Bifurcation delay - the case of the sequence: stable focus - unstable focus - unstable node

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Bifurcation delay - the case of the sequence: stable focus - unstable focus - unstable node

Eric Benoît

To cite this version:

Eric Benoît. Bifurcation delay - the case of the sequence: stable focus - unstable focus - unstable

node. Discrete and Continuous Dynamical Systems - Series S, American Institute of Mathematical

Sciences, 2009, 2 (4), pp.911-929. �hal-00354175�

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stable fous - unstable fous - unstable node

Eri Benoît

∗

January19,2009

Abstrat

Letusgiveatwodimensionalfamilyofrealvetorelds.Wesupposethatthereexistsastationarypoint

where the linearized vetor eld hassuessively a stable fous, anunstable fous and anunstable node.

Whenthe parametermovesslowly,a bifurationdelay appears dueto theHopf bifuration. Thestudied

questioninthisartileistheontinuationofthedelayafterthefous-nodebifuration.

AMSlassiation: 34D15,34E15,34E18,34E20,34M60

Keywords: Hopf-bifuration,bifuration-delay,slow-fast,anard,Airy,relief.

1 Introdution

"Singularperturbations"isastudieddomainfrommanyyearsago. Sine1980,manyontributionswerewritten

beause new toolswere applied to the subjet. The main studied objetsare the slow fast vetor elds also

known assystems with two time-sales. Wewill givethe problem herewith amorepartiular point of view:

thebifuration delay,asin artiles[8,2,9,7℄. Wewritethestudiedsystem: εX˙ =f(t, X, ε)^,^whereε^is^a^real

positiveparameterwhih tendstozero. Forabetterunderstandingoftheexpressiondynami bifurationit is

bettertowrite thesystemafteraresalingofthevariable:

X˙ = f(a, X, ε)

˙

a = ε

wherea^is^a"slowlyvarying"parameter.

The main objetsin this study are theeigenvaluesof the linearpartof equation X˙ =f(a, X,0) ^near ^the

quasi-stationarypoint. Indeed,theygiveaharaterizationofthestabilityoftheequilibriumofthefastvetor

eldatthispoint. Theaimofthisstudyistounderstandwhathappenswhenthestabilityofaquasi-stationary

pointhanges. Abifurationourswhenatleastoneoftheeigenvalueshasanullrealpart.

Inthisartilewerestritourstudytotwo-dimensionalrealsystems. Inthissituation,thegeneribifurations

are: thesaddle-nodebifuration,theHopfbifuration andthefous-nodebifuration.

Thesaddle-nodebifurationissolvedbytheturningpointtheory: whentherealpartofoneoftheeigenvalue

beomes positive, there is no delay and a trajetory of the systems leaves the neighborhood of the quasi-

stationary point when it reahes the bifuration. For this study, the study of one-dimensional systems is

suient: we have a deomposition of the phase spae where only the one-dimensionalfator is interesting.

Thereexistmanyartilesonthissubjet,wewillbeinterestedpartiularlyby[3℄wherethemethod ofrelief is

used. Theartile[6℄introduesthegeometrialmethodsofFenihel's manifold.

TheHopfdelayedbifurationiswellexplainedin[10℄, wewillupgradetheresultsinparagraph2below.

Inafous-nodebifuration, thestabilityofthequasistationary pointdoesnothange,then, loally,there

isnoproblemofanardsorbifurationdelays. Indeed,whenthereis abifurationdelayat aHopf-bifuration

point,itis possibleto evaluatethevalueofthedelay,and themainquestionisto understandtheinuene of

thefousnodebifurationtothisdelay.

Inparagraph2,theHopfbifurationaloneisstudied,aswellasthefous-nodebifurationfollowingaHopf

bifurationinparagraph3.

