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IndexAadaptedcoordinates,tothedynamicsaroundL1oftheRTBP,69adaptedframetoaresponsetorusofaskew-product,23,85toatorus,20,194toaninvariantmanifoldofaxedpointofamap,12ofavectoreld,33adaptednorm,86,102algebraofpowerseries,40algebraicmanipulator,seespecicalgebraicmanipulatoralgorithmforthevalidationofberwisehyperbolicinvarianttori,103algorithm,online,seeonlinealgorithmalgorithmiccomplexity,42ofthecomputationofaninvariantmanifoldofaxedpointofamap,44ofaxedpointofavectoreld,44oftheRTBP,54algorithmsforthecomputationofinvariantmanifoldsofxedpointsofmaps,seeparameterizationmethod,computationofinvariantmanifoldsofxedpointsofmapsinvarianttoriofmaps,seeparameterizationmethod,meta-algorithmforthecomputationofinvarianttoriofmapsKAMtori,165normallyhyperbolicinvarianttori,seeNewtonstepforthecomputationofanormallyhyperbolicinvarianttorusresponsetoriofskew-productslargematrixmethods,seelargematrixmethodsprojectionmethod,92reducibilitymethod,95almostcomplexstructure,128ambientspace(ambientmanifold),8,11,19,79,122,192,193areapreservingmap,60Arnoldfamily,fattened,seefattenedArnoldfamilyasymptoticexpansion,57automaticdifferentiation,3,30automaticreducibility,129BBanachalgebrasofperiodicfunctions,99benchmarktablescomputationofresponsetori,105computationofthecentermanifoldofL1oftheRTBPingraphstyle,55computationoftruncatedproductsofpowerseries,45bifurcationsinsideaninvariantcylinder,232bifurcationsofquasi-periodictorinon-smooth,78,106smooth,78bootstrapintheregularity,81bottleneck,computational,seecomputationalbottleneckboundaryofthecentermanifoldofL1oftheRTBP,58breakdownofasaddlecurveina3DfattenedArnoldfamily,229asaddleresponsecurve,106anattractingcurveina2DfattenedArnoldfamily,215thegoldencurveinthestandardmap,170SpringerInternationalPublishingSwitzerland2016`A.
Haroetal.
,TheParameterizationMethodforInvariantManifolds,AppliedMathematicalSciences195,DOI10.
1007/978-3-319-29662-3259260Indexbundle(un)stable,27,81,85,105,111,195invariant,81invariantnormal,27,195normal,21,23tangent,21,194bundlemap,23,79CCalabiinvariant,177centermanifoldofL1oftheRTBPboundary,seeboundaryofthecentermanifoldofL1oftheRTBPboundary,errorestimates,seeerrorestimatesintheboundaryofthecentermanifoldofL1oftheRTBPcomputationof,seecomputationofahigh-orderexpansionofthecentermanifoldofL1oftheRTBPgrowthofthecoefcients,seegrowthofthecoefcientsofthecentermanifoldofL1oftheRTBPofaxedpointofamap,15ofaxedpointofavectoreld,35w.
r.
t.
