Singular Perturbations of Bifurcations With Multiple Independent Bifurcation Parameters

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Singular Perturbations of Bifurcations with Multiple Independent Bifurcation Parameters

Author(s): Robert W. KolkkaSource: SIAM Journal on Applied Mathematics, Vol. 44, No. 2 (Apr., 1984), pp. 257-269Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2101158 .

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SIAM J. APPL. MATH. ? 1984 Societyfor ndustrial nd Applied MathematicsVol. 44, No. 2, April 1984 003

SINGULAR PERTURBATIONS OF BIFURCATIONS WITHMULTIPLE INDEPENDENT BIFURCATION PARAMETERS*

ROBERT W. KOLKKAt

Abstract.There are severalnonlinear ifurcationroblemswhich nvolvemultiple ifurcationaram-eters "specified nputs").Often here s a correspondingperturbed" roblem ssociatedwith hebifurcationproblemwhichmodels mperfections.nstead of bifurcation,he perturbed roblem xhibits smoothbutrapid transition n the criticalrange of the bifurcation arameter.A new generalization f the singlebifurcation arametermethod of Matkowsky nd Reiss, SPB ("SingularPerturbationsf Bifurcations")[SIAM J.Appl. Math.,33 (1977), pp. 230-255] is employed o solve theproblem ftheslightlyrookedrotating lastica shaft) ubject to a dead load appliedat the ends.

1. Introduction. here are several problems hat rise n the study f nonlinearphenomenawhich xhibit he property f bifurcation. fundamentalusuallytrivial)

state existsforall values of some specified nputparameters,nd for certain pecificcombinations f values,nontrivial olutionsbranch from he basic state.Up to thepresent, heoverwhelming ajority fproblems reated ontained singlebifurcationparameter. herelativelyewmultiple arameter roblemswhich avebeenattempted,have been confined trictlyo numerical nalysis.However, t is difficulto ascertainthedependence f olutions n theparametersfa problem rom numericalomputa-tion,whileasymptoticnd perturbationmethodsusually larifyhisdependence.

A multipleperturbation arametermethodfor analyzingmultipleparameterbifurcation roblemshas recently een developedbyPlaut [1]. It was presented n

the contextof postbucklingnalysisof elasticsystems, ut the theorydeveloped isapplicable n generalto severaltypes fnonlinearbifurcationroblems.Bifurcation roblemsexhibit everal important eatures f the actual physical

systemwhich heymodel, however he sharptransitionsi.e., the actualbifurcations)rarelyoccurin experiments.n any physical ystem here s alwayssome degreeofimperfectionnherent n the system.The imperfectionsimpurities, eometric evi-ations, noise, etc.) accountforthe smoothtransitions bserved n experiments.Werefer o the bifurcation roblemwith mperfectionss theperturbed roblem. t ischaracterizedmathematicallyya parameter , and expressed s

(1.1) G[w; An; 8]=0.The operatorG is a nonlinear peratordefined n an appropriateHilbert pace offunctions,w is the solutionvector, nd An,n= 1, 2,... , m,are the m bifurcationparameters. he dependenceof w on spatialand time-like ariables s suppressednthisnotation.

The generaltheory oranalysis fproblems f thetype 1.1), wherem= 1 (i.e.a singlebifurcationarameter)wasoriginally iven n 2], and therehave beenseveralsubsequent pplications3]-[6] of thebasicmethod,whichyieldsuniformsymptoticrepresentationsf he olutions o theperturbed roblem or llvaluesof hebifurcation

parameter.Since theappearanceof[2], therehave been variousgeneralizationsf the basic

method.Rosenblatand Cohen [7] have analyzedthe problem n which teadystatebifurcation ranches re perturbed y timeperiodic mperfections.inayand Reiss

*Receivedby the editorsFebruary 4, 1983.t Department f Mathematical ciences, Rensselaer Polytechnic nstitute, roy, New York 12181.

Present ddress,Department fMathematical ciences, ndiana University-Purdueniversity,ndianapolis,Indiana46223.