∗

Laboratoire deMathématiques et Appliations, Université dela Rohelle, avenueMihelCrépeau, 17042 LA ROCHELLE

and/orprojetCOMORE,INRIA,2004routedesLuioles06902SOPHIA-ANTIPOLISourriel:ebenoituniv-lr.fr

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steadystateinthewhole domain,sothistrajetoryhasaninnitedelay. Theusedmethods arereal, andthe

systemhastobesmooth(atuallyonlyC²^). În^paragraphs^2.2êt^3.2,^weâvoid^this^very^speialhypothesis. It isheresupposedthatthesystemisanalyti,andwestudythesolutionsonomplexdomains. Unfortunately,I

havenotaproofforthemainresultofthisartile. Butitseemstomethattheproblemisinteresting,andthe

resultsareargumented.

WeuseNelson'snonstandardterminology(seeforexample[5℄). Indeed,almostallsentenesanbetranslated

in lassialterms,where ε îsônsidered âsâ^variableând ^notâsâ^parameter. Ôften, ^thetranslationis given onfootnotes.

2 The delayed Hopf bifuration

Theproblemis studiedand essentiallyresolvedin [10℄. Wegiveheretheproofsto improvetheresultsandto

xtheideasforthemainparagraphoftheartile. Themaintoolistherelief'stheoryofJ.L.Callot,explained

in[4℄.

Thestudiedequationis

εX˙ =f(t, X, ε) ⁽¹⁾

wheref îsânalytiônâ^domainDôf^C×^C²×^C.

Hypothesis and notations

H1 Thefuntionf îsânalyti. Ît^takes^real^values^when^theârgumentsâre^real.

H2 Theparameterε^is^real, ^positive,innitesimal 1

.

H3 Thereexists ananalytifuntion φ^,^dened ônâômplex^domainDt ^so^that f(t, φ(t),0) = 0^. ^Theûrve X =φ(t) îs âlled ^the ^slow ûrve ôf êquation^(1). ^We âssume^that ^the intersetion ofDt ^with ^the ^real

axisis aninterval]tm, tM[^.

H4 Letusdenoteλ(t)ândµ(t)^for^theeigenvaluesofthejaobianmatrixDXf^,ômputedât^point(t, φ(t),0)^.

Weassumethat ,fort^real,^the^signsôf^the^realândîmaginary^partsâre^given^by^the^table^below^:

t tm ^a tM

ℜ(λ(t)) ^- ⁰ ⁺

ℜ(µ(t)) ^- ⁰ ⁺

ℑ(λ(t)) ^- ^- ^-

ℑ(µ(t)) ⁺ ⁺ ⁺

Then,whent^inreases^fromt_m ^tot_M^,^thequasi-steadystateisrstanattrativefous,thenarepulsive fous,with aHopfbifurationat t=a^.

2.1 Input-output funtion when there exists a big anard

Inthissetion,weassumethatthereexistsabiganardX˜(t)î.e. â^solutionôfêquation⁽¹⁾^suh^that²X˜(t)≃φ(t)

foralltⁱⁿ^theS^-interiorôf]tm, tM[^. ^We^now^want^to^study^theôthers^solutionsôfêquation⁽¹⁾^byômparison

withX˜^.

Themaintoolforthatisasequeneofhangeofunknowm: rst,weperformatranslationonX^,^depending

ont ^to^put^the^bigânardôn^theâxis:

X = ˜X(t) + Y

Itgivesthesystem

εY˙ = g(t, Y, ε) ^with g(t, Y, ε) =f(t,X˜(t) +Y, ε)−f(t,X˜(t), ε)

1

Inlassialterms,weassumethatε^leavesⁱⁿâ^smallômplex^setor: |ε|^boundedândarg(ε)∈]−δ, δ[^.

