application of lyapunov exponents to strange attractors and intact & damaged ship ... ·...
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ApplicationofLyapunovExponentstoStrangeAttractorsandIntact&Damaged
ShipStability
WilliamR.Story
ThesissubmittedtotheFacultyoftheVirginiaPolytechnicInstituteandStateUniversityinpartialfulfillmentofthe
requirementsforthedegreeof
MasterofScienceIn
OceanEngineering
LeighMcCue,ChairAlanBrownWayneNeu
April29,2009
Blacksburg,VirginiaTech
Keywords:Stability,Capsize,Lyapunov,Attractor,Lorenz
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ApplicationofLyapunovExponentstoStrangeAttractorsandIntact&DamagedShipStability
WilliamR.Story
(ABSTRACT)
Thethreatofcapsizeinunpredictableseashasbeenarisktovessels,sailors,andcargosincethebeginningofaseafaringculture.Theeventisanonlinear,chaoticphenomenonthatishighlysensitivetoinitialconditionsanddifficulttorepeatedlypredict.Inextremeseastatesmostshipsdependonanoperatingenvelope,relyingontheoperator’sdetailedknowledgeofheadingsandmaneuverstoreducetheriskofcapsize.Whileinsomecasesthismitigatesthisrisk,thenonlinearnatureoftheeventprecludesanycertaintyofdynamicvesselstability.
ThisresearchpresentstheuseofLyapunovexponents,aquantitythatmeasurestherateoftrajectoryseparationinphasespace,topredictcapsizeeventsforbothintactanddamagedstabilitycases.Thealgorithmsearchesbackwardsinshipmotiontimehistoriestogatherneighboringpointsforeachinstantintime,andthencalculatestheexponenttomeasurethestretchingofnearbyorbits.Bymeasuringtheperiodsbetweenexponentmaxima,thelead‐timebetweenperiodspikeandextrememotioneventcanbecalculated.Theneighbor‐searchingalgorithmisalsousedtopredicttheseevents,andinmanycasesprovestobethesuperiormethodforprediction.
Inadditiontotheshipstabilityresearch,theLyapunovexponentsareusedinconjunctionwithbifurcationanalysistodetermineregionsofstablebehaviorinstrangeattractorswhenthesystemparametersarevaried.Theboundariesofstabilityareimportantforalgorithmvalidation,wherethesetransitionsbetweenstableandunstablebehaviormustbeaccountedfor.
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Acknowledgements• Dr.LeighMcCue,forherguidance,support,patience,andabilitytoputupwithmycrap
foralltheseyears.
• TheAOEfaculty,particularlyAlanBrownandWayneNeu,fortheirknowledgeandsupportthroughoutmyundergraduateandgraduatecareer.
• TheOceanEngineeringClassof2007,fortheirfriendship,humor,andguidancethroughout.
• TheOfficeofNavalResearch,fortheirsupportofacademicendeavorssuchasthese.
• WanWu,forherassistanceandworkwiththebifurcationanalysisviatheAUTOprogram,andherpatiencewiththeauthor.
• ThankstoBilalAyyub,EricPatterson,andArtReedfortheirsupportoftheworkofChapter1.ThatworkwassupportedbyONRGrantN000140810695andNSFCMM1‐0747973.
• RegardingtheworkofChapter2,theauthorwishtothankDanHaydenforhisworkinreducinganddocumentingtheDTMBModel5514experimentaldata,WilliamBelknapforsharingthe5514data,andAndrzejJasionowskiforprovidingthedamagedshipdata.ThisworkhasbeensupportedbyONRGrantN000140610551.
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TableofContents(ABSTRACT) II
ACKNOWLEDGEMENTS III
TABLEOFCONTENTS V
LISTOFFIGURES VII
LISTOFTABLES IX
CHAPTERS 1
1. INTRODUCTION 11.1 LYAPUNOVEXPONENTS 11.2 WOLFALGORITHM 21.3 SANOANDSAWADAALGORITHM 31.4 VERIFICATIONANDVALIDATION 41.5 LYAPUNOVAPPLICATIONTOSHIPCAPSIZE 52. IDENTIFICATIONOFPARAMETERTRANSITIONBOUNDARIESWITHLYAPUNOVEXPONENTS 72.1 BIFURCATIONANALYSIS 72.2 TIME‐SERIESLENGTH 72.3 LORENZSYSTEM 82.4 ROSSLERSYSTEM 92.5 SMALL‐SCALEPARAMETERVARIATIONS 102.5.1 LorenzSystem 102.5.2 RosslerSystem 122.5.3 RosslerHyperchaosSystem 142.6 LARGE‐SCALEPARAMETERVARIATIONS 162.6.1 LorenzSystem 162.6.2 RosslerSystem 292.6.3 RosslerHyperchaosSystem 403. APPLICATIONOFLYAPUNOVEXPONENTSTOINTACT&DAMAGEDSHIPSTABILITYCASES 413.1 APPLICATIONOFFTLESTODYNAMICSHIPMOTION 413.2 DAMAGEDSTABILITYOFACOMMERCIALPASSENGERRO‐ROSHIP 413.2.1 PeriodMeasurement 423.2.2 NeighborMeasurement 483.3 APPLICATIONTONOTIONALHULLFORM5514CAPSIZECASES 534. APPLICATIONOFNEIGHBORSEARCHINGMETHODTOREALTIMESHIPMOTIONS 634.1 MOTIVATION 634.2 EXPERIMENTALSETUP 634.2.1 Data‐collection 634.2.2 Algorithm/Datamodification 644.3 REAL‐TIMENEIGHBORCOUNTINGRESULTS 66 66
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5. CONCLUSIONS 695.1 VERIFICATIONANDVALIDATION 695.2 APPLICATIONTOSHIPCAPSIZE 695.3 FUTUREWORK 69
APPENDIXA 70
1. FIGURES 702. TABLES 75
APPENDIXB:CHOOSINGD.O.F.PARAMETERSFORBESTNEIGHBORS/FTLERESULTS 76
REFERENCES 81
1. CHAPTER1 812. CHAPTER2 82
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ListofFiguresFIGURE1:ELONGATIONOFELLIPSEAXESASANEXPONENTIALFUNCTIONOFLYAPUNOVEXPONENTS
(ADAPTEDFROMOTTET. AL 1994) 2FIGURE2:LYAPUNOVSPECTRUMCONVERGENCEFORTHEROSSLERATTRACTOR 8FIGURE3:LORENZOSCILLATORSYSTEMFOR
�
σ = 10,R = 28.0,beta = 8/3 9FIGURE4:ROSSLEROSCILLATORFOR
�
a = 0.15,b = 0.2,c = 10 10FIGURE5:LORENZLYAPUNOVSPECTRUMCHANGESFORSMALL‐SCALECHANGEIN
�
σ 11FIGURE6:LYAPUNOVSPECTRUMCHANGESFORSMALL‐SCALEVARIATIONSINAFORWOLFSYSTEM 12FIGURE7:ROSSLERLYAPUNOVSPECTRUMCHANGESFORSMALL‐SCALEVARIATIONSINCFORWOLF
SYSTEM 13FIGURE8:ROSSLERHYPERCHAOSLYAPUNOVSPECTRUMCHANGESFORSMALL‐SCALECHANGESINAFOR
WOLFVARIATION 14FIGURE9:LORENZLYAPUNOVSPECTRUMFORCHANGESIN
�
σ FORTHEWOLFVARIATION(
�
σ = 16, R = 45.92, b = 4.0) 17FIGURE10:LORENZFIRSTLYAPUNOVEXPONENTCHANGEANDHOPFBIFURCATIONFORLARGE–SCALE
CHANGEIN
�
σ 17FIGURE11:LORENZSECONDLYAPUNOVEXPONENTCHANGEFORLARGE‐SCALECHANGESIN
�
σ 18FIGURE12:LORENZTHIRDLYAPUNOVEXPONENTCHANGESFORSMALL‐SCALEVARIATIONINSIGMA 19FIGURE13:LORENZFIRSTLYAPUNOVEXPONENTCHANGEANDPHASE‐SPACEFORLARGE–SCALECHANGE
IN
�
σ FORTHECLASSICVARIATION 20FIGURE14:LORENZCLASSICVARIATIONSTEADY‐STATEPHASE‐SPACEFOR
�
σ =2.0 21FIGURE15:LORENZCLASSICVARIATIONUNSTRUCTUREDPHASE‐SPACEFOR
�
σ =19.0 22FIGURE16:LORENZLYAPUNOVSPECTRUMCHANGESFORRINTHEWOLFVARIATION
(
�
σ = 16, R = 45.92, b = 4.0) 23FIGURE17:LORENZLYAPUNOVSPECTRUMCHANGESFORRINTHECLASSICVARIATION
(
�
σ = 10, R = 28.0, b = 8 /3) 24FIGURE18:LORENZFIRSTLYAPUNOVEXPONENTCHANGESFORLARGE‐SCALEVARIATIONSINB 25FIGURE19:LORENZSECONDLYAPUNOVEXPONENTCHANGESFORLARGE‐SCALEVARIATIONSINB 26FIGURE20:LORENZTHIRDLYAPUNOVEXPONENTCHANGESFORLARGE‐SCALEVARIATIONSINB 27FIGURE21:LORENZCLASSICALVARIATIONTRANSITIONBETWEENCHAOTICANDSTABLEBEHAVIORFOR