257



258 ROBERT W. KOLKKA

[8] have extended hemethod n thecase where thelinearized peratorevaluated atthe criticalpointhas two linearly ndependent olutions.This propertyed to sig-nificantlyifferentesults. n thiswork we extendthemethod o the case ofseveralindependent ifurcationarameters.We analyzethecase of two ndependent ifurca-

tionparameters orthe sake ofsimplicitynd clarity fpresentation,o we have

(1.2) G[w; A, u;5] = O.

The extension o threeormore ndependent ifurcationarameterss a trivialmatteras shown n [1].

It shouldbe notedthat n [1] some important esults ftheperturbed roblemare obtainedand are in complete greementwith hecurrentnalysis.However,themethodemployed n [1] does not yielduniform symptotic epresentations f thesolutions o theperturbed roblemfor ll values ofthebifurcationarametersA and

,u,whereasthecurrentmethoddoes.2. Formulation.We consider heproblem f a slightlyrooked initial tress-free

wrinkling)haft ubjectedto a driven peed ofrotationw about the ongitudinalxisof theshaft, nd time-independentnd thrusts , as illustratednFigure2.1.

1t1 1

J t

.~A . .* g

I ,\ toI

-~~ - 777Th7XFIG. 2.1

The exactElastica (Bernoulli-Euler)theory s employed, nd we seek dynamicequilibriumolutions, .e., solutions hatare time-independenthenviewed fromcoordinate ystem otatingwith the shaft.The ends of the shaft re constrained nsucha fashion o assurethattheshaft recesseswith he

driving ngularvelocity.This systemwe consider s a conservativeystem, nd the governing quationsmaybe derivedvia a variational rinciple r simple quilibriumonsiderations. itherway, hegoverningquationfor he additional ransverse isplacementW(X), whereX is arc length,s

(2.1a)

F _ XX 11 WX+8Wo A



BIFURCATIONS WITH MULTIPLE INDEPENDENT PARAMETERS 259

X as a subscript enotesdifferentiationithrespect o X. The constants , I, and pare theYoung's modulus, ross-sectionalmoment farea, and massperunit ength,respectively.WO(X) is the formof the initialstress-free rinkling,nd 5(>O) itsmagnitude. he simply upportedboundary onditions re

(2. 1b) W(O) = Wxx 0)= W(l) = wxx (1)= ?,

where1 s the ength fthe assumed nextensible)haft. mployinghenondimension-alization

W(X) WO(X) P12 pwo214 X(2.2) w(x)= , g(x) = 1X, A= EI ' X=l

we obtain thedimensionless roblem,

(2.3a) [(i) +A ( wI+5g) I) /-kw =

(2.3b) w= w"=O, x=O, 1,

where *)' denotesdifferentiationith espect ox.Themagnitudef the mperfectionis assumedto be small. Withthis ssumption, etainingerms f at mostthird egreeleads to theproblemwe solve,

(2.4a) w""(1 w' ) +4 w"'w"w' w"3+ Aw"(1 3w' ) w+ A8g"-gg = 0.

(2.4b) w= w"=O, x=O, 1.Thewrinklingistribution(x) is taken o be compatiblewith heboundary onditions(2.4b) and to satisfy ecessarycontinuity equirements,o that g admits to theappropriate igenfunctionxpansions.Thus (1.2) is givenby 2.4), and we refer o itas theperturbed roblem.

3. The bifurcationroblem.The bifurcationroblem sgivenby 1.2) with = 0,i.e.,

(3.1) G[w; A, ;O]--F[w; A,

]=0

whereF is givenby

(3.2) F[w;A,,u]=w""(1+w'2)+4w"'w"w'+w"3+Aw"(1+3w'2)-,uw=O,

subjectto theboundary onditions2.4b).Following laut 1],weseekasymptoticolutions f 3.2) via the doubleperturba-

tionparameter xpansion,

1 22

w(x)=

Ew1(X)+-

[? 1X)+ 2EWl2(X)+ q

W22W]2!(3.3a)

+- [E wWi(x) + r w222(X)+ 3 ? W122(X)+ 3qE W211(X)]+ 0(s4),3!

(3.3b) A= Ac+A1E +A27 (A 1E2+ 2A12E71 +222772)

+ O(S),2!