2

Withoutnonstandardterminology,abiganardisasolutionofequation(1)dependingontheparameterε^suh^that

∀t∈]tm, tM[, lim

ε>0,ε→0

X˜(t, ε) =φ(t)

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ThematrixDXf(t, φ(t),0)^has^twoômplexônjugate^distinteigenvalues(seehypothesisH4),thenthereexists alineartransformation P(t)^whih ^transforms^the^jaobian^matrixⁱⁿâânonial^form. ^We ^dene^the^hange

ofunknown

Y =P(t)Z

Thenewsystem,hasthefollowingform (wewroteonlytheinterestingterms):

εZ˙ = h(t, Z, ε) ^with h(t, Z, ε) =

α(t) −ω(t) ω(t) α(t)

Z+O(ε)Z+O(Z²) ^, λ(t) =α(t)−iω(t)

Thenexthange isgivenbythepolaroordinates:

Z =

rcosθ rsinθ

εr˙ = r(α(t) +O(ε) +O(r)) εθ˙ = ω(t) +O(ε) +O(r)

Thelastoneisanexponentialmirosope 3

:

r = expρ ε

ρ˙ = α(t) +O(ε) +e^ρ^εk1(r, θ, ε)

εθ˙ = ω(t) +O(ε) +e^ρ^εk2(r, θ, ε) ⁽²⁾

While ρ îs ^non ^positive ând ^non innitesimal, r îs exponentially small and the equation (2) gives a good approximationof ρ ^with ρ˙ =α^. ^When ρ ^beomes innitesimal, with a moresubtle argument (see [1℄)using dierentialinequations,weanprovethatr^beomes^noninnitesimal. Thisgivesthepropositionbelow:

Proposition 1 Let us assumehypothesis H1 toH4(Hopf bifuration) for equation(1). However, we assume

that there exists aanard X(t)˜ ^going âlong⁴ ^the ^slow ûrve ât ^least ôn ]tm, t_M[^. ^Then îf X(t) ^goes âlong ^the

slow urveexatly 5

on ]te, t_s[ ^with[te, t_s]⊂]tm, t_M[^,^then Z ts

te

ℜ(λ(τ))dτ = 0

Theinput-output relation(betweente ^andts⁾^is ^dened ^byRts

te ℜ(λ(τ))dτ = 0^. Ît îs ^desribed^by îts^graph

(seegure1). Inthisase,thisrelationisafuntion.

Figure1: Theinput-outputrelationforequation(3)whenthereexistsabiganard.

3

Allthepreeedingtransformationswereregularwithrespettoε^.^This^lastôneîs^singularâtε= 0^.

4

AsolutionX˜(t, ε)^goesâlong^the^slowûrveât^leastôn]t1, t2[îf

∀t∈]t¹, t2[, lim

ε>0,ε→0

X(t, ε) =˜ φ(t)

5

AsolutionX˜(t, ε)^goesâlong^the^slowûrveêxatly ôn]te, ts[îfît ^goesâlong^the^slowûrveât^least ôn ]te, ts[^,ândîf^the

interval]te, ts[^is^maximal^for^this^property.

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Inthisparagraph, t^beomesômplex,ⁱⁿ ^the ^domainD^t^. ^Weâssume^that ^forâllt ⁱⁿ D^t^, ^the^twoeigenvalues

λ(t)ândµ(t)âre ^distint. Îtîs â^neessaryôndition^toâpply^Callot's^theoryôf^reliefs.

WedenethereliefsRλ ^andRµ ^by:

F_λ(t) = Z t

a

λ(t)dt ^, R_λ(t) =ℜ(Fλ(t)) Fµ(t) =

Z t a

µ(t)dt ^, Rµ(t) =ℜ(Fµ(t))

It iseasyto seethat λ(t) =µ(t)^,^and Fλ(t) =Fµ(t)^, ^thenRλ(t) = Rµ(t)^. ^The^two^funtionsRλ ^andRµ

oinideonthereal axis. WewilldenoteR(t)^.

Denition1 Wesay thatapath γ:s∈[0,1]7→ Dt ^goes^down ^the ^relief Rλ îf ândônly îf d

dsRλ(γ(s))<0 ^for

allsⁱⁿ [0,1]^.

Denition2 Letus give apoint te ^suh ^that (te, φ(te),0) ∈ D^. ^We ^say ^that Dt îsâ ^domain ^below te îf ând

only if for all t ⁱⁿ ^the S^-interior ôf Dt^, ^there êxist^two ^paths ⁱⁿ Dt^, ^from t_e ^tot^,^the ^rst ône ^goes ^down ^the

relief R_λ ^and^the ^seond^one^down R_µ^.

