Asymptotic Solutions of Strongly Nonlinear Systems of Differential Equations

278 Pages·2013·1.592 MB·English

by  V V Kozlov, Stanislav D Furta, Lester Senechal

Springer Monographs in Mathematics
Forfurthervolumes:
http://www.springer.com/series/3733
(cid:127)
Valery V. Kozlov
Stanislav D. Furta
Asymptotic Solutions
of Strongly Nonlinear
Systems of Differential
Equations
123
ValeryV.Kozlov StanislavD.Furta
SteklovMathematicalInstitute FacultyforInnovativeandTechnological
RussianAcademyofSciences Business
Moscow RussianAcademyofNationalEconomy
Russia andPublicAdministration
Moscow
Russia
Translation of the 2nd Russian original edition entitled “Asimptotiki reshenij sil’no
nelinejnykh sistem differentsial’nykh uravnenij”, published jointly by RCD (Regular and
Chaotic Dynamics), Izhevsk, and IKI (Institut Kompyuternykh Issledovanij), Moscow, in
2009
ISSN1439-7382
ISBN978-3-642-33816-8 ISBN978-3-642-33817-5(eBook)
DOI10.1007/978-3-642-33817-5
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2012954017
MathematicsSubjectClassification(2010):34D05,34DXX
(cid:2)c Springer-VerlagBerlinHeidelberg2013
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Translator’s Note
Thetranslatorhasworkedwiththetheoryofdifferentialequationsanditsapplica-
tionsformanyyears,andyetthetranslationofthepresentbookpresentedsignificant
challenges, for from the outset new territory needed to be explored and many
new ideas assimilated. At each iteration, in reviewing the translation, one or two
subtletieswerediscoveredthatcausedachangeinmeaningoremphasis.Hopefully
thisiterativeprocesshasconvergedtoaresultthatfaithfullyrepresentstheRussian
original,albeitwithoutitseloquence.
The translator has taken the opportunity to prepare an introductory survey
lectureforthematerialofthebookthatisdirectedtowardcolleaguesfromvarious
disciplines (since much of the stimulus for the range of problems considered in
thebookcomesfromthephysicalsciences)andwillwelcomeopportunitiesforits
dissemination.Questionsandcorrectionsforthebookcanbesenttothededicated
e-mailaddress:[email protected].
Newmarket,NH,USA LesterJ.Senechal
Badenweiler,Germany
v
(cid:127)
Preface
For us, the authors of the monographAsymptotic Solutionsof Strongly Nonlinear
Systems of Differential Equations—the first Russian edition (1996) by Moscow
University Press and the second edition (2009) by the publisher R&C Dynamics
(http://shop.rcd.ru)—thedecision by Springer-Verlag to publish an English trans-
lation of the book is an important event. It is not merely an offspring born from
thepangsofcreativestruggleandthusafavorite,fortherearemanybooks,butwe
presenttotheWesternreaderamonographthatisaveryspecialbook.
Inwhatwayspecial?First,evenatthemomentthefirstRussianeditionappeared
in1996,itwasthefruitofadecadeandahalfofresearch.Thefirstpublicationof
V.V.Kozlovonthesubjectgoesbackto1982[103,104,117]andisdedicatedtoa
very important fundamentalproblem: the inversion of Lagrange’s theorem on the
stabilityofequilibrium—notatrivialtask,overwhichresearchershadstruggledfor
morethanhalfacentury.Theideabehindthat1982paperbelongstoN.G.Chetaev:
“To provethe instabilityof equilibriumis to findjust onetrajectoryof the system
thattendstotheequilibriumpositionastimedecreasesindefinitely”.Buttheword
“find” is easily said! The 1982paper showedthat the solution of the equationsof
motionofa naturalmechanicalsystem,with somerestrictions,canbeconstructed
in the form of generalized power series whose convergence is proved by quite
sophisticatedmethodsoffunctionalanalysis.
For a long time our effortswere devoted to this issue and matters related to it.
Yearspassed,andatlastitdawneduponusthatthemethod—developedtoaddressa
veryfamous,interesting,andyetparticularproblem—wasinfactuniversal,giving
scientists the opportunity to “view from above” the problem of the asymptotic
behaviorofdifferentialequationsintheneighborhoodofanonelementarysingular
point.Inaddition,itwasdiscoveredthatthetechniqueusedinthe1982paper,which
seemedsosuccessfulandevenelegant,hasadeepconnectionwiththefundamental