CHANGESINB 28FIGURE22:ROSSLERFIRSTLYAPUNOVEXPONENTCHANGESFORLARGESCALECHANGESINA 29FIGURE23:ROSSLERFIRSTLYAPUNOVEXPONENTCHANGESFORLARGESCALECHANGESINA 30FIGURE24:ROSSLERTHIRDLYAPUNOVEXPONENTCHANGESFORLARGESCALEVARIATIONSINA 31FIGURE25:ROSSLEROSCILLATORFOR
�
a = 0.05,b = 0.2,c = 10 32FIGURE26:ROSSLERLYAPUNOVSPECTRUMCHANGESFROMLARGE‐SCALEVARIATIONSINB 33FIGURE27:FIRSTLYAPUNOVEXPONENTCHANGESFORSMALL‐SCALEVARIATIONSINC 34FIGURE28:SECONDLYAPUNOVEXPONENTCHANGESFORSMALL‐SCALEVARIATIONSINC 35FIGURE29:THIRDLYAPUNOVEXPONENTCHANGESFORSMALL‐SCALEVARIATIONSINC 36FIGURE30:ROSSLERFIRSTLYAPUNOVEXPONENTCHANGEANDPHASE‐SPACEFORLARGE–SCALE
CHANGEINC 37FIGURE31:ROSSLEROSCILLATORPERIODICITYFOR
�
a = 0.2,b = 0.2,c = 4 38FIGURE32:ROSSLEROSCILLATORPERIODICITYFOR
�
a = 0.2,b = 0.2,c = 20 39FIGURE33:DAMAGEDSTABILITYRUN101.FROMTOPTOBOTTOM:ROLLVS.TIME,PERIODVS.TIME,FTLE
VS.TIME 42FIGURE34:DAMAGEDSTABILITYRUN101.CLOSEUPOFFTLEVALUESANDPERIODMEASUREMENT 43FIGURE35:DAMAGEDSTABILITYRUN101.FTLEANDPERIODMEASUREMENTSVS.TIME 44FIGURE36:ROLLVS.TIMEANDFTLEPERIODFORCAPSIZERUN402 46FIGURE37:MARKEDPERIODINDICATORFORLARGESTAMPLITUDEMOTION,RUN402 47FIGURE38:DAMAGEDSTABILITYRUN101ROLLVS.NUMBEROFNEIGHBORS. 48FIGURE39:DAMAGEDSTABILITYRUN101ZOOMOFNEIGHBORCOUNTING 49
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FIGURE40:DAMAGEDCASERUN101ROLLVS.FLAG 50FIGURE41:DAMAGEDCASERUN101ROLLVS.DANGERINDICATOR 51FIGURE42:HULLFORM5514RUN216ROLLVS.NUMBEROFNEIGHBORS 54FIGURE43:HULLFORM5514RUN216ROLLVS.PERIOD 55FIGURE44:HULLFORM5514RUN216ROLLVS.ROLLVELOCITYBASINOFSTABILITYFORPERIOD
INDICATORS 56FIGURE45:HULLFORM5514RUN216ROLLVS.ROLLVELOCITYBASINOFSTABILITYFORNEIGHBORHOOD
INDICATORS 57FIGURE46:HULLFORM5514RUN327NEIGHBORHOODLOSS 58FIGURE47:HULLFORM5514RUN327ROLLVS.ROLLVELOCITYNEIGHBORHOODLOSSMARKERS 59FIGURE48:HULLFORM5514RUNS220,331,333ROLLVS.ROLLVELOCITYNEIGHBORHOODLOSSMARKERS
60FIGURE49:MOOGMOTIONPLATFORM 63FIGURE50:CROSSBOWTILTSENSORMOUNTEDONMOTIONPLATFORM 64FIGURE51:DAMAGEDSTABILITYRUN101,DATARECORDEDFROMMOOGPLATFORM 66FIGURE52:ROSSLERHYPERCHAOTICATTRACTORFOR
�
a = 0.25,b = 3.0,c = 0.05,d = 0.5 70FIGURE53:LORENZLYAPUNOVSPECTRUMFORSMALL‐SCALECHANGESINR 71FIGURE54:LORENZLYAPUNOVSPECTRUMFORSMALL‐SCALECHANGESINB 71FIGURE55:ROSSLERHYPERCHAOSLYAPUNOVSPECTRUMCHANGESFORSMALL‐SCALECHANGESINBFOR
WOLFSYSTEM 72FIGURE56:ROSSLERHYPERCHAOSLYAPUNOVSPECTRUMCHANGESFORSMALL‐SCALECHANGESINCFOR
WOLFSYSTEM 73FIGURE57:ROSSLERHYPERCHAOSLYAPUNOVSPECTRUMCHANGESFORSMALL‐SCALECHANGESIND FOR
WOLFSYSTEM 74FIGURE58:DAMAGEDSTABILITYRUN101ROLLVSTIMEANDNON‐DIMENSIONALIZEDFTLEPERIOD
MEASUREMENT 76FIGURE59:DAMAGEDSTABILITYRUN101ROLL/ROLLVELOCITYVS.TIMEANDNON‐DIMENSIONALIZED
FTLEPERIOD 77FIGURE60:DAMAGEDSTABILITYRUN101PITCHVS.TIMEANDNON‐DIMENSIONALIZEDFTLEPERIOD 78FIGURE61:DAMAGEDSTABILITYRUN101PITCH/PITCHVELOCITYVS.TIMEANDNON‐DIMENSIONALIZED
FTLEPERIOD 79FIGURE62:DAMAGEDSTABILITYRUN101PITCH/PITCHVELOCITY&ROLL/ROLLVELOCITYVS.TIMEAND
NON‐DIMENSIONALIZEDFTLEPERIOD 80
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�
ListofTablesTABLE1:STANDARDDEVIATIONSFORLYAPUNOVSPECTRUMFORSMALL‐SCALECHANGESIN
�
σ 11TABLE2:STANDARDDEVIATIONSFORLYAPUNOVSPECTRUMFORSMALL‐SCALECHANGESINA 12TABLE3:STANDARDDEVIATIONSFORROSSLERLYAPUNOVSPECTRUMFORSMALL‐SCALECHANGESINC
13TABLE4:STANDARDDEVIATIONSFORLYAPUNOVSPECTRUMFORSMALL‐SCALECHANGESINA 15TABLE5:STANDARDDEVIATIONFORLORENZLYAPUNOVSPECTRUMFORLARGE‐SCALECHANGESIN
�
σ 17TABLE6:STANDARDDEVIATIONSFORLORENZLYAPUNOVSPECTRUMFORSMALL‐SCALECHANGESINR24TABLE7:STANDARDDEVIATIONSFORLORENZLYAPUNOVSPECTRUMFORSMALL‐SCALECHANGESINB27TABLE8:STANDARDDEVIATIONSFORROSSLERLYAPUNOVSPECTRUMFORSMALL‐SCALECHANGESINA
31TABLE9:STANDARDDEVIATIONSFORROSSLERLYAPUNOVSPECTRUMFORSMALL‐SCALECHANGESINB
33TABLE10:STANDARDDEVIATIONSFORROSSLERLYAPUNOVSPECTRUMFORSMALL‐SCALECHANGESINC
39TABLE11:LEADTIMEFORPERIODCORRELATIONOFMAXIMUMROLLAMPLITUDES 45TABLE12:LEADTIMEFORNEIGHBORCORRELATIONOFMAXIMUMROLLAMPLITUDES 52TABLE13:LEADTIMEFORNEIGHBORHOODLOSSCORRELATIONOFHULLFORM5514CAPSIZECASES 61TABLE14:LEADTIMEFORREAL‐TIMENEIGHBORHOODLOSSCORRELATIONOFDAMAGEDSTABILITY
CASES 67TABLE14:STANDARDDEVIATIONSFORLORENZLYAPUNOVSPECTRUMFORSMALL‐SCALECHANGESIN
SIGMA 75TABLE15:STANDARDDEVIATIONSFORLORENZLYAPUNOVSPECTRUMFORSMALL‐SCALECHANGESINB
75TABLE16:STANDARDDEVIATIONSFORROSSLERLYAPUNOVSPECTRUMFORSMALL‐SCALECHANGESINB
75TABLE17:STANDARDDEVIATIONSFORROSSLERHYPERCHAOSLYAPUNOVSPECTRUMFORSMALL‐SCALE
CHANGESINB 75TABLE18:STANDARDDEVIATIONFORROSSLERHYPERCHAOSLYAPUNOVSPECTRUMFORSMALL‐SCALE
CHANGESIND 75
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1
Chapters1. Introduction
1.1 LyapunovExponents
Themotionofasinglepointonanattractorcanbedefinedaschaoticifexhibitssensitivitytoinfinitesimallysmallchangesininitialconditions(Ottet. al 1994).Asimplebuttellingexampleofthisconditionwouldbetoplaceaballonahillandgiveitasmallpushinonedirection.Nomatterhowprecisethepushmaybe,theballwillalwaysfollowadifferentorbitdownthehillbecauseoftheminisculedifferencesintheforcebeingappliedandtheterrainitfollows.Theelaborateorbitstructurethatcomesasaresultofvastnumberofpossibleorbits,aswellasthe“stretching”ofminutedisplacementsoftheorbit(initialconditionsensitivity),canbemodeledwithLyapunovexponents(Ottet. al 1994).
ThestartingpointfordefiningaLyapunovexponentisaflowfield:
x = v x( ) (1)
Inadditiontothisflowfield,atrajectory x t( ) isdefined,aswellassmalldeviationsfromthattrajectory,δx .Afterformingamatrixofderivatives, Lij =
∂vi∂x j
,anequationforthe
changingnatureoftheflowcanbedefinedas:
δ x = L x t( )( )δx (2)
Therefore,forallinitialtrajectoriesandinitialdisplacementsamaximalLyapunovexponentforthesystemcanbedefinedasfollows(EckhardandYao,1993):
λ∞ = limT→∞
1Tlog
δx t( )δx 0( ) (3)
Thisexponentisnormallyassumedtoexistonanattractor,anditshouldbenotedthatforsomecasesanattractormaynotexistforthesystem.Aswasmentionedpreviously,theexponentmeasuresthestretchingofnearbyorbitsinphasespace;thisstretchingcancomeintheformanexpandingorcontractingnature,andmaybestbevisualizedbyaballofinitialconditionpoints.Becauseofthelocaldeformations,orstretching,oftheflow,theballofinitialconditionpointsink dimensionswillbecomeakdimensionalellipsoidwhoseaxesaredeformingexponentiallyasdefinedbytheseLyapunovexponents(Wolfet al 1985).
