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经典数学丛书
S.维金斯 (S.Wiggins) 著 / 世界图书出版公司 / 2013-05 / 平装
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非线性动力系统和混沌应用导论(第2版)
《非线性动力系统和混沌应用导论(第2版)》是一部高年级的本科生和研究生学生学习应用非线性动力学和混沌的入门教程。《非线性动力系统和混沌应用导论(第2版)》的重点讲述大量的技巧和观点,包括了深层次学习本科目的必备的核心知识,这些可以使学生能够学习特殊动力系统并获得学习这些系统大量信息。因此,像工程、物理、化学和生物专业读者不需要另外学习大量的预备知识。新的版本中包括了大量有关不变流形理论和规范模的新材料,拉格朗日、哈密尔顿、梯度和可逆动力系统的也有讨论,也包括了哈密尔顿分叉和环映射的基本性质。本书附了丰富的参考资料和详细的术语表,似的《非线性动力系统和混沌应用导论(第2版)》的可读性更加增大。
seriesprefaceprefacetothesecondeditionintroduction1equilibriumsolutions,stability,andlinearizedstability1.1equilibriaofvectorfields1.2stabilityoftrajectories1.3maps1.4someterminologyassociatedwithfixedpoints1.5applicationtotheunforcedduffingoscillator1.6exercises2liapunovfunctions2.1exercises3invariantmanifolds:linearandnonlinearsystems3.1stable,unstable,andcentersubspacesoflinear,autonomousvectorfields3.2stable,unstable,andcentermanifoldsforfixedpointsofnonlinear,autonomousvectorfields3.3maps3.4someexamples3.5existenceofinvariantmanifolds:themainmethodsofproof,andhowtheywork3.6time-dependenthyperbolictrajectoriesandtheirstableandunstablemanifolds3.7invariantmanifoldsinabroadercontext3.8exercises4periodicorbits4.1nonexistenceofperiodicorbitsfortwo-dimensional,autonomousvectorfields4.2furtherremarksonperiodicorbits4.3exercises5vectorfieldspossessinganintegral5.1vectorfieldsontwo-manifoldshavinganintegral5.2twodegree-of-freedomhamiltoniansystemsandgeometry5.3exercises6indextheory6.1exercises7somegeneralpropertiesofvectorfields:existence,uniqueness,differentiability,andflows7.1existence,uniqueness,differentiabilitywithrespecttoinitialconditions7.2continuationofsolutions7.3differentiabilitywithrespecttoparameters7.4autonomousvectorfields7.5nonautonomousvectorfields7.6liouville'stheorem7.7exercises8asymptoticbehavior8.1theasymptoticbehavioroftrajectories8.2attractingsets,attractors,andbasinsofattraction8.3thelasalleinvarianceprinciple8.4attractioninnonautonomoussystems8.5exercises9thepoincare-bendixsontheorem9.1exercises10poincaremaps10.1case1:poincar6mapnearaperiodicorbit10.2case2:thepoincaremapofatime-periodicordinarydifferentialequation10.3case3:thepoincaremapnearahomoclinicorbit10.4case4:poincar6mapassociatedwithatwodegree-of-freedomhamiltoniansystem10.5exercises11conjugaciesofmaps,andvaryingthecross-section11.1case1:poincar6mapnearaperiodicorbit:variationofthecross-section11.2case2:thepoincaremapofatime-periodicordinarydifferentialequation:variationofthecross-section12structuralstability,genericity,andtransversality12.1definitionsofstructuralstabilityandgenericity12.2transversality12.3exercises131agrange'sequations13.1generalizedcoordinates13.2derivationoflagrange'sequations13.3theenergyintegral13.4momentumintegrals13.5hamilton'sequations13.6cycliccoordinates,routh'sequations,andreductionofthenumberofequations13.7variationalmethods13.8thehamilton-jacobiequation13.9exercises14harniltonianvectorfields14.1symplecticforms14.2poissonbrackets14.3symplecticorcanonicaltransformations14.4transformationofhamilton'sequationsundersymplectictransformations14.5completelyintegrablehamiltoniansystems14.6dynamicsofcompletelyintegrablehamiltoniansystemsinaction-anglecoordinates14.7perturbationsofcompletelyintegrablehamiltoniansystemsinaction-anglecoordinates14.8stabilityofellipticequilibria14.9discrete-timehamiltoniandynamicalsystems:iterationofsymplecticmaps14.10genericpropertiesofhamiltoniandynamicalsystems14.11exercises15gradientvectorfields15.1exercises16reversibledynamicalsystems16.1thedefinitionofreversibledynamicalsystems16.2examplesofreversibledynamicalsystems16.3linearizationofreversibledynamicalsystems16.4additionalpropertiesofreversibledynamicalsystems16.5exercises17asymptoticallyautonomousvectorfields17.1exercises18centermanifolds18.1centermanifoldsforvectorfields18.2centermanifoldsdependingonparameters.18.3theinclusionoflinearlyunstabledirections18.4centermanifoldsformaps18.5propertiesofcentermanifolds18.6finalremarksoncentermanifolds18.7exercises19normalforms19.1normalformsforvectorfields19.2normalformsforvectorfieldswithparameters19.3normalformsformaps19.4exercises19.5theelphick-tirapegui-brachet-coullet-iooss19.6exercises19.7liegroups,liegroupactions,andsymmetries19.8exercises19.9normalformcoefficients19.10hamiltoniannormalforms19.11exercises19.12conjugaciesandequivalencesofvectorfields19.13finalremarksonnormalforms20bifurcationoffixedpointsofvectorfields20.1azeroeigenvalue20.2apureimaginarypairofeigenvalues:thepoincare-andronov-hopfbifurcation20.3stabilityofbifurcationsunderperturbations20.4theideaofthecodimensionofabifurcation20.5versaldeformationsoffamiliesofmatrices20.6thedouble-zeroeigenvalue:thetakens-bogdanovbifurcation20.7azeroandapureimaginarypairofeigenvalues:thehopf-steadystatebifurcation20.8versaldeformationsoflinearhamiltoniansystems20.9elementaryhamiltonianbifurcations21bifurcationsoffixedpointsofmaps21.1aneigenvalueofi21.2aneigenvalueof-1:perioddoubling21.3apairofeigenvaluesof1viodulus1:thenaimark-sackerbifurcation21.4thecodimensionoflocalbifurcationsofmaps21.5exercises21.6mapsofthecircle22ontheinterpretationandapplicationofbifurcationdiagrams:awordofcaution23thesmalehorseshoe23.1definitionofthesmalehorseshoemap23.2constructionoftheinvariantset23.3symbolicdynamics23.4thedynamicsontheinvariantset23.5chaos23.6finalremarksandobservations24symbolicdynamics24.1thestructureofthespaceofsymbolsequences24.2theshiftmap24.3exercises25theconley-moserconditions,or“howtoprovethatadynamicalsystemischaotic”25.1themaintheorem25.2sectorbundles25.3exercises26dynamicsnearhomoclinicpointsoftwo-dimensionalmaps26.1heterocliniccycles26.2exercises27orbitshomoclinictohyperbolicfixedpointsinthree-dimensionalautonomousvectorfields27.1thetechniqueofanalysis27.2orbitshomoclinictoasaddle-pointwithpurelyrealeigenvalues27.3orbitshomoclinictoasaddle-focus27.4exercises28melnikov'smethodforhomoclinicorbitsintwo-dimensional,time-periodicvectorfields28.1thegeneraltheory28.2poincaremapsandthegeometryofthemelnikovfunction28.3somepropertiesofthemelnikovfunction28.4homoclinicbifurcations28.5applicationtothedamped,forcedduffingoscillator28.6exercises29liapunovexponents29.1liapunovexponentsofatrajectory29.2examples29.3numericalcomputationofliapunovexponents29.4exercises30chaosandstrangeattractors30.1exercises31hyperbolicinvariantsets:achaoticsaddle31.1hyperbolicityoftheinvariantcantorsetaconstructedinchapter2531.2hyperbolicinvariantsetsinr“31.3aconsequenceofhyperbolicity:theshadowinglemma31.4exercises32longperiodsinksindissipativesystemsandellipticislandsinconservativesystems32.1homoclinicbifurcations32.2newhousesinksindissipativesystems32.3islandsofstabilityinconservativesystems32.4exercises33globalbifurcationsarisingfromlocalcodimension——twobifurcations33.1thedouble-zeroeigenvalue33.2azeroandapureimaginarypairofeigenvalues33.3exercises34glossaryoffrequentlyusedtermsbibliographyindex
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