Table Of ContentEncyclopedia of
Complexity and Systems Science Series
Editor-in-Chief: Robert A. Meyers
Giuseppe Gaeta
Editor
Perturbation
Theory
Mathematics, Methods and Applications
A Volume in the Encyclopedia of
Complexity and Systems Science,
Second Edition
Encyclopedia of Complexity and
Systems Science Series
Editor-in-Chief
RobertA.Meyers
The Encyclopedia of Complexity and Systems Science series of topical vol-
umes provides an authoritative source for understanding and applying the
concepts of complexity theory, together with the tools and measures for
analyzing complex systems in all fields of science and engineering. Many
phenomenaatallscalesinscienceandengineeringhavethecharacteristicsof
complexsystems,andcanbefullyunderstoodonlythroughthetransdisciplin-
aryperspectives,theories,andtoolsofself-organization,synergetics,dynam-
ical systems, turbulence, catastrophes, instabilities, nonlinearity, stochastic
processes, chaos, neural networks, cellular automata, adaptive systems,
genetic algorithms, and so on. Examples of near-term problems and major
unknowns that can be approached through complexity and systems science
include:Thestructure,historyandfutureoftheuniverse;thebiologicalbasis
ofconsciousness;theintegrationofgenomics,proteomicsandbioinformatics
assystemsbiology;humanlongevitylimits;thelimitsofcomputing;sustain-
abilityofhumansocietiesandlifeonearth;predictability,dynamicsandextent
ofearthquakes,hurricanes,tsunamis,andothernaturaldisasters;thedynamics
ofturbulentflows;lasersorfluidsinphysics,microprocessordesign;macro-
molecularassemblyinchemistryandbiophysics;brainfunctionsincognitive
neuroscience; climate change; ecosystem management; traffic management;
and business cycles. All these seemingly diverse kinds of phenomena and
structure formation have a number of important features and underlying
structures in common. These deep structural similarities can be exploited to
transferanalyticalmethodsandunderstandingfromonefieldtoanother.This
uniqueworkwillextendtheinfluenceofcomplexityandsystemsciencetoa
muchwideraudiencethanhasbeenpossibletodate.
Giuseppe Gaeta
Editor
Perturbation Theory
Mathematics, Methods and
Applications
A Volume in the Encyclopedia of Complexity
and Systems Science, Second Edition
With99Figuresand4Tables
Editor
GiuseppeGaeta
DipartimentodiMatematica
UniversitàdegliStudidiMilano
Milano,Italy
ISSN2629-2327 ISSN2629-2343(electronic)
EncyclopediaofComplexityandSystemsScienceSeries
ISBN978-1-0716-2620-7 ISBN978-1-0716-2621-4(eBook)
https://doi.org/10.1007/978-1-0716-2621-4
©SpringerScience+BusinessMedia,LLC,partofSpringerNature2022
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeor
part of the material is concerned, specifically the rights of translation, reprinting, reuse of
illustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,
andtransmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,or
bysimilarordissimilarmethodologynowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthis
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt
fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthis
bookarebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernorthe
authorsortheeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontained
hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwith
regardtojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerScience+BusinessMedia,
LLCpartofSpringerNature.
Theregisteredcompanyaddressis:1NewYorkPlaza,NewYork,NY10004,U.S.A.
Series Preface
TheEncyclopediaofComplexityandSystemScienceSeriesisamultivolume
authoritative source for understanding and applying the basic tenets of com-
plexity and systems theory as well as the tools and measures for analyzing
complexsystemsinscience,engineering,andmanyareasofsocial,financial,
andbusinessinteractions.Itiswrittenforanaudienceofadvanceduniversity
undergraduateandgraduatestudents,professors,andprofessionalsinawide
range of fields who must manage complexity on scales ranging from the
atomicandmoleculartothesocietalandglobal.
Complexsystemsaresystemsthatcomprisemanyinteractingpartswiththe
ability to generate a new quality of collective behavior through selforga-
nization, e.g., the spontaneous formation of temporal, spatial, or functional
structures. They are therefore adaptive as they evolve and may contain self-
drivingfeedbackloops.Thus,complexsystemsaremuchmorethanasumof
theirparts.Complexsystemsareoftencharacterizedashavingextremesensi-
tivity to initial conditions as well as emergent behavior that are not readily
predictableorevencompletelydeterministic.Theconclusionisthatareduc-
tionist(bottom-up)approachisoftenanincompletedescriptionofaphenom-
enon.Thisrecognitionthatthecollectivebehaviorofthewholesystemcannot
be simply inferred from the understanding of the behavior of the individual
componentshasledtomanynewconceptsandsophisticatedmathematicaland
modeling tools for application to many scientific, engineering, and societal
issues that can be adequately described only in terms of complexity and
complexsystems.
