Perturbation Theory: Mathematics, Methods and Applications PDF
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Encyclopedia 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
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