Thenumberofexponentsforthesystemisdeterminedbythenumberofstatevariablesgoverningthesystem.Generallyforanycontinuoustime‐dependentdynamicsystemwithoutafixedpoint,therewillbeazeroexponentreflectingtheslowlychangingprincipalaxis,apositiveexponentreflectinganexpandingaxis,andanegativeexponentreflectingacontractingaxis(Wolfet al 1985).
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Figure1:ElongationofellipseaxesasanexponentialfunctionofLyapunovexponents(adaptedfromOttet. al 1994)
Thenumberofexponentsforthesystemisdeterminedbythenumberofstatevariablesgoverningthesystem.Generallyforanycontinuoustime‐dependentdynamicsystemwithoutafixedpoint,therewillbeazeroexponentreflectingtheslowlychangingprincipalaxis,apositiveexponentreflectinganexpandingaxis,andanegativeexponentreflectingacontractingaxis(Wolfet al 1985).
Afinite‐timeLyapunovexponent(FTLE)issimplytheLyapunovexponentdefinedoverashorttimeinterval,ratherthanoveraninfinitecontinuoustimeseries.Itcanbedefinedas(EckhardtandYao,1993):
λ x t( ),δx 0( )( ) = 1Tlog
δx t + T( )δx t( ) (4)
TheFTLEallowsforamoremeaningfulmeasureofreal‐timechangesoftheexpandingandcontractingnatureoftheellipticalaxes,andistheformoftheLyapunovexponentthatwillbeusedintheanalysisofthedamagedstabilitycases.TheLyapunovexponentasdefinedoveraninfinitecontinuoustimeseries,whereascapsizeisafinitetimeevent.IntheFTLEcalculationtheJacobianisbeingcalculatedlocallyateachinstantinthetimeseries,andthustheFTLEisreactingtothechangesastheyoccur.
1.2 WolfAlgorithm
OneapproachusedtocalculatetheLyapunovexponentspectrumforthisworkisthealgorithmdevelopedbyWolfet al(1985),whichdeterminestheexponentsdirectlyfromtheequationsofmotion.Wolf’smethodfollowsthelong‐termchangesalongaprincipleaxis,or“fiducialtrajectory”,inordertocalculatethelargestpositiveexponentvalues,andmaintainsspaceorientationusingaGram‐Schmidtreorthonormalizationprocedure(Wolfetal1985).
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1.3 SanoandSawadaAlgorithm
ThealgorithmdevelopedforLyapunovexponentsbySanoandSawada(1985)isusedtocalculateboththeLyapunovspectrumandFTLEvalues.TheSanoandSawadaapproachbeginswiththesamestepsaspresentedinequations1‐3,butalsodefinesalinearoperator,At :
δx t( ) = Atδx 0( ) (5)
GiventhetimeseriesmeasuredatthediscretetimeintervalΔt , x j = x t0 + j −1( )Δt( ) ,thek‐dimensionalellipsoidasdescribedabovecanbedefinedbyadisplacementvector yi ,andadisplacementvectoroveratimeintervalτ = mΔt ,givenby zi .ThedetailsofthederivationofthesevectorsarethoroughlyoutlinedinSanoandSawada(1985).Withthesevectorsdefined,theevolutionoftheellipsoidcanberepresentedby:
zi = AJ yi (6)
Wherethematrix AJ isanapproximationoftheflowmap At ,fromequation5.Usingaleast‐
error‐algorithm,whichminimizestheaverageofthesquarederrornormbetween zi andAJ y
iwithrespecttoallcomponentsofthematrix AJ (SanoandSawada,1985),theLyapunovexponentscanbefoundasfollows:
λi = limn→∞
1nτ
ln Ajeij
j=1
n
∑ (7)
Inthisequationn isthenumberofdatapoints,ande isasetoforthonormalbasisvectorsthatarerenormalizedusingtheGram‐Schmidtprocedure(SanoandSawada,1985).
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1.4 VerificationandValidation
“[I]n a meaningful though overly scrupulous sense, a ‘Code’ cannot be Validated, but only a Calculation (or range or calculations with a code) can be Validated. However, it is clear that physical problems and their solutions present moreorless continuum responses in their parameter spaces. Although parameter ‘transition’ boundaries do occur, at which solution properties can change discontinuously or rapidly, these parameter transition boundaries are at least countable, and are usually few. The determination of the parameter transition boundaries is the task of the entire professional community (experimental, theoretical, computational) working in the subject area”(Roache,1998,pp.280‐281).
ChaoticattractorsareoftenusedasanalgorithmverificationbenchmarkforthecalculationoftheLyapunovexponentspectrum.Verificationistheprocessbywhichtheresearcherdemonstratesthatthenumericalmodelimplementationandsolutionmatchesthedevelopedtheoreticalmodel;thisprocessworkshandinhandwithvalidation,whichconfirmsthatthemodelisanaccuraterepresentationofthephysicalrealityofthesystem(McCue,2008).
PreviousworkdonebyRosensteinet al(1992)investigatedLyapunovsensitivitytovariouschangesmadeintheLorenzsystem,asamethodforverifyinganewalgorithmforthecalculationofthelargestLyapunovexponent.Theworkinvestigatedtheeffectofembeddingdimension,timeserieslength,reconstructiondelay,andadditivenoiseonthespectrumofexponents,butdidnotinvestigatetheeffectofattractorparametervariationonthespectrum.
TherehasbeenlittleresearchdoneinregardtoLyapunovsensitivitytosmall‐scalechangesintheparametersofthreeandfour‐dimensionalstrangeattractorsfromtheverificationandvalidationperspective.Byinvestigatingtheeffectsoftheseparametervariationsonthecomputedattractors,thetransitionboundariesbetweenstableandunstableregionsoftheattractorcanbedetermined;thesetransitionshavebeenexploredbefore,butneverwithrespecttotheLyapunovspectrumasaV&Vtool.Theseboundariesareanimportantpieceofalgorithmvalidation,wherethesensitivitytoexperimentalerrorcanleadtoincorrectresultsifthesetransitionsarenottakenintoaccount.
TheuseofbothWolfandSano/Sawadaalgorithmsisakeypartoftheverificationandvalidationofsimulatedvs.experimentalresults.Infieldsofresearchsuchasshipmotions,datasetsarecreatedbothfromexperimentaltanktestingaswellasdirectlyfromtheequationsofmotion.Lyapunovexponentsarearobusttoolforvalidatingbothtypesofmodels;thereshouldbeexcellentagreementbetweenexponentvaluesiftheunderlyingphysicsoftheexperimentalmodelagreeswiththeexperimentaldata.
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1.5 LyapunovApplicationtoShipCapsize
Shipcapsizeisoftenachaoticphenomenonwithcapsize/non‐capsizeconditionsdemonstratinghighsensitivitytoinitialconditions.Manymathematical,statistical,andnumericalmethodshavebeenemployedtodeterminethelikelihoodofshipcapsizeinspecifiedseaconditions.However,becauseofthechaoticnatureoftheproblemitisexceedinglydifficulttoconsistentlyandrobustlypredictshipcapsizeinaseriesofrandomwaves.Additionallythereistheissueofrarity,wheredisparatetimeintervalsbetweenrollperiodandlossofstabilityleadtodifficultiesinnumericalsimulationsforcapsizecases.Theaveragetimebeforestabilityfailureisverylargecomparedtonaturalrollperiod,andthereforesetsofreconstructedwavedatamustbeverylongtocaptureallpossibledynamics,presentinganumericalchallengewhenworkingwiththecomparativelysmalltimescaleofrollperiod(Belenky2007).Recentinnovationinhulldesignhasbeenheighteningtheawarenessoftheseissues,andhaspushedfurtherinvestigationintothenatureofshipcapsize.Thisveryrealproblemiswherethemathematicalstudyofchaoticprocessesmayone‐dayallowforthereal‐timepredictionofwhetherashipisfacingimminentcapsize.
Capsizeresearchiscurrentlybeingperformedbothexperimentallyandnumericallyatgovernmentandacademicresearchinstitutionsworldwide,withpowerfulnumericaltoolsbeingdevelopedforthepurposeofanalyzingnonlinearshipmotions.TheLAMPprogramwasusedtocompletenumericaltime‐domainworkinthe1990’sbyLin andYue(1990,1993),andbeganin1988asaDARPAprojectforthesimulationofnonlinearshipmotions(Belenky2002).TheuseofthistoolhasbeencontinuedbyBelenky,Weems,andLiutet. al.(2002)tosimulatecriteriasuchaswater‐on‐deck,impactandwhipping,andwave‐loads.Thenonlinearstrip‐theorycodeFREDYNhasbeensignificantlyusedbyDeKatet. al. (2000,2001)tomakemotionpredictionsforbothintactanddamagedstability,aswellasprogressivefloodingandsloshing.