styleofparameterization,16,36center-(un)stablemanifoldofaxedpointofamap,15ofaxedpointofavectoreld,35classicalKAMmethods,121,151,182cocycle,seelinearskew-productcohomologicalequationforanormallyhyperbolicinvarianttorus,197foraninvariantmanifoldofaxedpointofamap,13foraninvariantmanifoldofaxedpointofavectoreld,34foraninvarianttorusofamap,21forKAMtoriofexactsymplectomorphisms,131implementation,167forKAMtoriofHamiltonianvectorelds,181forlowerdimensionaltoriofexactsymplectomorphisms,183forresponsetoriusingtheprojectionmethod,90usingthereducibilitymethod,94fromLindstedtseries,158normal,seenormalcohomologicalequationtangent,seetangentcohomologicalequationcollinearlibrationpointsoftheRTBP,54collisionofinvariantbundlesnon-smooth,106,110smooth,110commentsontheimplementationofanalgebraicmanipulatorofdense,multivariatepowerseries,39treeandvectorimplementation,45approximationofparameterizationsoftori,203computerassistedproofsinanalysis,97largematrixmethods,seecomputationalbottleneckinlargematrixmethodsthecomputationofinvariantmanifoldsofxedpointsofmaps,18thecomputationofinvarianttoriofmaps,27compatibletriple,128complexication,38complexityalgorithmic,seealgorithmiccomplexityofanelementaryfunction,42computationof2DKAMtoriintheFroeschlemap,172a2DnormallyhyperbolicinvariantcylinderintheFroeshlemap,232afakeStrangeNon-chaoticAttractor,115ahigh-orderexpansionofthecentermanifoldofL1oftheRTBPparameterizedinamixedstyle,62parameterizedingraphstyle,55apartialnormalformaroundtheL1pointoftheRTBP,69anexpansionoftheLorenzmanifold,48attractingtoriina2DfattenedArnoldfamily,211Earth-Moontransittrajectories,seeEarth-MoontransittrajectoriesinvariantbundlesfortheHarpermap,110inversesofmatrixvaluedmaps,201torusdiffeomorphisms,201Lindstedtseriesforthegoldencurveinthestandardmap,157periodicorbitsofhighperiodinthecentermanifoldofL1oftheRTBP,61PoincaremapsofthecentermanifoldofL1withamixed-styleparameterization,62saddletoriina3DfattenedArnoldfamily,220,225saddletoriofthequasi-periodicallyforcedstandardmap,105theglobalizationoftheLorenzmanifold,seeLorenzmanifold,globalizationthegoldencurveinthestandardmap,168computationtime,seebenchmarktablesIndex261ofthedomainofvalidityofapartialnormalformaroundL1oftheRTBP,71oftheexpansionsofthecentermanifoldofL1oftheRTBPinamixedstyle,62ofthepartialnormalformaroundL1oftheRTBP,69computationalbottleneckinlargematrixmethods,seelargematrixproblem,90inthecomputationofPoincaremapsofthecentermanifoldofL1oftheRTBP,59inthemanipulationoftruncatedpowerseries,18,40computationalcostofanalgorithmformanipulatingpowerseries,42ofthecomputationofKAMtori,seeef-cientimplementationofthecomputationofKAMtoriofthesolutionoftheinvarianceequationofaninvariantmanifoldofaxedpointofavectoreld,44w.
r.
tthestyleofparameterization,44ofthetruncatedproductofmultivariatepowerseries,42computationalefciency,seeefcientimplementationcomputeralgebrasystem,30computerassistedproofs,seevalidation,3,7,97conservativecontextseeHamiltoniandynamics,1conservativesystemcontinuous,seeHamiltoniansystemdiscrete,seesymplectomorphismcost,computational,seecomputationalcostcrossresonancesbetweeneigenvaluesofxedpointsofmaps,16,17ofxedpointsofvectorelds,35,36cubictangency,216Dderivative,radial,seeradialderivativedevilstaircase,211,221,234DFT,seeDiscreteFourierTransformDiophantinecondition,24,94,96,138,183,210discreteFouriertransform,164discretequasi-periodicSchr¨odingeroperator,109dissipativecontext,26quasi-periodicinvarianttoriinthe,seequasi-periodicinvarianttorusinthedissipativecontextdissipativesystem,217dynamicalsystemdiscrete,seediscretedynamicalsystemnon-autonomous,seenon-autonomousdynamicalsystemEEarth-Moonsystem,53,65Earth-Moontransittrajectories,72efcientcomputationofPoincaremapsonanenergylevelofthecentermanifoldofL1oftheRTBP,59ofresponsetoriinthevergeofbreakdown,77ofthecentermanifoldofthecollinearpointsoftheRTBP,31ofthecompositionofthereduceddynamicswiththeparameterization,18withperiodicfunctions,inthegridandFourierrepresentations,206efcientimplementationavoidinglargematrixproblems,28ofthecompositionofpowerserieswithelementaryfunctionsusingtheradialderivative,41ofthecomputationofKAMtori,163oftheparameterizationmethodforinvariantmanifoldsofxedpointsofvectorelds,30oftheproductofhomogeneouspolynomials,45eigenvaluesattheoriginintheLorenzsystem,seeLorenzsystem,eigenvaluesattheorigininternal,seeinternaleigenvaluesnormal,seenormaleigenvaluesofthecollinearlibrationpoints,seecollinearlibrationpointsoftheRTBP,eigenvaluestangent,seetangenteigenvalueselementaryalgorithmmanipulatingmultivariate,truncatedpowerseries,42function,30,42functionofpowerseries,40map,44vectoreld,30equationcohomological,seecohomologicalequationinvariance,seeinvarianceequationergodicrotation,123,129,177errorestimatesfor2DKAMtoriintheFroeschlemap,174fora2DnormallyhyperbolicinvariantcylinderintheFroeshlemap,232262Indexerrorestimates(cont.