(3.3c) -= -c+q




where 2=(r + )2. The perturbationarameterE is defined y

(3.4) 2 = (w, w),

where f,g) is an appropriatennerproduct, ivenby

(3.5) ( f,g) f(x )g(x) dx.0

The perturbationarameter 1 sdefined y 3.3c). The leadingorder ermsn 3.3b,c)

A, tL lie on a path,

(3.6) r(AcC) = 0

intheA, t plane,and r is determined ythe inearization f 3.2). The linearizationof (3.2) is

(3.7) w"" Aw"1-tw = 0,

togetherwith heboundary onditions2.4b), whichwill be takento be understoodfrom hispointon,yields hepaths longwhichbifurcationakesplace.The nontrivialeigenfunctionsre

(3.8) w(x) = On(x) = 2 sinnlrx), n = l, 2,3,*

Substitutionf (3.8) into 3.7) givesthesequence ofpaths,

(3.9) r(A ,--(n7Tr)4 A n,7 r2-U = O n =1, 2, ***

alongwhich ifurcationakesplace.We cantakeA, 00withoutny ossofgeneraliz-ation,physics ictates hat u 0 anyway. he critical ath s defined s theone whichhas thepoint hat iesclosest o theorigin. he criticalnteger ,rsdeterminedrom

(3.10) ncr min minpn(A,H)n A,gK

wherepn(Ak,) is the distancefrom heorigin f a pointA,u which ies on thepath

rn(A,A). In this ase we find hatncr= 1 and thus

(3.11) r(A (lc)=r (kc, to = X -Akc2 -1C = ?

Equation (3.11) is simply straightine (Fig. 3.1) which ntersectshe axes at theirrespective single" bifurcation oints.Straight ines need not always be the casehowever, speciallywhendealingwithnonconservative roblems s shown n[1].

Substitution f (3.3) into (3.2) leads to a sequence of linear boundaryvalueproblems o solve for he unknownsw1(x),wij(x),Wijk(x),nd Aj,Aij,, , k= 1, 2, etc.The first rderequationsatisfies




A

l4

IT2~ ~ ~ ~~~~~~~~~~~~I .

FIG. 3.1

whereL(A, 1ui)fs defined y

(3.13) L(A,A)f f"f Akf'-Zf.

The solutionforw1, hus, s

(3.14) wl(x) = 4(x) =1/2sin lTx).

The second orderequationsare found o be

(3.15a) L(A, ,c) w11=-2A I,

(3.15b) L(Aci JcW22 0,

(3.15c) L(AC, A) w12 -A24"+ 4'.

We see that 3.15a, c) are the nhomogeneous orms f 3.12) evaluated tthesingular

point, and the Fredholmalternativerequires that the inhomogeneous ermsbeorthogonal o all solutions fthecorresponding omogeneous djointproblem.Thelinearproblem onsideredhereis self-adjoint,o we have

(3.16) -2A1(4", 0) = 0,

(3.17) -A2(4", 0)+(4, 4)=0,

(3.18) Al= 0,

(3.19) 12= 2IT

The problems orwij(x), i, = 1, 2, with 3.18), (3.19), become

(3.20) L(AC,AJ)wi= 0,

so we take




Proceeding n thisrecursive ashion,we obtainthe third rderequations

(3.22a) L(AC,wc) lll = - 3Al4?"- 6?,,.. I2- 24"'.". ' - 64"3- 9A,4C"42,

(3.22b) L(AC, c) W2220,

(3.22c) L(AC, c)w122 - A220",

(3.22d) L(Ac,1c) w211 - 2A 2 )"

Invoking heFredholm lternative n (3.22), we find hat

(3.23) All = 2T4 - 3ACT2> O,

(3.24) A22 ?

(3.25) A12 O-

Higher orderterms re calculated n exactly he same manner.Summarizingesults p to thepresent rder,we have from3.14), (3.18), (3.19),

(3.21), (3.22), (3.24) and (3.25),

(3.26) w(x) = ?+(X) + O(E ),

(3.27) A=AC 2 0(s3)IT

(3.28) =c + q.