Lyapunovexponentshaveseenlimiteduseinthefieldofnavalarchitectureandshipdynamics.SomeoftheearliestworkwasdonebyPapoulias,investigatingthebehaviorofamooringsystemfortankersinathreedegree‐of‐freedommodel.TheLyapunovspectrumwasusedinthiscasetoconfirmtheonsetofchaoticbehavior,andthusinstabilityinthemodel(Papoulias,1987)EarlyworkbyFalzaranocalculatedtheLyapunovspectrumforthecapsizecaseofthefishingvesselPattiB.Itwasconcludedthattheexponentcanserveasbothaqualitativeandquantitativemeasureofchaos,withapositivevaluebothconfirmingchaoticbehaviorandassociatinganumberwiththatexpansion(Falzarano1990).Spyrou’sworkinvestigatedLyapunovexponentsinconnectionwithlarge‐amplitudeshipmotioninquarteringwaves;rudderanglewasusedasacontrolparameter,withafocusonoscillatorybehaviorandtransitionstochaoticregions.Theexponentswereusedinconjunctionwithbifurcationanalysistodetectthetransitionboundarybetweenstableandchaoticbehaviorinacontrols‐fixedship,focusingonpositivevaluesofthefirstexponent(Spyrou1996).WorkbyMurashigeandcollaboratorsexaminedtheroleofchaoticbehaviorinafloodingbox‐bargemodelinwaves;themodelwascoupledtwo‐dimensionallywithrollandflooding.Theirresultsdeterminedthattherollresponseofafloodedvesselcanexhibitchaoticbehaviorinregularwaves,supportedbytheexistenceofapositiveLyapunovexponent(Murashige1998a;1998b;2000).Arnoldetal.correlatedmeasurementsofthepositive
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Lyapunovexponentinthespectrumtocapsizeresultsinaonedegree‐of‐freedomrollmodel.Theirresultsconcludedthatfortheirmodeltheattractordisappearsinacapsizecase,usuallywhilethefirstexponentisnegative;thissignifiesastableperiodicattractorratherthanachaoticregion.Theresearchproducedsomecapsizecasesthatweretheresultofapositiveexponent,demonstratingthatthenumericalmodelcanexistasachaoticattractorbeforecapsize(Arnold2003).
RecentworkbyMcCueet.al.hasusedLyapunovexponentsfordeterministicresearchonshipmotionsandcapsize.Theworkusedafinite‐timeformoftheexponent(FTLE)toexamineitsabilitytopredictcapsizecasesinasingledegree‐of‐freedomrollmodel.TheresearchconcludedthattheLyapunovapproachcanaccuratelypredictimpendingcapsizeinregularandrandomseas(McCue2005).FutherworkbyMcCue,Bassler,andBelknapcontinuedtheuseofFTLE’stoindicatecapsizeinexperimentaltimeseriesdatafromwavetanktestsconductedattheNavalSurfaceWarfareCenterinCarderock,MD.Thisresearchisacontinuationofthatwork,examiningthefeasibilityofFTLE’sforbothintactanddamagedstability,andtheapplicationofdifferentalgorithmstoimprovetheresponsetimeforcapsizeprediction.
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2. IdentificationofParameterTransitionBoundarieswithLyapunovExponents
2.1 Bifurcationanalysis
Thesensitivitytoparameterchangesforstrangeattractorscanbeachievedthroughanumberofmethods,themosttraditionalofwhichisbifurcationanalysis.ThedefinitionofthisanalysisasoutlinedbyCrawford(1989):
“Bifurcation theory studies these qualitative changes in the phase portrait, e.g., the appearance or disappearance of equilibria, periodic orbits, or more complicated features such as strange attractors. The methods and results of bifurcation theory are fundamental to an understanding of nonlinear dynamical systems…”
BifurcationanalysiswasusedtofurthervalidatetheLyapunovtransitionboundariesforlarge‐scaleparameterchanges.TheAUTOprogram,originallydevelopedbyDoedel(2008),isafreewaresoftwarepackagewithbuilt‐inalgorithmsforcalculatingHopfbifurcationsfortheLorenzsystem.ThesebifurcationsoccurduringthetransitionalphasesoftheLorenzoscillator,fromstabletounstablebehavior,andareusefulinverifyingthetransitionscapturedbytheLyapunovexponentspectrum.
2.2 Time‐SeriesLength
Bothtime‐serieslengthandtime‐steparecriticalinordertoallowforthelong‐termconvergenceoftheLyapunovexponent.AbarbaneletalinvestigatedtheeffectoflocalLyapunovexponentsandtheirgovernanceofsmallperturbationsalonganorbitbasedonafinitenumberofsteps.TheirworkconcludedthatasL,thenumberofstepsalongtheorbit,growstoinfinity,variationsaboutthemeanoftheLyapunovexponentsapproacheszero(Abarbaneletal1991).FurtherworkstudyingtheeffectsoftimeserieslengthwithregardstoexponentdeviationwascompletedbyRosensteinetal;theyalsoexploredvariationsinembeddingdimension,reconstructivedelay,andadditivenoiseusingthesameLorenzsysteminvestigatedbyWolfetal.ThefindingsconcludedthatthebestresultsforLyapunovcalculationwereachievedusingalongtime‐seriesandcloselyspacedsamples;theyalsosawsimilarresultsusinglongobservationtimeandwidely‐spacedsamples(Rosensteinetal,1992).Alldatasetsinthisstudyare2000secondsinlengthwithastepsizeof0.1seconds,providingasamplesizeof20,000points.Generallyforanycontinuoustime‐dependentdynamicsystemwithoutafixedpoint,therewillbeazeroexponentreflectingtheslowlychangingprincipleaxis,apositiveexponentreflectinganexpandingaxis,andanegativeexponentreflectingacontractingaxis(Wolfetal1985).Convergenceforthepositiveandzeroexponentsoccursquicklyalongthetimeseries,whilethethirdexponent,whichisgenerallyconsideredtobethemostunstableofthethree,takesanumberofiterationstoconverge,asseeninFigure2.Bytheconclusionofthetimeseries,allthreeexponentsoscillatearoundtheirfinalvalueontheorderof0.1‐0.3%.
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Figure2:LyapunovspectrumconvergencefortheRosslerattractor
2.3 LorenzSystem
TwodifferentvariationsontheLorenzsystemwereinvestigated,eachhavingbeenpreviouslysolvedfortheLyapunovspectrausingdifferentapproaches.Thealgorithmusedforthisworkwasverifiedagainstthepublishedvalues.Thefirstsystemisbasedonafamiliarmodelthatwastheresultofastudyofconvectionintheloweratmosphere(Abarbaneletal1991).TheseparametervaluesfortheLorenzsystemwereusedbyWolfetalintheirdirectmethodalgorithm,andhavebeenusedmorerecentlyinotherapproachestodetermineLyapunovspectraforstrangeattractors.TheWolfsystemisdefinedbythefollowingparameters:
�
σ = 16, R = 45.92, b = 4.0TheparametersofthesecondsystemareasdefinedbyLorenzhimselfinhisoriginalworkontheattractorasfollows(Lorenz1963):
�
σ = 10, R = 28.0, b = 8 /3.Thissystemiswidelyconsideredtobetheclassicexampleofa
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Lorenzoscillator,asseeninFigure3.
Figure3:Lorenzoscillatorsystemfor
�
σ = 10,R = 28.0,beta = 8/3
2.4 RosslerSystem
ThethreedimensionalRosslerattractor,asproposedbyRosslerin1976,isathree‐dimensionalsystemthathasapositive,zero,andnegativeexponentspectrumwhenstable;twovariationsinparametervalueswerecompared,thosebyWolfet.al.(
�
a = 0.15, b = 0.20, c = 10.0),asseeninFigure4,andthosebySanoandSawada
�
(a = 0.20, b = 0.20, c = 5.7) .TheRosslerhyperchaossystem,a4‐Dhyperchaoticflowproposedin1979andseeninFigure52inAppendixA,wasalsoinvestigated;itcontainsasecondpositiveexponentreflectingtheexpandingaxisoftheextraunseendimension.Itisextremelysensitivetoparameterinputs,andthereforethereisgenerallythefollowingaretheonlyparametersusedfornumericalstudy(a=0.25,b=3.0,c=0.05,d=0.5).
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Figure4:Rossleroscillatorfor
�
a = 0.15,b = 0.2,c = 10
2.5 Small‐scaleParameterVariations
Thedefinitionof“small‐scale”forthisstudywereparameterperturbationsof10%oftheparametervalueineachdirection.
2.5.1 LorenzSystem
2.5.1.1
�
σ (small‐scale)
Parameterchangesofthisscaleshowsmallchangesintheexponentvalues,butatnopointaretheysignificantenoughtoforcethesystemintoadifferencephaseororbit.Figure5showstheLyapunovexponentspectrumchangesforbothWolfandClassicalvariations,notingthatthedifferenceinparametersbetweenthetwoaccountsfortheapparentgapinmeasurementvalue.
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Figure5:LorenzLyapunovspectrumchangesforsmall‐scalechangein
�
σ
ThethirdexponentshowsappreciablechangeforboththeWolfandClassicalvariations,butitslinearnatureshowsbeliesnoseriesphasechangeortransitionalperiod.Thestandarddeviationsforthe
�
σ caseareseeninTable1:
Variation Exponent1
�
σ Exponent2
�
σ Exponent3
�
σ Wolf 0.007445 0.0006189 0.9992
Sano/Sawada 0.005233 0.0009968 0.5216Table1:StandarddeviationsforLyapunovspectrumforsmall‐scalechangesin
�
σ
2.5.1.2 R(small‐scale)
TheR‐parameterhaslittletonoeffectonthesystemforchangesofthisdegree.AscanbeseeninFigure50andTable14ofAppendixA,theWolfvariationshowsalittlemorechangeinthethirdexponentthantheClassicvariation,thoughbothdeviationsaresmall;thoughthethirdexponenthasbeenthemosterraticofthethree,forRvariationsitremainsmoderatelyconstant.Thesecondexponentvariationisagainonthescaleofcomputationalerror,oscillatingaroundzero,andthefirstexponentincreasesinalinearlypositivedirectionforthedurationofthecalculations.
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2.5.1.3 b(small‐scale)
Allthreeexponentstrendsimilarlytochangesinsigma,withthethirdexponentdecreasinglinearly,thefirstexponentlinearlyincreasingtoasmalldegree,andthesecondoscillatingaroundzero.Thereisnoappreciabletransitionorphasechange.
2.5.2 RosslerSystem
GiventhesimilaritybetweenparametersoftheWolfandSano/Sawadasystems,andlittlenumericalchangeineithersystemforsmall‐scaleparameterchange,onlytheWolfvariationwasused.
2.5.2.1 a (small‐scale)
TheRosslersystemshowednoobviousphasechangesortransitionboundariesfora:
Figure6:Lyapunovspectrumchangesforsmall‐scalevariationsinaforWolfsystem
System Exponent1
�
σ Exponent2
�
σ Exponent3
�
σ Wolf 0.01860 0.001806 0.03411
Table2:StandarddeviationsforLyapunovspectrumforsmall‐scalechangesina
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Allthreeexponentdeviationsarerelativelyuniform,withnoincreasingordecreasingtrendsamongthem.Allthreeexponentsfluctuatealongthelengthoftheparameterchange,andthesystemisstablethroughout,withnotransitions.
2.5.2.2 b (small‐scale)
Changesinb haveevenlessofanaffectonthesystem,ascanbeseeninFigure51andTable16ofAppendixA.
2.5.2.3 c(small‐scale)
Changesinparameterc reflectthoseseenintheLorenzsystem:
Figure7:RosslerLyapunovspectrumchangesforsmall‐scalevariationsincforWolfsystem
Variation Exponent1
�
σ Exponent2
�
σ Exponent3
�
σ Wolf 0.005389 0.0002207 0.3316
Table3:StandarddeviationsforRosslerLyapunovspectrumforsmall‐scalechangesinc
Thefirsttwoexponentsshowverylittledeviationalongthelengthofchangesinc. Thethirdexponentshowsaslightlynegativelinearchange,butnothingsignificantenoughtobeconsideredtransitionbehavior.
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2.5.3 RosslerHyperchaosSystem
2.5.3.1 a(small‐scale)
TheinitialsystemparametergivenbyWolfwasa=0.25,andchangeswereattemptedfor10%ofthisvalueineachdirection.However,forearlyvaluesa=0.225‐0.23,thesystemwouldnotconvergetoastablefourthexponent.Theconvergencebeganata=0.2325andcontinuedthrougha=0.2575.AfterthispointtheODEsolverfailedtocomputeforallfurthertimeiterations.Figure8showsthevariationsalonga thatwouldconvergetoastablefourthexponent:
Figure8:RosslerHyperchaosLyapunovspectrumchangesforsmall‐scalechangesinaforWolfvariation
Whencomparingallthreeattractorsystems,theRosslerHyperchaossystemisthemostreactiveofallthreewhenmakingsmall‐scalevariations,atleastinregardstothefourthexponent.Thisexponentisextremelysensitivetoparameterchange,andisthedrivingforcebehindtheconvergencefailurefortheattractor.Thestandarddeviationsbelietheerraticnatureofthenegativeexponent:
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Variation Exponent1
�
σ Exponent2
�
σ Exponent3
�
σ Exponent4
�
σ Wolf 0.006431 0.003710 0.0006582 13.41
Table4:StandarddeviationsforLyapunovspectrumforsmall‐scalechangesina
Thedeviationsinthefirstthreeexponentsareonthesamescaleassimilarmeasurementsfortheothertwoattractorsystems.Giventhisresult,itdoesnotappearthatthesystemisundergoinganymajorphasechangeortransitionperiod.Thenegativeexponentisclassicallythemostunstableofthethree,andthusallthreeexponentsmustbetakenintoaccountwhenconsideringwhetherthesystemiscrossingatransitionboundary.
2.5.3.2 b(small‐scale)
Thoughthenegativeexponentshowsslightlyoscillatorybehavior,theotherthreeexponentsremainconstant,similartothecaseforchangesina. Therefore,thereappeartobenoapparenttransitionboundaries;thedataforbcanbefoundinAppendixA,Figure52andTable17.
2.5.3.3 c (small‐scale)
Thechangesinparameterc mirrorthoseina;insteadofapositivelytrendingerraticnegativeexponent,ittrendsrapidlynegative.Aswiththeotherparameters,thereislittletonochangeinthefirstthreeexponents,andthereforenoreasontosuspectaphasechangeinthesystem.SeeFigure53andTable18inAppendixAforthevisualsanddeviations.
2.5.3.4 d(small‐scale)
Unliketheothertwoattractors,theRosslerHyperchaossystemhasafourthdimension,andthereforeafourthparametertovary.Thechangesinparameterd causeoscillationsinthefourthexponentsimilartochangesintheb parameter,aswellasanincreasingtrendsimilartochangesintheaparameter;theexponentfailedtoconvergeforanyvalueofdhigherthan0.545.Thefirstthreeexponentsareagainprimarilyuniformthroughouttheirlength,sonotransitionboundarieswereobserved.SeeAppendixAforfurtherdata.
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2.6 Large‐ScaleParameterVariations
Theparameterperturbationsonthelargescalewereontheorderof101oftheoriginalparametervalue,withapproximately20differentparametervaluesforeachsystem.Forexample,theRosslerattractor’soriginalWolfparameterswerea=0.15,b=0.2,c=10.0.Theaparameterwasvariedfrom0.5‐0.35inastepsizeof0.5,withextrapointsnearthe0.35mark;thiswasduetotheinstabilityoftheattractorbeyondavalueofa=0.38,atwhichpointtheMATLABODEsolverfails.Theothertwoparametersareheldconstantwhiletheaparameterisbeingvaried.Similarly,a andcareheldconstantwhilebisvariedfrom0.05‐1.0,anda andbareheldconstantwhilecisvariedfrom1‐20.ThiswiderangeforeachparametervalueallowedforthequalificationofanysignificantchangesinLyapunovexponentsolutionduetoincreasingchaosintheattractor,thusdeterminingtheparametertransitionboundaries.
2.6.1 LorenzSystem
2.6.1.1
�
σ (large‐scale)
The
�
σ valuewasthemostsignificantdriverforchangesintheLyapunovspectraoftheLorenzoscillator,particularlyinthethirdexponent;asmentionedearlier,thisexponentistheleaststableofthethree,andthereforetheonemostpronetochangesinthesystem.Figure9showsthespectrumprogressioninallthreeexponentsforchangesin
�
σ fortheWolfvariation:
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Figure9:LorenzLyapunovspectrumforchangesin
�
σ fortheWolfVariation(
�
σ = 16, R = 45.92, b = 4.0)
Theclassicalsystembehavesinaverysimilarmanner,withlarge‐scalechangesinsigmapromptingarapiddecreaseinthevalueofthethirdexponent.Thestandarddeviationsforthethreeexponentsshowthatthepositiveexponent(1)isaffectedmorethanthezeroexponent(2)forallthreesystems,withthenegative(3)exponentvaryingbyamuchlargerfactor,asseeninTable5:
Variation Exponent1
�
σ Exponent2
�
σ Exponent3
�
σ Wolf 0.5444 0.1002 6.612Classic 0.7171 0.3538 6.577
Table5:StandarddeviationforLorenzLyapunovspectrumforlarge‐scalechangesin
�
σ
ThefirstandsecondexponentsshowqualitativelysimilarparametertransitionboundariesinbothLyapunovspectrumandbifurcationanalysis,asseeninFigures10and11:
Figure10:LorenzfirstLyapunovexponentchangeandHopfbifurcationforlarge–scalechangein
�
σ
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Figure11:LorenzsecondLyapunovExponentchangeforlarge‐scalechangesin
�
σ
Bothexponentsshowarapidincreaseforearlysigmachanges,transitioningtoarelativeplateauofstabilitywithlittletonosignificantchange,andfinallyadecreaseatincreasingvaluesofsigma;theinitialvaluesforbothvariationsrestinthestableareaoftheparameterrange.Theresultsindicatethataparametertransitionboundarydoesexistatbothlowandhighendsofthesigmaspectrum,withhighinstabilityinLyapunovvaluesforsmallvaluesofsigma.TheHopfbifurcationsmarktheboundariesinmostcases,thoughforthefirstexponentitoccursearlierthantheLyapunovtransitionindicates.
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Thethirdexponentshowsalineardecreaseas
�
σ increases:
Figure12:LorenzthirdLyapunovexponentchangesforsmall‐scalevariationinSigma
Thereisonlyaslighttransitionaryperiodforthethirdexponent,withbothvariationsstabilizingforthemiddlerangeofsigmavalues;theClassicvariationappearstotransitionagainattheendofthesigmarange.Againthebifurcationsindicatethetransitionperiods,thoughmoresubtlethanthoseindicatedbythefirstandsecondexponents.
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Figure13:LorenzfirstLyapunovexponentchangeandphase‐spaceforlarge–scalechangein
�
σ fortheClassicvariation
TheLorenzphasespacetransitionsthroughthreedistinctformsasthe
�
σ parameterischangedintheClassicalvariation.Thefirstphase,seeninFigure14,isaunitcycle;inthisregionof
�
σ theequationsofmotionhaveasteadystatesolution.
