Singular Perturbations of Bifurcations with Multiple Independent Bifurcation Parameters Author(s): Robert W. Kolkka Source: SIAM Journal on Applied Mathematics, Vol. 44, No. 2 (Apr., 1984), pp. 257-269 Published by: Society for Industrial and Applied Mathematics Stable URL: http://www.jstor.org/stable/2101158 . Accessed: 23/12/2010 15:01 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=siam . . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Society for Industrial and Applied Mathematics is collaborating with JSTOR to digitize, preserve and extend access to SIAM Journal on Applied Mathematics. http://www.jstor.org
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8/8/2019 Singular Perturbations of Bifurcations With Multiple Independent Bifurcation Parameters
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 .
Accessed: 23/12/2010 15:01
Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless
you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you
may use content in the JSTOR archive only for your personal, non-commercial use.
Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at .http://www.jstor.org/action/showPublisher?publisherCode=siam. .
Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed
page of such transmission.
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of
content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms
of scholarship. For more information about JSTOR, please contact [email protected].
Society for Industrial and Applied Mathematics is collaborating with JSTOR to digitize, preserve and extend
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.
[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
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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
where *)' denotesdifferentiationith espect ox.Themagnitudef the mperfectionis assumedto be small. Withthis ssumption, etainingerms f at mostthird egreeleads to theproblemwe solve,
(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.,
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
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BIFURCATIONS WITH MULTIPLE INDEPENDENT PARAMETERS 261
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
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BIFURCATIONS WITH MULTIPLE INDEPENDENT PARAMETERS 263
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
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
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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].
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BIFURCATIONS WITH MULTIPLE INDEPENDENT PARAMETERS 265
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.
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
8/8/2019 Singular Perturbations of Bifurcations With Multiple Independent Bifurcation Parameters
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.
8/8/2019 Singular Perturbations of Bifurcations With Multiple Independent Bifurcation Parameters
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
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BIFURCATIONS WITH MULTIPLE INDEPENDENT PARAMETERS 269
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.