Theorem2(Callot) Let us assume that Dt îs â ^domain ^below t_e^. Â ^solution X(t) ôf êquation ⁽¹⁾ ^with

an initial ondition X(te) innitesimally lose to φ(te) îs ^dened ât ^least ôn ^the S^-interior ôf Dt ^where ît îs

innitesimally losetoφ(t)^.

Letusapplythistheoremtothefollowingexample,hosenasthetypialexamplesatisfyinghypothesisH1

toH4(Hopfbifuration).

εx˙ = tx+y+εc1

εy˙ = −x+ty+εc2

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Theeigenvaluesareλ=t−iêtµ=t+i^. ^The^levelûrvesôf^the^two^reliefsRλ(t) = ¹2(t−i)²ândRµ(t) = ¹2(t+i)²

aredrawnongure2.

Figure2: Thelevelurvesofthetworeliefsof equation(3),andadomainbelowt_e

Generially 1

thereis nosurstabilityat pointt =i^(see ^[3℄^for^the^denition ^ofsurstability). Consequently, wehavethefollowingresultsforallequationssuhthatthereliefsareonthesametypeasthose ofgure2:

Denition3 Letusgive tc â^point² ^wherê ^the êigenvalue^vanishes: λ(tc) = 0^. ^The^value ôf ^the^relief ât^point tc îsâ^ritial ^valueôf ^the^relief Rλ^. ^The ^bump³ îs^the^real^number t^∗ ^bigger^thana^,^minimal^suh^thatRλ(t^∗)

isaritial value. Theanti-bumpisthe real numbert^∗∗ ^smaller^thana^,^maximal^suh^thatRλ(t^∗∗)^is^a^ritial

value.

1

Idonotknowtheexatgenerihypothesis. Wehavetoombinetheonstraintsgivenbythesurstabilitytheoryof[3℄andthe

fatthattheequation(1)isreal

2

Insomeases,itispossiblethattc^is^innite. ^ForεX˙ =

„ sint cost

−cost sint

«

X+O(X²) +O(ε)^we^havetc= +i∞^.

3

Thename"bump"isatranslationofthefrenhname"butée"

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Forequation(3),thebumpist^∗= 1^and^the^anti-bumpt^∗∗=−1^.

Theorem3 Atrajetoryof equation(1)an goalong the slowurve X=φ(t)êxatly ôn]te, t_s[ îfândônlyîf

oneofthe following isveried:

t_e< t^∗∗ ^and t_s=t^∗ t_e=t^∗∗ ^and t_s> t^∗

t^∗∗ < t_e< a ^and a < t_s< t^∗ ^and R(te) =R(ts)

Thistheoremisillustratedbythegraphoftheinput-outputrelation,drawnongure3.

Figure3: Theinput-outputrelationforequation(3)

3 Delayed Hopf bifuration followed by a fous-node bifuration

Thestudiedequationis

εX˙ =f(t, X, ε) ⁽⁴⁾

wheref îsânalytiônâ^domainDôf^C×^C²×^C,ând^satises^the^followinghypothesis:

Hypothesis and notations

HFN1 Theanalytifuntionf ^takes^real^values^when^the^arguments^are^real.

HFN2 Theparameterε^is^real, ^positive,innitesimal.

HFN3 There exists an analyti funtion φ^, ^dened ôn âômplex ^domain Dt ^suh ^that f(t, φ(t),0) = 0^. ^The

urveX =φ(t)îsâlled^the^slowûrveôfêquation^1. ^Weâssume^that^theintersetionofDt^with^the^real

axisis aninterval]tm, t_M[^.

HFN4 Letusdenoteλ(t)ândµ(t)^for^theeigenvaluesofthejaobianmatrixDXf^,ômputedât^point(t, φ(t),0)^.