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Figure14:LorenzClassicvariationsteady‐statephase‐spacefor
�
σ =2.0
Afterthetransitiontochaoticbehavior,theoscillatorphasespacelookslikeFigure2,withtwodistinctlobesandclassicattractorbehavior.Afterthesecondtransitionthephasespacereturnstoastablefixed‐pointattractor,andtheorbitsallconvergetoonepoint,asseeninFigure15:
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Figure15:LorenzClassicvariationunstructuredphase‐spacefor
�
σ =19.0
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2.6.1.2 RVariation
ChangesintheRparameteraffectthesystemfarlesssignificantlythanthoseofsigma,bothinoverallLyapunovvariationandinrecognitionofclearparameterboundarytransitionswithbifurcationanalysis.ThechangeinRforbothvariationscanbeseeninFigures16and17:
Figure16:LorenzLyapunovspectrumchangesforRintheWolfVariation(
�
σ = 16, R = 45.92, b = 4.0)
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Figure17:LorenzLyapunovspectrumchangesforRintheClassicVariation(
�
σ = 10, R = 28.0, b = 8 /3)
AsseeninthepreviousfiguresandTable6,thereislittlechangeamongsttheexponentscomparedtothosechangesmadewithSigmavariations.
Variation Exponent1
�
σ Exponent2
�
σ Exponent3
�
σ Wolf 0.1319 0.0005465 0.1322Classic 0.1227 0.0002564 0.1231
Table6:StandarddeviationsforLorenzLyapunovSpectrumforsmall‐scalechangesinR
Thoughthereisaslightlinearincreaseforthepositiveexponent(1)inbothcases,andconverselyamorepronouncedlineardecreaseforthenegativeexponent(3),therearenocleartransitionsinphasespace.
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2.6.1.3 b(large‐scale)
Forvariationsinthebparameter,theClassicvariationshowedsignificantdifferencesinbothparametertransitionandoverallexponentbehavior.Thethreeexponentvariationfiguresandstandarddeviationtablearedetailedbelow:
Figure18:LorenzfirstLyapunovexponentchangesforlarge‐scalevariationsinb
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Figure19:LorenzsecondLyapunovexponentchangesforlarge‐scalevariationsinb
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Figure20:LorenzthirdLyapunovexponentchangesforlarge‐scalevariationsinb
Variation Exponent1
�
σ Exponent2
�
σ Exponent3
�
σ Wolf 0.2104 0.0003961 1.572Classic 0.5659 0.1569 0.6625
Table7:StandarddeviationsforLorenzLyapunovspectrumforsmall‐scalechangesinb
TheClassicvariationshowssignificantdeviationsinallthreeexponentsthroughtherangeofbetavalues,quicklytrendingnegativeasthebetavalueincreases.ThoughthehigherbetavaluesforthisvariationareonthesamescaleastheWolfvariation,thecombinationofhighbetavalueswiththeothertwoparameterscausesthesystemtobecomeunstable,ascanbeseeninFigure18.Forthefirsttwoexponentsthereisanotabletransitionboundaryatb=3,wheretheLyapunovexponentvaluesrapidlytrendbelowzero.TheWolfvariationshowsnoneoftheseboundaries,andremainsstablethroughouttherangeofparametervalues.TheClassicvariationbifurcationsitsintheLyapunovtransitionzoneforallthreeexponents,whiletheWolfbifurcationslightlyprecedesthetransition.
Forvalidationinterestthesmallrangeofpermissibleb valuesfortheClassicvariationshouldbenoted.Asthetransitionoccurs,theLorenzsystemquicklyunravelsfromclassicchaoticbehaviortothestablefixed‐pointconvergencestructureseeninFigure15.Thefollowingfigureshowsthesystemastransitionsbetweenthetwophases:
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Figure21:LorenzClassicalvariationtransitionbetweenchaoticandstablebehaviorforchangesinb
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2.6.2 RosslerSystem
2.6.2.1 a (large‐scale)
LiketheLorenzsystem,theRosslerattractorisathree‐dimensionalsystemthatcanbreakdownfromitstypicalstructurewhenparameterchangesbecometoopronounced.ItshouldbenotedthatAUTOhaddifficultycalculatingthebifurcationpointsforthissystem.Thismayhavebeenduetotheneedforanegativeaparametertofindthesteadystatesolution,fromwhichthechaoticbifurcationsmaybecalculated.ChangesinLyapunovspectrafromvariationina canbeseeninFigures22‐24:
Figure22:RosslerfirstLyapunovexponentchangesforlargescalechangesina
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Figure23:RosslerfirstLyapunovexponentchangesforlargescalechangesina
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Figure24:RosslerthirdLyapunovexponentchangesforlargescalevariationsina
SimilarlytotheLorenzsystem,changesinthefirstparameteraremostevidentinthethirdexponent.Itshowsthehighestlevelofvariationamongthethree,asseeninthestandarddeviationvalues:
Variation Exponent1
�
σ Exponent2
�
σ Exponent3
�
σ Wolf 0.09301 0.03894 0.7853
Sano/Sawada 0.0798 0.09224 0.4550Table8:StandarddeviationsforRosslerLyapunovspectrumforsmall‐scalechangesin
a
OnesignificantchangefromtheLorenzoscillatoristhemovementofthethirdexponent:ratherthantrendinglinearlynegative,ithasapositivetrendasaincreasesforbothsystems.Also,forsmallvaluesofa, thesystembeginstoconvergetoafixedpoint,atwhichpointallexponentsarenegative.Asaincreasesto0.1,theLyapunovdistributionis(0,‐,‐),signifyingaunitcycleofperiod1;after0.1thedistributionreturnstoafamiliar(+,0,‐),astandardstrangeattractor.Figure25showstheattractoratana valueof0.05,whereitconvergingfromafixedpointtoaunitcycle.
Thesecondandthirdexponentsshowclearparametertransitionboundaries,thoughtherangeofexponentstabilityismirrored.Thesecondexponentstabilizesasa increases,levelingofftoastablevaluearounda=0.2,whilethethirdismoderatelystableatlowvalues,
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onlytoincreaseafterthe0.2mark.Thefirstexponentistheleastdefinedofthethree,withnoapparenttransitionareasatanypointintherangeofparametervalues.
Figure25:Rossleroscillatorfor
�
a = 0.05,b = 0.2,c = 10
2.6.2.2 b (large‐scale)
VariationsinbfortheRosslersystemwereatorbelowtheorderofaccuracyfortheLyapunovalgorithm.AsseeninFigure26,thevaluesforallthreeexponentswererelativelystableacrosstheentirerangeofchangesinb:
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Figure26:RosslerLyapunovspectrumchangesfromlarge‐scalevariationsinb
Theoscillationsseeninthethreeexponentswereonalevelsmallenoughtobeontheorderofaccuracy,andthereforethebparametershowsreasonablestabilityhasnoapparenttransitionboundariesatanypointinthemeasuredrangeofvalues:
Variation Exponent1
�
σ Exponent2
�
σ Exponent3
�
σ Wolf 0.02961 0.009072 0.03233
Sano/Sawada 0.02859 0.02681 0.06090Table9:StandarddeviationsforRosslerLyapunovspectrumforsmall‐scalechangesin
b
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2.6.2.3 c(large‐scale)
Forthisparameter,thesystemevolvedsimilartothatoftheLorenzoscillator.Thethirdexponentunderwentafamiliarlineardecreaseasc increased,thefirstexponentstayedpositiveforalmosttheentirespectrum,andthesecondexponentoscillatedaround0formostoftheduration,seeninFigure27:
Figure27:FirstLyapunovexponentchangesforsmall‐scalevariationsinc
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Figure28:SecondLyapunovexponentchangesforsmall‐scalevariationsinc
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Figure29:ThirdLyapunovexponentchangesforsmall‐scalevariationsinc
Thefirstexponentshowsoscillatorybehavior,withnoapparenttransitionboundaries;itappearsthattheexponentslightlystabilizesbetweenc=10andc=15,butthenbeginstooscillateagain.Thesecondexponentshowsmoredelineatedbehavior,withparametertransitionboundariesatbothlowandhighendsofthespectrumofc values.Therearenoapparenttransitionsforthethirdexponent,whichshowslinearbehaviorthroughoutitslength.
Notably,bothvariationsundergobriefperiodsofperiodicitythroughouttheextentofparametervariation,returningtoachaoticstateaninstantlater.Figure30showsthephasespacebehaviorfordifferingexponentvalues:
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Figure30:RosslerfirstLyapunovexponentchangeandphase‐spaceforlarge–scalechangeinc
Forsmallvaluesofc, thesystemexhibitsperiodicity.Figure31wasrunfor500seconds,withan0.001stepsizeforthetimeinterval:
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Figure31:Rossleroscillatorperiodicityfor
�
a = 0.2,b = 0.2,c = 4
ComparethepreviousfiguretoFigure32,whichruninthesamemannerforac valueof20:
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Figure32:Rossleroscillatorperiodicityfor
�
a = 0.2,b = 0.2,c = 20
Forthiscasetheattractorexhibitschaoticbehavior,anddoessoformostvaluesofc. ItisapparentthattheresultsofFigure32arenotaresultofthenumberoforbits,butrathertheirpaththroughphasespace.
ThestandarddeviationsforchangesinthisparameterappearverysimilartotheLorenzcase,withsmallchangesinthefirsttwoexponentsfollowingbysignificantchangesinthethird:
Variation Exponent1
�
σ Exponent2
�
σ Exponent3
�
σ Wolf 0.0387 0.1457 5.948
Sano/Sawada 0.04787 0.04840 5.863Table10:StandarddeviationsforRosslerLyapunovspectrumforsmall‐scalechanges
inc
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2.6.3 RosslerHyperchaosSystem
Thehyperchaossystemisextremelysensitivetoparameterinputs,andassuchisimpossibletochangeonalargescaleliketheothertwosystems.Asseeninsection2,thereisaverysmallrangeforwhichthesystemcanbesolvedviaanODEsolverinMATLAB.Itappearsthatthesystemhasnotransitionboundaries,andonlyexistsforaverysmallrangeofparametervalues.
