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CHARACTERISTICMATRICESFORLINEARPERIODICDELAYDIFFERENTIAL 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 extended 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],