Weassumethat ,fort^real,^the^signsôf^the^realândîmaginary^partsâre^given^by^the^table^below^:

t tm ^a ^b tM

ℜ(λ(t)) ^- ⁰ ⁺ ⁺ ⁺

ℜ(µ(t)) ^- ⁰ ⁺ ⁺ ⁺

ℑ(λ(t)) ^- ^- ^- ⁰ ⁰

ℑ(µ(t)) ⁺ ⁺ ⁺ ⁰ ⁰

Then, when t înreases ôn ^the ^real înterval ]tm, t_M[^, ^we ^have ^suesively ân âttrative ^fous, â ^Hopf

bifuration att =a^, â^repulsive^fous, â^fous-node ^bifurationât t=b ândâ^repulsive^node. Ât ^point t=b^,^the^twoeigenvaluesoinide. Weassumethatλ(t) =µ(t)ônlyât ^pointb^. Âtually,ⁱⁿ^theômplex

plane, the two eigenvalues are the two determinations of a multiform funtion dened on a Riemann

surfaewithasquarerootsingularityatpointb^.

However, there is asymmetry: if the funtion

√

is dened with a ut-o on the positivereal axis, it

satises

√s=−√

s^and^we^then^have

µ(t) = λ(t)

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R_λ(t) =ℜ Z t

a

λ(t)dt

and R_µ(t) =ℜ Z t

a

µ(t)dt

arethetwodeterminationsofamultiformfuntionwithasquarerootsingularityatpointt=b^. ^However,

there is a symmetry: if the funtion

√

is dened with a ut-o on the positive real axis, it satises

√s= −√

s ând ^we ^have^then: Rµ(t) = Rλ(t)êxept ôn ^the ût-o^half ^line [b,+∞[^. ^Fôr ^real t > b^,

wehoosedeterminationsofsquareroot suhthat λ(t)< µ(t)^. ^Weâssume^thatRλ ^hasâûnique^ritial

pointwithritialvalueRc^. ^Weâssume^that Rλ(b)< Rc^. Ânêxampleîs^givenând^studiedⁱⁿ^paragraph

3.2.1.

3.1 Input-output funtion when there exists a big anard

Weassume now that there exists abig anard X(t)˜ î.e. â ^solutionôf êquation ⁽¹⁾ ^suh ^that X˜(t) ≃φ(t) ^for

all t ⁱⁿ ^the S^-interior ôf ]tm, tM[^. ^The ^study ^below îs ^similar ^to ^paragraph ^2.1. ^The âdded ^diulty îs ^the

oinideneofthetwoeigenvaluesatpointb ^whih^do^not^allow^todiagonalizethelinearpart.

ThersthangeofunknownisX = ˜X(t) +Z ^whih^moves^the^bigânard ôn^theâxisX = 0^: εZ˙ = A(t)Z+O(ε)Z+O(Z²) ^, A(t) =D_Xf(t, φ(t),0)

Letusdenote

α(t) β(t) γ(t) δ(t)

theoeientsofthematrixA(t)^. Âsⁱⁿ^paragraph^2.1,^the^hangeôfûnknowns Z =

rcosθ rsinθ

, r = expρ ε

givesthenewsystem:

ρ˙ = α(t) cos²θ+ (β(t) +γ(t)) cosθsinθ+δ(t) sin²θ+O(ε) +e^ρ^εk1(r, θ, ε)

εθ˙ = γ(t) cos²θ+ (δ(t)−α(t)) cosθsinθ−β(t) sin²θ+O(ε) +e^ρ^εk2(r, θ, ε) ⁽⁵⁾

Fornonpositiveρ^(more^preisely,^forinnitesimalr^),^the^seondêquationîsâ^slow-fastêquation. Îts^slowûrve

isgivenby

θ = arctan





δ(t)−α(t)± q

α(t)²−2α(t)δ(t) +δ(t)²+ 4β(t)γ(t) 2β(t)





Ithastwobranheswhenλândµâre^reals, ôneîsâttrative,^theôtherîs^repulsive: ^see^gure^4.