ExamplesofGrandScientificChallengeswhichcanbeapproachedthrough
complexity and systems science include: the structure, history, and future of
theuniverse;thebiologicalbasisofconsciousness;thetruecomplexityofthe
genetic makeup and molecular functioning of humans (genetics and epige-
netics)andotherlifeforms;humanlongevitylimits;unificationofthelawsof
physics;thedynamicsandextentofclimatechangeandtheeffectsofclimate
change;extendingtheboundariesofandunderstandingthetheoreticallimits
ofcomputing;sustainabilityoflifeontheearth;workingsoftheinteriorofthe
earth;predictability,dynamics,andextentofearthquakes,tsunamis,andother
natural disasters; dynamics of turbulent flows and the motion of granular
materials;thestructureofatomsasexpressedintheStandardModelandthe
formulation of the Standard Model and gravity into a Unified Theory; the
v
vi SeriesPreface
structureofwater;controlofglobalinfectiousdiseases;andalsoevolutionand
quantificationof(ultimately)humancooperativebehaviorinpolitics,econom-
ics,businesssystems,andsocialinteractions.Infact,mostoftheseissueshave
identified nonlinearities and are beginning to be addressed with nonlinear
techniques,e.g.,humanlongevitylimits,theStandardModel,climatechange,
earthquake prediction, workings of the earth’s interior, natural disaster
prediction,etc.
The individual complex systems mathematical and modeling tools and
scientific and engineering applications that comprised the Encyclopedia of
ComplexityandSystemsSciencearebeingcompletelyupdatedandthemajor-
itywillbepublishedasindividualbookseditedbyexpertsineachfieldwhoare
eminentuniversityfacultymembers.
Thetopicsareasfollows:
AgentBasedModelingandSimulation
ApplicationsofPhysicsandMathematicstoSocialScience
CellularAutomata,MathematicalBasisof
ChaosandComplexityinAstrophysics
ClimateModeling,GlobalWarming,andWeatherPrediction
ComplexNetworksandGraphTheory
ComplexityandNonlinearityinAutonomousRobotics
ComplexityinComputationalChemistry
ComplexityinEarthquakes,Tsunamis,andVolcanoes,andForecastingand
EarlyWarningofTheirHazards
ComputationalandTheoreticalNanoscience
ControlandDynamicalSystems
DataMiningandKnowledgeDiscovery
EcologicalComplexity
ErgodicTheory
FinanceandEconometrics
FractalsandMultifractals
GameTheory
GranularComputing
IntelligentSystems
NonlinearOrdinaryDifferentialEquationsandDynamicalSystems
NonlinearPartialDifferentialEquations
Percolation
PerturbationTheory
ProbabilityandStatisticsinComplexSystems
QuantumInformationScience
SocialNetworkAnalysis
SoftComputing
Solitons
StatisticalandNonlinearPhysics
Synergetics
SystemDynamics
SystemsBiology
SeriesPreface vii
EachentryineachoftheSeriesbookswasselectedandpeerreviewsorganized
by one of our university-based book Editors with advice and consultation
providedbyoureminentBoardMembersandtheEditor-in-Chief.Thislevelof
coordination assures that the reader can have a level of confidence in the
relevance and accuracy of the information far exceeding than that generally
found on the World Wide Web. Accessibility is also a priority and for this
reason each entry includes a glossary of important terms and a concise
definition of the subject. In addition, we are pleased that the mathematical
portions of our Encyclopedia have been selected by Math Reviews for
indexing in MathSciNet. Also, ACM, the world’s largest educational and
scientificcomputingsociety,recognizedourComputationalComplexity:The-
ory,Techniques,andApplicationsbook,whichcontainscontenttakenexclu-
sively from the Encyclopedia of Complexity and Systems Science, with an
awardasoneofthenotableComputerSciencepublications.Clearly,wehave
achievedprominence atalevelbeyond ourexpectations, butconsistentwith
thehighqualityofthecontent!
PalmDesert,CA,USA RobertA.Meyers
December2022 Editor-in-Chief
Volume Preface
Theideabehindperturbationtheoryisthatwhenwearenotabletodetermine
exactsolutionstoagivenproblem,wemightbeabletodetermineapproximate
solutionstoourproblemstartingfromsolutionstoanapproximateversionof
the problem, amenable to exact treatment. Thus, in a way, we use exact
solutions to an approximate problem to get approximate solutions to an
exactproblem.
It goes without saying that many mathematical problems met in realistic
situations, in particular as soon as we leave the linear framework, are not
exactly solvable – either for an inherent impossibility or for our insufficient
skills. Thus, perturbation theory is often the only way to approach realistic
nonlinearsystems.