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3. ApplicationofLyapunovExponentstoIntact&DamagedShipStabilityCases
3.1 ApplicationofFTLEstoDynamicShipMotion
TheLyapunovExponentapproachwasusedfortwodifferentscenarios:damagedstabilitydataforacommercialpassengerRo‐Roshipmodel,andtheintactstabilityofnotionaldestroyerDTMBhullmodel5514.Bothanalysesuserollandpitchdatathathasbeennormalizedwithrespecttothemeanandstandarddeviation;theroll‐velocityandpitchvelocitywasthencalculatedbasedonthesenormalizedvalues.
3.2 DamagedStabilityofaCommercialPassengerRo‐RoShip
ThedataforthedamagedstabilityanalysiswasprovidedbyDr.AndrzejJasionowskioftheShipStabilityResearchCenteroftheUniversitiesofStrathclydeandGlasgow(Jansionowski,2001).ThemodeltestswereperformedattheDennyTankattheUniversityofStrathcyldeona1:40scalemodelofapassengerRo‐Rovehicle(Jansionowsky,2001).
Forthisdataset,FTLEtimehistorieswerecalculatedalongwiththeperiodbetweenneighboringFTLEmaxima,andbothplottedvs.timeasshowninFigures1and2.Theperiodcalculationswereemployedinanattempttoprovideinstantaneousqualitativeandquantitativemethodsfordeterminingthetimeforadvancewarningofextremeshipmotions.Theperiodmeasurementsaremadeusingareversedifferencemethod,inordertosimulatereal‐timedatacollection.Thismethodsearchesbackwardsinthetimeseriestofindneighboringpointstopopulatethedisplacementvectors yi and zi ;thebackwardsapproximationwasusedtomorecloselyapproximatearealisticon‐boardscenariowheretheonlyavailabledatawouldbeloggedtime‐historiesforpreviousshipmotions.
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3.2.1 PeriodMeasurement
Figure33showsthefullrangeofroll,period,andFTLEdatathatwascalculatedforDamagedStabilityRun101:
Figure33:DamagedStabilityRun101.Fromtoptobottom:Rollvs.Time,Periodvs.Time,FTLEvs.Time
ThefourLyapunovexponentsmeasuredforthissystemwere
�
λ1 = 0.4882, λ2 = −0.08540, λ3 = −0.6922, λ4 = −2.854 ,qualitativelyidentifyingthesystemwithoneexpandingaxisandtwocontractingaxesintheballofinitialconditionpoints;thesecondexponentistheslowlychangingprincipalaxis,andwouldlikelytrendtoazerovalueinaninfinitetimeseries.Ingeneralapositiveexponentreflectsachaoticalsystem,azeroexponentidentifiesastableorbit,andanegativeexponentcharacterizesaperiodicorbit;however,theexistenceofanypositiveexponentidentifiesitasachaotic,ratherthanstableorperiodic.
Inthisanalysis,themeasuredperiodvalueistheΔt betweenneighboringFTLEmaxima,calculatedwithabackwardsapproximation;thisdeltavalueisindicatedbytheredarrowinFigure34:
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Figure34:DamagedStabilityRun101.CloseupofFTLEvaluesandperiodmeasurement
ThoughitappearsfromthescaleofFigure33thatthereisaperiodmeasurementateverytimestep,periodmeasurementsonlyoccurateachmaxima;thegreaterthespacingbetweenmaximapoint,thegreatertheperiod.TheperiodmeasurementoftheFTLE’siscloselylinkedtothedrop‐outpointsintheFTLEmeasurements;thesedrop‐outsoccurwherethecodecannotfindenoughneighboringpointstofillthe yi and zi vectors,andthealgorithmautomaticallyappliesanarbitrarilyhighvaluetotheFTLE,asseeninFigure35:
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Figure35:DamagedStabilityRun101.FTLEandPeriodmeasurementsvs.Time
Figure35makesitapparentthatthelargestperiodmeasurementsaredirectlytiedtolackofneighbors,ratherthanany“stretching”oftheFTLEvalues;thoughdirectlymeasuringthenumberofneighborsprovedtobeabettersolution,theperiodmeasurementsprovidesimplevisualcuesforextrememotion,andwereadequateadvanceindicatorsforlargerollamplitudes,asillustratedinTable11.Themaximumrollamplitudesforeachtimeserieswererecorded,alongwiththetimeatthatpointandthetimeoftheprecedingperiodspike.
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RunID Max.RollAmplitude(Degrees)
TimeofMax.Roll(Seconds)
TimeofPeriodMax.(Seconds)
Lead‐Time
Run101 ‐10.368 575.44 542.97 32.47
Run101 ‐15.983 1083.19 1022.67 60.52
Run102 ‐13.98 1144.13 1112.50 31.63
Run366 ‐11.613 1050.93 1055.14 ‐4.21
Run398 ‐18.296 715.87 685.51 30.36
Run399 ‐16.49 1122.41 1108.92 13.49
Run400* ‐29.345 1993.9 1959.95 33.95
Run400* ‐29.345 1993.9 2001.70 ‐7.80
Run401 ‐12.52 502.4 436.48 65.92
Run402 ‐17.46 500.5 436.91 63.59
Table11:Leadtimeforperiodcorrelationofmaximumrollamplitudes
Thepredictiveresultsshowagreatdealofvariation;theaveragelead‐timeis31.99seconds,thestandarddeviation26.26seconds,andthevariance689.6,withtheaveragebeingslightlyskewedtowardsthelargervalues.However,therearesomecaseswheretheperiodspikesarenotpredictiveatall;theyaremerelyreactingtothelargemotionsaftertheyoccur,asrepresentedbythenegativevaluesinthetable.Someofthisinconclusivelyisduethevariationsinneighborvectorsfordifferentrollseries;theneighborvectorscanvarygreatlybasedontheprecedingshipmotion.Forexample,ifaseriesofdataundergoeslargeamplitudemotionstwiceduringitsduration,thenthesecondmotionwillfindmoreneighboringpointstopopulatethevectorsandonlyaveryextremerollorpitchmotionwillcausealossinthenumberofneighboringpoints.Inashipboardapplication,thecodecouldpotentiallyhaveavastnumberofdatapointstosortthroughtofindneighboringpoints.Withalargedatabaseofdataathand,onlysignificanteventswouldcontainrollvalueswhereveryfewneighborscouldbefound,e.g.irregularlarge‐amplitudeshipmotions.Anothercauseofthesenegativevaluesisthetime‐delayinperiodcalculation,wherealossofneighborstakesanumberoftime‐stepsbeforeitisreflectedintheperiodspike,ascanbeseeninthemajorperiodspikesofFigure35.
Accuratelydeterminingwhichperiodspikeisaflagforthelargeamplituderollisthemostsignificantchallengeoftheperiodmeasurementtechniques.Onatypicalrun,eachlarge‐amplituderollmotioncancreatemultiplelargespikesinFTLEperiod.Run402isagoodexampleofthedifficultyinherentinusingtheperiod‐measurementmethodasapredictorforthemostextrememotions:
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Figure36:Rollvs.TimeandFTLEperiodforCapsizeRun402
Thefigureaboveisqualitativelysimilartotheperiod‐measurementresultsformostoftheanalyzeddatasets;periodspikeswereseenearlyinallruns,asaresultoftheinitiallackofdatafromwhichtopullneighboringpoint.Examinationoftherolltimeseriesmakesitapparentthatmanyoftheperiodspikesareeitherreactingtoorslightlypredictinglargelocalvariationsinroll.Whiletheselocalvariationsareimportant,thisstudyismostconcernedwithpredictingtheextremevariations,andthereforethedatainTable11wascompiledwiththelargestrollvalueinmind;forthecaseofRun402,thelargestamplitudeoccurs500secondsintotherun,andthefirstmajorprecedingperiodspikeat436seconds,asseenbelow:
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Figure37:Markedperiodindicatorforlargestamplitudemotion,Run402
Itisapparentthatsomeoftheperiodspikesarereactingtotheshipmotions,butdifficulttoascertaintheirpredictivenature.Thespikeat243secondscouldbeanindicatorforthelarge‐amplitudemotionstocome,oritcouldbereactingtothequickrolloscillationatthatpointinthetimeseries.Theperiodmarkersprovedtobeinconclusivepredictorscomparedtosimilarpredictionsmadebycalculatingthenumberofneighbors,giventhetime‐delayinherentinareverse‐approximationmethod.Ultimately,neighborhoodmeasurementsprovetobeasuperiorpredictivemethod.
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3.2.2 NeighborMeasurement
Whiletheperiodmeasurementsdoanadequatejobofpredictingtheextremerollmotions,Figure35showsthatthelargeperiodspikesarereactingtothelossofFTLEneighbors,whichisinturnreactingtotheupcominglargemotionamplitudes.Thisobservationledtoamodificationofthealgorithmthatsolvesonlyforneighboringdatapoints,ratherthantheactualFTLEvaluesthemselves.Thethresholdforthenumberofneighborswassetat50;ifmorethan50neighboringpointsarefoundtofillthe yi and zi vectors,thenthecodecontinuestoiterate.Belowthisvaluetheneighboringpointsarecountedandgraphedinrelationtotherollmotions.Thefollowingfigurepresentsatypicalrunwiththecountingofneighborvalues:
Figure38:DamagedStabilityRun101Rollvs.NumberofNeighbors.