When θ ^goes âlongâ ^branh ôf ^the^slowûrve, ^(and ^when r îs innitesimal),an easy omputation shows that ρ˙ îs înnitely ^lose ^to ône ôf ^the eigenvaluesλ ôr µ^. ^The ^repulsive ^branh ôrresponds ^to ^the ^smallest

eigenvalue(whih isreal positive). Whent < b^, ^theângleθ ^movesînnitely^fast,ând ânâveraging^proedure

isneededtoevaluatethevariationofρ^:

hρ˙i =

Rθ1+2π θ1

˙ ρ θ˙dθ Rθ1+2π

θ¹ 1 θ˙dθ

Aneasyomputationshowsnowthat,intheS^-interior^of^the^domaint < b^,ρ <0^,^we^have hρ˙i ≃ α(t) +δ(t)

2 = ℜ(λ(t)) =ℜ(µ(t))

Let us give an initial ondition (t, θ) ^between ^the ^two ^branhes ôf ^the ^slow ûrve ând ρ ^negative ^non

innitesimal(intheexample,weantaket= 0.8^,θ= 0^,ρ=−0.03^). ^For^inreasingt^,^the^urve(t, θ(t))^goes

alongtheattrativebranhoftheslowurve,whileρ^believes^negative^non innitesimal. Fordereasingt^,^the

solutiongoesalongtherepulsivebranh,thenθ^moves^innitely^fast^whileρ^believes^negative^noninnitesimal.

Consequently, we know the variation of ρ(t) ^(see ^gure ^4). Âs ⁱⁿ ^paragraph ^2.1, â^more ^subtle ârgument îs

neededtoprovethatwhenρ^beomesinnitesimal,thevariabler^beomes^noninnitesimalandthetrajetory

X ^leaves^theneighborhoodoftheslowurve.

Fromthisstudy, allthebehavioursof ρ(t)âre^known,^dependingôn^theînitial ôndition. ^Theyâre^drawn

ongure5.

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Figure4: Oneofthetrajetoriesofsystem0.002 ˙X =

t 1 t−0.3 t

X ^drawn ^with^the ^variables(θ, ρ)^. ^The

slowurveis alsodrawn

Figure 5: Thepossiblebehavioursofρ(t)^.

Proposition 4 Let us give an equation of type (6) with hypothesis HFN1 to HFN5. Assume also that there

exists a big anard X(t)˜ ^going âlong ^the ^slow ûrve ôn ^the ^whole înterval ]tm, tM[^. Îf â ^trajetory X(t) ^goes

alongthe slow urveexatly onan interval ]te, ts[^with [te, ts]⊂]tm, tM[^,^then Z ts

te

ℜ(λ(τ))dτ ≤ 0 ≤ Z ts

te

ℜ(µ(τ))dτ

Conversely, iftheinequalities abovearesatised,thereexistsatrajetorygoing alongthe slowurveexatly

on]te, t_s[^.

Theinput-outputrelationisdesribedbyitsgraph,drawnongure6.

We ouldgivemorepreise results ifweonsider the two variablesr ^and θ ^for^the input-output relation.

Indeed,whenthepoint(te, ts)îsⁱⁿ^theînteriorôf^the^graphôf^theinput-outputrelation,weknowthat,attime ofoutput,θ îs^goingâlong^theâttrative^slowûrve^whihôrresponds^to^theûnique^fast^trajetory^tangent^to

theeigenspae ofthebiggesteienvalueµ^.

3.2 The fous-node bifuration is a bump

Hereisthemainpartofthisartile. Today,Iamnotabletoprovetheexpetingresults,butIhavepropositions

inthisdiretion. Toexplaintheproblem,I willgiveonjetures.

Letusdenetheanti-bump t^∗∗ ând^the^two^bumpst^∗_λândt^∗_µ âsⁱⁿ ^denition^3:

Rλ(tc) = Rµ(tc) = Rλ(t^∗∗) = Rµ(t^∗∗) = Rλ(t^∗_λ) = Rµ(t^∗_µ)