cid:67)n)p isaJordanchain
of(I−µT)x =0inµ=µ∗ ifandonlyif
x0=E0[y0,0], y0=Γ+x0
x1=E0[y1,0]+E1[y0,0], y1=Γ+x1−Γ+E1[y0,0]
. .
.. .. (4.21)
xp−1=E0[yp−1,0]+...+Ep−1[y0,0], yp−1=Γ+xp−1−Γ+E1[yp−2,0]−...
−Γ+Ep−1[y0,0],