Itisimplicitintheverynatureofperturbationtheorythatitcanonlywork
onceaproblemwhichisbothsolvable–onealsosays“integrable”andinsome
sense“near”totheoriginalproblem–canbeidentified(itshouldbementioned
inthisrespectthattheissueof“how near isnear enough”isa delicate one).
Quite often, the integrable problem to be used as a starting point is a linear
one–maybeobtainedasthefirst-orderexpansionaroundatrivialorhowever
knownsolution–andnonlinearcorrectionscanbecomputedtermbytermvia
a recursive procedure based on expansion in a small parameter (usually
denoted as ε by tradition); the point is that at each stage of this procedure,
one should only solve linear equations, so that the procedure can, at least in
principle,becarriedoveruptoanydesiredorder.Inpractice,oneislimitedby
time,computationalpower,andtheincreasingdimensionofthelinearsystems
tobesolved.
But limitations are not only due to the limits of the humans – or the
computers–performingtheactualcomputations:infact,somedelicatepoints
arise when one considers the convergence of the ε series involved in the
computationsandintheexpressionofthesolutionsobtainedbyperturbation
theory.
These points – that is, the power of perturbation theory, its basic features
andtools,anditslimitationsinparticularwithregardtoconvergenceissues–
arediscussedintheentry▶“PerturbationTheory”byG.Gallavotti.Thisentry
alsostressestherolewhichproblemsoriginatinginphysicshadinthedevel-
opment of perturbation theory; and this is not only in historical terms (the
computation of planetary orbits) but also in more recent times through the
workofPoincaréfirstandthenviaquantumtheory.
ix
x VolumePreface
ThemodernsettingofperturbationtheorywaslaiddownbyPoincaréand
goesthroughtheuseofwhatistodayknownasPoincarénormalforms;these
areacornerstoneofthewholetheoryandhence,implicitlyorexplicitly,ofall
the entries presented in this section of the Encyclopedia. But they are also
discussed in detail, together with their application, in the entry ▶“Normal
FormsinPerturbationTheory”byH.Broer.
The latter deals with the general problem, that is, with evolution differ-
ential equations (dynamical systems) with no special structure; or in appli-
cations originating from physics or engineering, one is often dealing with
systems that (within a certain approximation) preserve energy and can be
writteninHamiltonianform.Inthiscase,asemphasizedbyBirkhoff,onecan
moreefficientlyconsiderperturbationsoftheHamiltonianratherthanofthe
equationsofmotion(theadvantageoriginatinginthefactthattheHamilto-
nian is a scalar function, while the equations of motion are a system of 2n
equations in 2n dimensions). The normal form approach for Hamiltonian
systems,andmoregenerallyHamiltonianperturbationtheory,isdiscussedin
theentry▶“HamiltonianPerturbationTheory(andTransitiontoChaos)”by
H.BroerandH.Hanßmann.Thisalsodiscussestheproblemoftransition–as
some control parameter, often the energy, is varied – from the regular
behavior of the unperturbed system to the chaotic (“turbulent” if we deal
withfluidmotion)behaviordisplayedbymanyrelevantHamiltonianaswell
asnon-Hamiltoniansystems.
Asmentionedabove,inallthemattersconnectedwithperturbationtheory
and its applications, convergence issues play an extremely important role.
Theyarediscussedintheentry▶“ConvergenceofPerturbativeExpansions”
by S. Walcher, both in the general case and for Hamiltonian systems. The
interplay between perturbations – and more generally changes in some
relevant parameter characterizing the system within a more general family
ofsystem–andqualitative(notonlyquantitative)changesinitsbehavioris
ofcourseofgeneralinterestnotonlyinthe“extreme”caseoftransitionfrom
integrable to chaotic behavior but also when the qualitative change in the
behavior of the system is somehow more moderate. Such a change is also
knownasabifurcation.Albeitthereisnoentryspecificallydevotedtothese,
the reader will note that the concept of bifurcation appears in many, if not
most,oftheentries.
The behavior of a “generically perturbed” system depends on what is
meantby“generically.”Inparticularifwedealwithanunperturbedsystem
which has some degree of symmetry, this may be an“accidental” feature –
maybe due to the specially simple nature of integrable systems such as the
onechosenasanunperturbedone–butmightalsocorrespondtoarequire-
ment by the very problem we are modeling; this is often the case when we
deal with problems of physical or engineering origin, just because the
fundamental equations of physics have some degree of symmetry. The
presence of symmetry can be quite helpful – for example, in reducing the
effectivedegreesoffreedomofagivenproblem–andshouldbetakeninto
account in the perturbative expansion. Moreover, the perturbative expan-
sionscanbemadetohavesomedegreeofsymmetrywhichcanbeusedinthe
solutionoftheresultingequations.Thesemattersarediscussedatlengthin