The dependenceof theamplitude on theparametersA, ,u can now be ascertained

bysimply liminating from3.27) and (3.28), the result s

(3.29) E = ?E(A + ) =i

- 43)AIT2+p,-IT4)1/2IT O 34

and ssketchednFig.3.2.We alsonotethat nthe imit s either arameterpproacheszero,we recover he ndividual ingleparameter ifurcations,

(3.30) E(A, 0) =+ (A T2)1/2;IT

(3.31) ?(0, ) = 3 ( _TIT

RESPONSE IAGRAM

A

V/C(A, )u) > O

FIG. 3.2(A,)<

FIG. 3.2




The definitions f the standard lasses of singleparameterbifurcations; uper-critical, ranscritical,nd subcritical re extended nthe obviouswayas shown nFig.3.3.Thustheproblem3.2) exhibits supercriticalifurcationromhepath 3.11).

(a) (b) (c)

SUPERCRITICAL TRANSCRITICAL SUBCRITICAL

FIG. 3.3

4. The outer xpansions.The outer xpansions re theasymptoticpproximationto the solution f theperturbed roblem wayfrom heneighborhood f thecriticalpath 3.11).

We proceed in a mannerdirectly nalogous to the systematics f the singlebifurcationarameter ase, completedetailsof which re given n[2]-[6]. Bifurcationsub-branch urfaces re defined s the wounbuckled lat

urfaces (A, u) 0,AIT2 +

/

-

IT4< 0, and AlT2+ , _ T4 >0, and as thetwo buckled onfigurations,(A, ,) > O,E (A,g) < 0, forAT2 + g _- T4> 0. The flat urfaces re referredoas theprimaryranchsurfaces, nd the buckledsheets > 0, E > 0, as thesecondary ranch urfaces.

Asymptoticepresentationsf theform

00

(4.1) w(x; A,;)= wo(x;A,y)+ wi X;A,j=1

are sought,wherewo(x;A,g) is theparticularub-branch urface boutwhichwe are

expanding. ubstitution f (4.1) into (2.4) generates sequence of linearboundaryvalue problems orthedeterminationf wj(x; A,). The problemforw1 s

(4.2) L(A, /) w1 M(A, /, wo)w1 k-Ag"+ Kg,

where

M(A, g, wo)w,= w'2wl + 4wi fwI' + (4w' w' + 3w+ +3Aw,2) wf

+(2 wl"'Wo 4w..w + 3Aw w') wI

The primary ranch urfaces re wo 0, so (4.2) simplifieso

(4.4) L(A, k)wP=--Ag"+ lg,

wherethesuperscript is attached o w1to denoteprimarylikewise superscriptwill aterbe attached odenotesecondary). rom theform f 4.4), we seek a solutionvia an eigenfunctionxpansionoftheform,

00

(4.5) WPx; A,/x) c= E CX(x),n=1




where0, (x), givenby 3.8), satisfies

(4.6) 4) + On' O'nn ?

and the eigenvalues -n re givenby(4.7) o=n= (ng)4_(ng)2.

The coefficientsnare readily btainedfrom ubstitutionf 4.5) into 4.4); the result1S

(4.8) Cn CnA, E) = (n)4-A (nT)2_

and'

(4.9) -= (g9, kn).

We expressthe resultwl as

(4.10) w (x; A,A

where

00

(4.11) T(x; A, y)=E CnA,A) OnX)n =2

Thus we see thatwl becomesunbounded n theneighborhood f the criticalpath,from hefirstermn 4.10). Equation 4.10) is theouter xpansion or heflat urfaceswo0 O(AT2+,1-7r4<0, AiT2+ -Tr4>0) validawayfrom heneighborhood fthecritical ath.

The secondarybranch urfaceshave wo(x;A, ) = (A, 1u)4(x), so theproblem(4.2) forw (x; A,/x)becomes

(4.12) L(A, k)wl + 9M(A,,

4)wl-Ag"+w g.To solve for wi, we define n appropriate igenvalueproblem,motivated rom heform f (4.12) as

(4.13) L(A, I+ 2(A )m = TMOrn m= 1,2,* .

The eigenvalues rmand eigenfunctions i/mx) are determined asymptotically becausetheproblem s nonconstantoefficient,nd we are interestedn ? small anyway)viaa regularperturbationxpansion n E; that s

2

(4.14a) 4mx) = 4OmX) + -- (X) + 2! qn Xm O?2!