works of such great classics as Lyapunov and Poincare´. This “view from above”
whichwehaveperhapsonlybythegraceofGod,gaveaccesstoanimmensenumber
ofapplications.Todaywescarcelycanfindanexamplefromanyofthepreviously
studied critical cases of the stability of equilibrium of autonomous systems of
ordinarydifferentialequationsforwhichsufficientconditionsforinstabilitycannot
vii
viii Preface
beobtainedbythemethoddeveloped.Moreover,wewereabletoobtainconditions
for instability in some previously unexplored cases and it was found that the
methodworkswellforperiodicandquasi-periodictime-dependenceofsystemsof
differentialequations.Wealsomanagedtomakesignificantprogressontheproblem
ofinvertingLagrange’stheoremonthestabilityofequilibriumandits“littlesister”:
Routh’s theorem,for which—as we know by the existence of the phenomenonof
gyroscopic stabilization—the converse is simply not true without the imposition
of additional conditions. Finally, the method provides enhanced results on the
instability of equilibrium for systems with time lags. Work on the method led (as
so often happens)to unexpectedsecondaryeffects: it turned outthat the behavior
ofsolutionsin the neighborhoodof a trajectory,whose existenceis guaranteedby
themethod,canindicatewhensystemsarenonintegrableorchaoticinonesenseor
another.
We are absolutely certain that our published work ends not with a period but
rather with a comma and that the possibilities that the method offers are far from
exhausted. We therefore think of the publication of our monograph by Springer-
VerlagasaninvitationtoWesternscholarstocontinueresearchinthisdirection.To
the great Russian poet Sergei Yesenin belong the words: “big things can only be
seenata distance”.We thushopethatourreaderscanbringafreshperspectiveto
themethodologythatwehavedevelopedandextendtherangeofproblemsthatcan
besolvedwithit.
We are deeply grateful to Springer-Verlag and in particular to Ruth Allewelt,
withoutwhoseperseverancethe preparationtime forthe book’spublicationmight
havebecomeasymptoticallyinfinite,andtothe translator,LesterSenechal,whose
interestandassistancehavehelpedpreserveourhopesforthebook’srealization.
Moscow,Russia V.V.Kozlov
S.D.Furta
Contents
1 Semi-quasihomogeneousSystemsofDifferentialEquations............ 1
1.1 Formal Asymptotic Particular Solutions of Semi-
quasihomogeneousSystemsofDifferentialEquations................ 1
1.2 ProblemsofConvergence............................................... 13
1.3 ExponentialMethodsforFindingNonexponentialSolutions......... 24
1.4 Examples................................................................ 41
1.5 GroupTheoreticalInterpretation....................................... 55
2 TheCriticalCaseofPureImaginaryRoots............................... 77
2.1 Asymptotic Solutions of AutonomousSystems of
DifferentialEquationsintheCriticalCase ofmPairs
of Pure Imaginary and n(cid:2)2m Zero Roots of the
CharacteristicEquation................................................. 77
2.2 PeriodicandQuasiperiodicSystems................................... 92
2.3 HamiltonianSystems ................................................... 108
3 SingularProblems ........................................................... 131
3.1 Asymptotic Solutions of Autonomous Systems
ofDifferentialEquationsintheCriticalCase ofZero
RootsoftheCharacteristicEquation................................... 131
3.2 ConcerningIteratedLogarithms........................................ 143
3.3 SystemsImplicitwithRespecttoHigherDerivatives
andKuznetsov’sTheory................................................ 152
4 InversionProblemfortheLagrangeTheorem
ontheStabilityofEquilibriumandRelatedProblems .................. 169
4.1 OnEnergyCriteriaforStability........................................ 169
4.2 RegularProblems....................................................... 189
4.3 SingularProblems....................................................... 200
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