)forattractingtoriina2DfattenedArnoldfamily,212forhaloorbits,63fororbitsintheLorenzmanifold,52forplanarLyapunovperiodicorbits,58forsaddletoriina3DfattenedArnoldfamily,222forsaddletoriofthequasi-periodicallyforcedstandardmap,105forthedeterminationofafundamentaldomainofaninvariantmanifoldofaxedpointofavectoreld,39inthe(reduced)rstintegral,39intheinvarianceequation,39intheorbit,39forthegoldencurveinthestandardmap,169forverticalLyapunovperiodicorbits,63intheboundaryofthecentermanifoldofL1oftheRTBP,60errorestimationoftheglobalizationoftheLorenzmanifold,50errorfunction,oftheinvarianceequationofaninvarianttorusofamap,20errortermoforderkofthecohomologicalequationofaninvariantmanifoldofaxedpointofamap,13ofavectoreld,34estimationofadomainofvalidityoftheexpansionsofthecentermanifoldofL1oftheRTBP,ingraphstyle,56oftheexpansionsofthepartialnormalformaroundL1oftheRTBP,70,72ofafundamentaldomainofaninvariantmanifoldofaxedpointofavectoreld,39ofafundamentaldomainoftheLorenzmanifold,48oftheaccuracyoftheexpansionsofthecentermanifoldofL1intheRTBP,inamixedstyle,65exactsymplectomorphism,124,129,135FfastFouriertransform,163,167faststablemanifoldinsidetheLorenzmanifold,48ofaxedpointofamap,16ofaxedpointofavectoreld,36fattenedArnoldfamily2D,2083D,217FFT,seeFastFourierTransformFFTW3,167FHIT,seeFiberwiseHyperbolicInvariantTorusberwisehyperbolicinvarianttorus,80validationalgorithm,102Finslernorm,79,82,86,102xedpointsinsideaninvariantcurve,221insideaninvariantcylinder,232ofthe2DfattenedArnoldfamily,209ofthe3DfattenedArnoldfamily,218oftheLorenzsystem,seeLorenzsystemxedpointsoftheRTBP,seerestrictedthree-bodyproblem,xedpointsFouriercoefcients,89,90,95,165Fouriermethods,89,207Fouriermodel,99Fourierseries,89,131framecomplementaryLagrangian,127Lagrangian,126symplectic,126frame,adapted,seeadaptedframeFroeschlemap,172,230fundamentaldomain,39oftheLorenzmanifold,48GGevreyexpansions,56graph,styleofparameterizationofanormallyhyperbolicinvarianttorusofadissipativesystem,26ofaninvariantmanifoldofaxedpointofamap,15ofaninvariantmanifoldofaxedpointofavectoreld,36ofaninvariantmanifoldofamap,10Greene'scriterion,168gridpointmethods,27,164,204,207gridpointmethodsversusspectralmethodsadigression,207comparisonofresultsinthecomputationofasaddlecurve,226comparisonofresultsinthecomputationofanattractingcurve,222gridspaceforperiodicfunctions,204growthofthecoefcientsofthecentermanifoldofL1oftheRTBP,56Index263HhalofamiliesofperiodicorbitsoftheRTBP,55Hamiltoniandynamics,24Hamiltoniansystem,178Harpermap,109hyperbolicsplitting,27,185,196Iimplementation,seecommentsontheimplementationofinternaldynamicscomputationof,seecomputationoftheinternaldynamicsof2DinvariantcylindersintheFroeschlemap,236ofanormallyhyperbolicinvariantmanifold,192ofanormallyhyperbolicinvarianttorus,194parameterizedingraphstyle,26parameterizedinnormalformstyle,26ofaresponsetorusofaskew-product,23,80ofaKAMtorusofasymplectomorphism,24,25,123ofaninvariantmanifoldofaxedpointofamap,11parameterizedingraphstyle,15ofaninvariantmanifoldofaxedpointofavectoreld,33ofaninvariantmanifoldofamap,9parameterizedingraphstyle,10parameterizedinnormalformstyle,10ofaninvariantmanifoldofavectoreld,9parameterizedgraphstyle,10parameterizedinnormalformstyle,10ofaninvarianttorusofamap,19ofattractingtoriinthe2DfattenedArnoldfamily,213ofsaddletoriinthe3DfattenedArnoldfamily,223oftheLorenzmanifold,47rotationnumber,seerotationnumberoftheinternaldynamicsinternaleigenvaluesforaninvariantmanifoldofaxedpointofamap,16ofavectoreld,36internalresonancesbetweeneigenvaluesofaxedpointofamap,15ofaxedpointofavectoreld,35interplanetarysuperhighway,67intervalarithmetic,98invarianceequationforanormalbundleofaninvarianttorus,27,195foraninvariantmanifoldofaxedpointofamap,11ofaxedpointofavectoreld,33ofaxedpointofavectoreld,computa-tionalcost,seecomputationalcostofthesolutionoftheinvarianceequationofaninvariantmanifoldofaxedpointofamap,9ofamap,parameterizedingraphstyle,10ofavectoreld,9ofavectoreld,parameterizedingraphstyle,10foraninvarianttorusofamap,20,192forquasi-periodicinvarianttori,122,125,129forresponsetoriinskew-products,80invariantcurvesinthe2DfattenedArnoldfamily,seecomputationofattractingtoriina2DfattenedArnoldfamilyinthe3DfattenedArnoldfamily,seecomputationofsaddletoriina3DfattenedArnoldfamilyinthestandardmap,seecomputationofthegoldencurveinthestandardmapinsideaninvariantcylinder,234ofthePoincaremaponthecentermanifoldofL1oftheRTBP,60invariantgraph,76,80invariantmanifoldofaxedpointofamap,11parameterizedingraphstyle,15ofamap,8ofavectoreld,9ofaninvariantobject,seewhiskerinvariantrotationaltorus,194invariantsection,80invarianttoriindissipativesystems,26,seeinvariantrotationaltorusLissajousandquasi-halo,60normallyhyperbolic,seenormallyhyperbolicinvarianttoriprimary,seeprimarytorusquasi-periodic,seequasi-periodicinvarianttorusresponse,seeresponsetorusrotational,seeinvariantrotationaltorussecondary,seesecondarytorusinvarianttorus,19,80isotropictorus,125264IndexKKAMtheorem,seetheoremofexistenceofKAMtoriKAMtheory,24,119Kantorovich-liketheorem,seetheoremofexistenceofFHITLLagrangiantorus,125,126,130largematrixmethods,27forthecomputationofresponsetoriofskew-products,88largematrixproblem,28,90linearskew-product,21,24,80Lissajousinvarianttori,60logisticmapquasi-periodicallyforced,seequasi-periodicallyforcedlogisticmapLorenzattractor,46,48Lorenzmanifold,47fundamentaldomain,seefundamentaldomainoftheLorenzmanifoldglobalization,48–51errorestimation,seeerrorestimationoftheglobalizationoftheLorenzmanifoldinternaldynamics,seeinternaldynamicsoftheLorenzmanifoldLorenzsystem,46classicalparametervalues,46eigenvaluesattheorigin,47xedpoints,46LyapunovfamilyofperiodicorbitsoftheRTBPplanar,54vertical,55Lyapunovnorm,seeadaptednormMmanifold(un)stable,see(un)stablemanifoldcenter,seecentermanifoldcenter-(un)stable,seecenter-(un)stablemanifoldinvariant,seeinvariantmanifoldparabolic,seeparabolicmanifoldmanifoldtubes(ofperiodicorbits),68manipulationofmatrix-valuedmapswithentriesgivenbyperiodicfunctions,206Melnikovconditions,96methodofrationalapproximations,89mixed,styleofparameterizationinaneighborhoodofL1oftheRTBP,65ofaninvariantmanifoldofaxedpointofamap,17ofaninvariantmanifoldofaxedpointofavectoreld,37modelmanifold,8,11,19,122,192NNewtonstepforthecomputationofanormallyhyperbolicinvarianttorus,196forthecomputationofaresponsetorusofaskew-product,24forthecomputationofaninvarianttorusofamap,20Newton'smethod,seeNewtonstepNHIM,seeNormallyHyperbolicInvariantManifoldNHIT,seeNormallyHyperbolicInvariantTorusnon-autonomousdynamicalsystem,22non-transittrajectories,67non-uniformhyperbolicity,110normalcohomologicalequationforanormallyhyperbolicinvarianttorus,198foraKAMtorusofaHamiltoniansystem,25foraninvariantmanifoldofaxedpointofamap,14foraninvariantmanifoldofaxedpointofavectoreld,34foraninvarianttorusofamap,22normaleigenvaluesforaninvariantmanifoldofaxedpointofamap,14ofaxedpointofavectoreld,34normalformofamaparoundaxedpoint,16ofavectoreldaroundaxedpoint,36normalform,styleofparameterizationLorenzmanifold,47ofanormallyhyperbolicinvarianttorusofadissipativesystem,26ofaninvariantmanifoldofaxedpointofamap,16ofaninvariantmanifoldofaxedpointofavectoreld,36ofaninvariantmanifoldofamap,10normalframe,127,130,153,195normallyhyperbolicinvariantcurveofattractingtype(attractingcurve),209ofsaddletype(saddlecurve),218normallyhyperbolicinvariantcylinder,230normallyhyperbolicinvariantmanifold,6,192normallyhyperbolicinvarianttori,188Index265numericalstabilityincomputinganinvariantmanifoldofaxedpointofavectoreld,37innormalformstyle,37Oonlinealgorithm,13,30Pparabolicmanifoldofaxedpointofamap,17parameterizationmethodcomputationofinvariantmanifoldsofxedpointsofmaps,11–17meta-algorithmforthecomputationofinvarianttoriofmaps,18–22precedents,1programofapplication,3vs.
largematrixmethods,28parameterization,stylesof,seestylesofparameterizationspartialnormalform,17aroundtheL1pointoftheRTBP,65periodicorbitshalofamiliesoftheRTBP,seehalofamiliesofperiodicorbitsoftheRTBPinsideaninvariantcylinder,232LyapunovfamiliesoftheRTBP,seeLyapunovfamilyofperiodicorbitsoftheRTBPofhighperiodoftheRTBP,seecomputationofperiodicorbitsofhighperiodinthecentermanifoldofL1oftheRTBPoninvariantcurves,211,221perturbativeresultvsaposterioriresult,77piecewiseLagrangianinterpolation,205Poincaredomainforaninvariantmanifoldofaxedpointofamap,16ofavectoreld,36PoincaremaponanenergylevelofthecentermanifoldofL1oftheRTBP,59computationalbottleneck,seecomputationalbottleneckinthecomputationofPoincaremapsofthecentermanifoldofL1oftheRTBPpowerseriesalgebraof,seealgebraofpowerseriesalgebraicoperationsof,seealgebraofpowerserieselementaryfunctionsof,seeelementaryfunctionsofpowerseriespowerseries,truncated,seetruncatedpowerseriesprimarytorus,25,120primitivefunctionofanexactsymplectomor-phism,124projectionmethod,90Qquadratictangency,216quasi-periodicinvarianttorus,123inthedissipativecontext,185ofaquasi-periodicallyforcedsystem,88ofanexactsymplectomorphism,125quasi-periodicmotion,119,162quasi-periodicallyforcedlogisticmap,114quasi-periodicallyforcedstandardmap,104quasi-periodicallyforcedsystem,23,88RR¨ussmannestimates,138,154radialderivative,41rationalapproximationmethod,seemethodofrationalapproximationsrealication,38reducedmatrixofthelineardynamicsaroundaresponsetorusofaskew-product,23ofthelineardynamicsaroundaninvarianttorusofamap,21reducibility,21equation,86,93,103toablock-diagonalcocycle,23,85,90,195toablock-triangularcocycle,21,129,194toaconstantcocycle,93reducibilitymethod,93renormalizationtheory,170resonancecrossingsoftheinternaldynamics,210,220resonancescross,seecrossresonancesintheinternaldynamics,seeresonancecrossingoftheinternaldynamicsinternal,seeinternalresonancesresponsetorus,23ofaquasi-periodicallyforcedsystem,seequasi-periodicinvarianttorusofaquasi-periodicallyforcedsystemrestrictedthree-bodyproblem,53algorithmiccomplexity,seealgorithmiccomplexityoftheRTBPcentermanifoldofL1,seecentermanifoldofL1oftheRTBPcollinearpoints,seecollinearlibrationpointsequationsofmotion,53xedpoints,54266IndexinvarianttoriLissajousfamily,seeLissajousinvarianttoriquasi-halofamily,seequasi-halofamilyofinvarianttoripartialnormalformaroundL1,seepartialnormalformaroundL1oftheRTBPperiodicorbitshalofamilies,seehalofamiliesofperiodicorbitsoftheRTBPLyapunovfamily,seeLyapunovfamilyofperiodicorbitsoftheRTBPRiemannianmetric,128,183rigorouscomputation,98rotationnumberoftheinternaldynamics,211,220,234RTBP,seerestrictedthree-bodyproblemSSchr¨odingeroperator,discretequasi-periodic,seediscretequasi-periodicSchr¨odingeroperatorsecondarytorus,25,121seminumericalalgorithms,30skew-productsystem,22,79slowstablemanifoldinsidetheLorenzmanifold,48ofaxedpointofamap,16ofaxedpointofavectoreld,36smalldivisors,131,137,182,183Sobolevseminorms,168,170specicalgebraicmanipulator,7,30implementation,seecommentsontheimplementationofanalgebraicmanipulatorspectralgap,81,110,113,192,193spectralmethods,207spectralmethodsversusgridpointmethods,seegridpointmethodsversusspectralmethodsspectrumofadiscretequasi-periodicSchr¨odingeroperator,109ofatransferoperator,81,193suddengrowth,106stablefoliation,195stablemanifoldofaxedpointofamap,15ofaxedpointofavectoreld,35ofaninvarianttorus,seewhiskerstrongorfast,seefaststablemanifoldweakorslow,seeslowstablemanifoldstandardmap,120,151,157,168strongstablemanifold,seefaststablemanifoldstylesofparameterizations,10,15,36graphstyle,seegraph,styleofparameteri-zationmixedstyles,seemixed,stylesofparameterizationnormalformstyle,seenormalform,styleofparameterizationsubgraph,styleofparameterizationofaninvariantmanifoldofaxedpointofamap,17symbolicalgebrasystem,7symplecticadaptedframe,126,132,153,179symplecticform,124symplectomorphism,124exact,seeexactsymplectomorphismTtangentcohomologicalequationforanormallyhyperbolicinvarianttorus,198foraKAMtorusofaHamiltoniansystem,25foraninvariantmanifoldofaxedpointofamap,14foraninvariantmanifoldofaxedpointofavectoreld,34,37foraninvarianttorusofamap,22tangenteigenvaluesforaninvariantmanifoldofaxedpointofamap,14ofaxedpointofavectoreld,34tangentframe,126,166theoremofexistenceofFHIT,82ofexistenceofKAMtori,140ofhyperbolicityofalinearskew-product,86torsionmatrix,21,23,25,143,166associatedwithanormalbundletoaninvarianttorus,27ofaLagrangiantorus,129ofaninvariantmanifoldofaxedpointofamap,12ofavectoreld,33torusautomorphism,19invariant,seeinvarianttorusisotropic,seeisotropictorusLagrangian,seeLagrangiantorusquasi-periodic,seequasi-periodicinvarianttorustransfermatrix,80transferoperator,80transferskew-product,seelinearskew-productIndex267transittrajectories,67Earth-Moon,seeEarth-Moontransittrajectoriestransversaleld,127treeimplementation,seecommentsontheimplementationofanalgebraicmanipulatorofdense,multivariatepowerseries,treeimplementationtrigonometricpolynomial,99,158,164,204truncatedFourierseries,89,164truncatedpowerseriesalgebraof,seealgebraoftruncatedpowerserieselementaryalgorithm,seeelementaryalgorithmforthecomputationofmultivariate,truncatedpowerseriesUuniformhyperbolicity,81,110unstablemanifoldofaxedpointofamap,15ofaxedpointofavectoreld,35ofaninvarianttorus,seewhiskerVvalidatednumerics,97validationofaberwisehyperbolicinvarianttorus,82ofasaddletorusofthequasi-periodicallyforcedstandardmap,108ofanattracting2–periodiccurveofthequasi-periodicallyforcedlogisticmap,116ofuniformhyperbolicityintheHarpermap,112ofuniformhyperbolicityofalinearskew-product,87vectorimplementation,seecommentsontheimplementationofanalgebraicmanipulatorofdense,multivariatepowerseries,vectorimplementationWweakstablemanifold,seeslowstablemanifoldwhiskersofanormallyhyperbolicinvariantcylinder,235ofapartiallyhyperbolicisotropictorus,181,235ofasaddleresponsetorus,17,86ofaninvariantobject,11