2

(4.14b) ~ m ~m + ETrn 2L! nO(3)

' Throughout his nalysiswe will ssumeg, i4O,theprocedure org, = 0 canbe modifiedn accordancewith 4].




Omitting hedetails, heresults re, asymptotically,

(4.15a) m(X) =(>(X) + 0 (_ 2) m = 1, 2, 3,***

Irm 7r4 A T2_

#? (7r6 4 e

2 + 0 (8 3) m=1,(4.15b) 9Ai= )

((mr)4_-A (m7r)2-_U + 0(?2), m=2,3,4,*.

The solutionforw' is givenby theeigenfunctionxpansion

00

(4.16) w' (x; A,,uk)= bmm x),m=1

where

(4.17) bm bm(A,/k) Agm gm

and

(4.18) gm-(g, rm) gm (g qm).

Re-expressing4.16) as

(4.19) w~(x;A) =7r4-A rIT2

- + [3T

(9(A1r4/4)]82(A, )

where00

(4.20) S(x; A, u) 2 bm(A, lk) tfmm(x),m=2

we note from he first erm of (4.19) thatthe outer expansionsforthe secondarybranch urfacesE > 0, < 0) likewisebecome unbounded n the mmediate eighbor-hood of thecritical ath.

In order to completethe solutionof theperturbed roblem,we mustconstructexpansionswhich re valid n thevicinityf the critical ath, nd which onnectwith

the outerexpansions s we exittheneighborhood f thecritical ath.Theyare calledthe nner xpansions nd are calculated n thenext section.

5. The inner xpansions.We unfold hebehavior fthe olution boutthe riticalpathbymeansofa double smallparameter xpansion,

w(x) = az1(x) + 2 [la2z11(x) 1832z22(x) + 2a,8Z12(x)]2!(5.1a)

+ 3a z111(x) +133Z222(x) + 3a132z122(X) + 33C82Z211(X)]+ 0(t4),3!

(5.1b) A A +AI28+ 1 (a2+0(a3)

(5.1c) y = c+ }3

(5.1d) aa3,

where, (a + p 2. The coefficient is the nnervariableof theMMAE (Method ofMatchedAsymptoticxpansions) 9]; it s a measureofdistance rom hecritical ath(3.11). The particularorm f 5.1 b,d) stems rom hefact hat hebifurcationroblem




A1(f) is theunique real root f < (*,

(5 11 ( _q) 2/3 1) 1/3

if exceeds *, then here re two additional ootsA2(W),A3(W).Their ign soppositeto thesignofA1(f). In addition, heserootssatisfy

(5.12) A2(W)- 0, IA3( I -> 00, (e oo, A2(*) =A3 ).

The relationship etweenf and the outervariablesA,,u,can now be obtainedfrom se oftheresult 5.5) in (5.1), and is found o be

2

(5.13) ATr2+ /-Ir4 2/3+ oW13)

The "unbuckling" ath,as itwouldbe referredo, is givenby 5.13), with = *, toleadingorderas

(5.14) *7r+ u* lr = 7

and is sketched nFig. 6.2.

6. The matching. everal inner nd outerexpansionshave been obtained.Theremainingetailofthe nalysisstosimply etermine owthey onnect.We accomplishthisdetailbythestandardprocedures ictatedbytheMMAE.

The asymptoticorms f theouterexpansions, rom4.10), (4.19), are givenby

(A,T2+ ~414k0(X)8(6.1) wpX; A,tu; ,8)~(l +IZ4-AIT2-

(6.2) w,X; A 7r; +) -_

3 2)1/2 (-24-AX

We now express 6.1), (6.2) in terms ftheinnervariables 5.1), theresult s

(6.3) wp 2l2g1 Oa

(6.4) WS-?12~a _____, O

(6.4) ~~~~~~(4r -3Ac7r2)1/2

The asymptotic orms f theamplitude fthe nner xpansions re givenby

lim A1(f) - g- , limA1(f) - /2_e<-00 f-->00 e< o4Tr4-3ACT2

(6.5) 2

limA2(W)- -2IT g1 limA3(W) - -/2

(<->00 f e<>o /474-3ACIT2

Simplecomparison f (6.3), (6.4) with 6.5) reveals which nner xpansions onnectwithwhich outer expansions,and is illustrated n Fig. 6.1. The uniformly alidasymptoticomposite xpansion s sketched nFig. 6.2.




MATCHING

01i

((w)) 0j 91>o

I3A ((W))X=(W,w)

Slice in (( w)), A, u space

FIG. 6.1. The curves0 are theouter xpansions orrespondingo theunbuckled onfiguration.hecurvesO' are theouter xpansions orrespondingothebuckled onfigurations.hebuckled olutions fthebifurcationroblemre indicated ythehatch-markedurves.Thedashedcurves re inner xpansions.The

curve , correspondsothe nner xpansionwith mplitudeA,(().

((W)) A

'6 0 ~ ~ ~ ~ ~ ~ >

FIG. 6.2. The composite xpansion esponse iagram.

7. Concludingremarks.We firstnote that in the limit as eitherparameterapproaches ero,the ndividual inglebifurcationarameter erturbed roblem esultsare recovered.The rolesofAand ,u may, fcourse,be interchanged ithout ffectingthe results.

The time-dependentinearized ersion f the rotatinglastica without mperfec-tions)has been analyzedby Huseyin 10]. The problem onsidered herewas simplytheLDSA (LinearizedDynamic tability nalysis) fthebasic statew 0. The resultstheredemonstrateheexistence f flutternstabilitytates,whichbifurcate rom he




basic state.The present nalysisdoes not rule out the existence f such states; theyare simply otsought.These nonlinear ibrating ifurcationtateswillbe thesubjectof futurework.

In conclusion,we remark hatthe analysisused hererequires finite ath along

whichbifurcationakes place. Problems nwhich he branch urfaces emaindiscon-nected from he flat urface w 0) for all values of theparametersA and ,t (i.e. atwo-dimensional ifurcation rom nfinity11]) do notadmitto thismethod.

8. Acknowledgments. he authorwishes o expresshisappreciation or he veryenlighteningiscussions ith rofessors aymondH. Plaut,Dept. ofCivilEngineering,VirginiaPolytechnicnstitute nd State University, lacksburg,VA., and Edward L.Reiss, Dept. ofEngineeringciences ndApplied Mathematics, orthwesternniver-sity, vanston, L.

REFERENCES

[1] R. H. PLAUT, Postbucklingnalysis ofcontinuous, lastic ystems ndermultipleoads,Parts 1 & 2,J.Appl. Mech., 46 (1979), pp. 393-403.

[2] B. J.MATKOWSKY AND E. L. REISS, Singular erturbationsfbifurcations,hisJournal, 3 (1977),pp. 230-255.

[3] J.TAVANTZIS, E. L. REISS AND B. J.MATKOWSKY, On smooth ransitionoconvection,hisJournal,34 (1978), pp. 322-336.

[4] E. L. REISS, Imperfectifurcation,dv. Seminar on Applications fBifurcation heory,AcademicPress,New York, 1976, pp. 37-72.

[5] W. B. DAY, Bucklingofa columnwithnonlinear estraintsndrandom nitialdisplacement, riefNote,J. Appl. Mech., 47 (1980), pp. 204-205.

[6] J. G. WATSON AND E. L. REISS, A statisticalheoryor mperfectifurcation,hisJournal, 2 (1982),pp. 135-147, Feb. 1982.

[7] S. ROSENBLAT AND D. S. COHEN, Periodically erturbedifurcation-1.Simplebifurcation,tud.Appl. Math., 63 (1980), pp. 1-23.

[8] L. R. SINAY AND E. L. REISS, Perturbedanel flutter: simplemodel, ubmitted.[9] A. H. NAYFEH, Perturbation ethods,JohnWiley,New York, 1973.

[10] K. HUSEYIN, Vibrationsnd Stability f MultipleParameter ystems,Noordhoff, roningen, heNetherlands, 978, pp. 142-152.

[11] S. ROSENBLAT AND S. H. DAVIS, Bifurcationromnfinity,hisJournal, 7 (1979), pp. 1-19.

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