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Figure39:DamagedStabilityRun101zoomofneighborcounting
Figure39givesabetterillustrationofwhatisoccurringastheneighborsarebeingcounted.Thebluedatalineistherollamplitude,thegreendatalinethenumberofneighbors.Thedropinneighborcountcontributesdirectlytothespikesinmeasuredperiodvalues;inthepreviousalgorithmacompletelossofneighbors(numberofneighborsdecreasesto0)causestheFTLEvaluesdropouttoasetvalueof‐500,andtheperiodamplitudeincreasesduetotheselargegapsbetweenFTLEmaxima.Figures38and39showwhytheneighborcountisultimatelymoreusefulthantheperiodmeasurements.Theredoesnotneedtobeacompletelossofneighborsforwarningflagstogoupregardinglackofneighboringpoints.Inthecaseabove,anyvaluethatisfallingbelowaneighborcountof50canbeseenasawarningflagwithrespecttolargeamplitudemotions.Whereasthetimeofmaxperiodforthisrunwasflaggedat1022.67seconds,thedropofneighborhoodcountbelow50neighborsoccursat1016.56seconds.Whilethisisnotahugeincreaseinlead‐time,9secondscanbeasignificantamountoftimeinregardstosplit‐seconddecision‐makingbyacaptainorcrew,andanyincreaseinwarningtimewillbetotheiradvantage.
AsseeninFigure38,lossofneighborsoccurserraticallyacrosstheentiretimeseries.Determiningwhichneighborhoodlosstomarkastheindicatorforaparticularmaximumamplitudeissomewhatofaqualitativedecision;thealgorithmmusttakeperiodsofstablebehaviorwheretherearenodrop‐outsintoaccount.Inanattempttoquantifythis
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neighborhoodlossandfindacomputationalsolutionthatwouldn’trequireavisualinspectionofthedata,asummationwasusedtoflaga“danger”marker.Foreverystepintimewherethenumberofneighborsfellbelow50,thevariable“flag”wasincreasedby1;therefore,themorestepsintimethatwereprogressingwithalackofneighbors,thesteepertheslopeoftheflagvariable,asseeninFigure40:
Figure40:DamagedCaseRun101Rollvs.Flag
Theflagvariable,increasinginvalueacrosstheentiretimeseries,experiencesdrasticincreasesinslopewherethereisalackofneighboringpoints,asaresponsetoincreasingshipmotionamplitudes.Whentheslopereachesacertainsteepness,asseeninFigure38justpastthe1000secondmark,a“danger”markerisflaggedasasignofincreasedamplitudemotion.Thedangermarkersprovideamoreconcreteloss‐of‐neighborindicator,andcanbeseenwithregardstorollmotioninFigure41:
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Figure41:DamagedCaseRun101Rollvs.Dangerindicator
Table12replicatestheresultsofTable11,usingthedangerindicatorratherthantheFTLEperiodasthemetricforlead‐time:
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RunID Max.RollAmplitude(Normalized)
TimeofMax.Roll(Seconds)
TimeofDangerFlag(Seconds)
Lead‐Time
Run101 ‐1.7587 575.44 539.80 35.64
Run101 ‐15.983 1083.19 1016.56 66.63
Run102 ‐13.98 1144.13 1110.39 33.74
Run366 ‐11.613 1050.93 1039.54 11.39
Run398 ‐18.296 715.87 671.82 44.05
Run399 ‐11.085 330.40 321.78 8.62
Run399 ‐16.49 1122.41 1103.50 18.91
Run400* ‐29.345 1993.90 1899.01 94.89
Run401 ‐12.52 502.40 340.12 101.99
Run402 ‐16.46 453.60 431.42 36.52
*CapsizeCase
Table12:Leadtimeforneighborcorrelationofmaximumrollamplitudes
Theaveragelead‐timefortheneighborhood‐loss“danger‐flag”methodis45.24seconds,a13.25secondimprovementovertheperiodmeasurementmethod.Thestandarddeviationandvariancebothincrease,to32.70secondsand1069.0respectively.However,moreimportantly,the“danger”spikesareamoreconcretequantitativeindicatorthantheperiodmeasurementmethod.Multiplelossesofneighborsisstillahurdle;liketheperiodspikes,insomerunsitcanbedifficulttodeterminewhich“danger”spikeisreactingtowhichlargeamplitudemotion,thoughtheclustersofspikestendtosignifyalargeramplituderollevent.Theflagsummationapproachremovesmuchoftheambiguityandsubjectivityoftheperiodandsimpleneighborcountingmethods,buttherearestillcaseswheremultiple“danger”spikesoccurbeforealargeamplitudeevent,andwhichonetodesignateasthetruewarningspikerequiresadecisiononpartofresearcher.Forfutureshipboardapplications,thealgorithmwouldneedtodeterminewhentosignalawarningwithoutanyhumaninput.
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3.3 ApplicationtoNotionalHullform5514CapsizeCases
Theperiodandneighborhoodmethodswereappliedtothe5514hullformdatainasimilarmannertothedamagedstabilitycase.Unlikethedamagedstabilitydata,allofthe5514runsthatwereanalyzedwerecapsizeruns.Inordertoprovideafull‐setofdatafortheneighbor‐findingprocess,all37differentrunswereanalyzedforneighborpoints,ratherthanattemptingtodrawneighborsfromthelimitedsetofdatacontainedinonecapsizerun.Thistechniqueprovidedanexcellentexampleofhowthisprocesscouldbeusedinreal‐worldapplications,wheretherewouldbemanyhoursofshiprollingdatatousefortheneighborsearchingprocess.
PreviousapproachesbyMcCueet al.exploredtheFTLEandLyapunovexponentvaluesexclusively,andusedroll/roll‐velocityandpitch/pitch‐velocityastheirstate‐spacevariables(McCueet al. 2006).Thisresearchfurtherstheirwork,withchangesinneighborcountingmethodsandapplicationofnewwarningalgorithms.
Aswiththedamagedstabilitycases,neighborhoodmeasurementswerebetterpredictorsforcapsizethanperiodindicators.Forthecapsizerunstheneighborhoodsizedroppedprecipitouslynearthebeginningoftherun;forthisstudyathresholdoffiftyneighboringpointswasused.Figure42illustratesatypicallossofneighborsforaHullform5514run,wheretheneighborhoodsizefallsbelowfiftyastheinstabilitiesapproach:
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Figure42:Hullform5514Run216Rollvs.NumberofNeighbors
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Figure43:Hullform5514Run216Rollvs.Period
Thecontrastbetweenthesetwofiguresreinforcesthestrengthoftheneighborhoodcountingcodeversustheperiodindicators.ForthecaseofRun216,thedropinnumberofneighborsprecedestheleadperiodspikeby5.5seconds,alead‐timeadvantagethatcarriedthroughallofthe5514capsizeruns.
Forrun216,thecapsizeeventoccursat11.88seconds,thefirstperiodspikeat9.75seconds,andthefirstdropofneighborhoodsizeat2.17seconds.Theperiodspikeresultsinalead‐timeof2.13seconds,andtheneighborhoodlossaleadof9.71seconds.Whileatfirstglancea2to9secondwarningappearstobeatrivialamountoftime,itisworthnotingthatboththefirstperiodspikeandlossofneighborsoccursintherealmofstabilityfortheship,asseeninFigures44and45:
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Figure44:Hullform5514Run216Rollvs.RollVelocitybasinofstabilityforperiodindicators
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Figure45:Hullform5514Run216Rollvs.RollVelocitybasinofstabilityforneighborhoodindicators
Theperiodmeasurementmethodforrun216hasthreemarkers,eachofwhichrepresentsasignificantperiodspikeinthetimeseries;notethatthelasttwomarkersinFigure43arenotgoodindicators,giventhattheshiphasalreadycapsizedbasedontherolltime‐history.Eachbasinofstabilityfortheneighborcountingmethodhastwohighlightedmarkers:themostoptimisticindicator,andaconservativealternative.Inthecaseoftheneighborhoodcountingmethod,bothmarkerssitwellinsidethebasin.Fortheperiodindicators,thefirstperiodspikesitswithinthebasin,butthesecondtwooccuraftercapsize,andwelloutsidethebasinofstability.Giventheseresults,theproceedingdiscussionwillonlyinvolvetheneighborhoodcountingmethod.Whiletheperiodindicatorsareaninterestingstudy,neighborhoodcountingconsistentlyprovidesalongerlead‐timeindicatorforcapsizecases.Figures46and47detailthelossinnumberofneighborsforrun327,another5514capsizecase:
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Figure46:Hullform5514Run327Neighborhoodloss
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Figure47:Hullform5514Run327Rollvs.RollVelocityneighborhoodlossmarkers
Againboththeoptimisticandconservativemarkerpointsliewellwithinthebasinofstabilityforthecapsizerun.Theleadtimeforthesepointswere4.91and4.12seconds,respectively,andareanumberofiterationsfromthepointwheretheshipdeviatesfromstablebehavior.Figure46isanexcellentexampleoftheconservativeandoptimisticmarkerpoints;thefirstlossofneighborsrecoversquickly,butthesecondconsistentlyfallsbelowthetwentyneighborthreshold.Figure48showsthecapsizecaserollvs.rollvelocitydatafortheotherthreecapsizeruns:
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Figure48:Hullform5514Runs220,331,333Rollvs.RollVelocityNeighborhoodlossmarkers
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Eachoftheothercasesshowneighborlossesoccurringwellwithintherealmofstabilityforthecapsizecase,oftenmanycyclesbeforetheshipfallsoutsideofthatstabilitybasin.ForthecaseofRun333itappearsthatthemarkersitsoutsideofthebasinofstability,butmuchcloserexaminationrevealsittobeananomalythatreturnstothebasinfornumerouscycles.Forthesecasestheactualleadtimegivenbythelossofneighborsislessthanwhatwasseenforthedamagedstability,butisstillofconsequence:
RunID TimeofCapsize(s) TimeofNeighborLoss(s) LeadTime(s)
Run216(Marker1) 11.88 2.17 9.71
Run216(Marker2) 11.88 4.17 7.71
Run220(Marker1) 35.63 2.04 33.59
Run220(Marker2) 35.63 5.79 29.84
Run327(Marker1) 9.29 4.38 4.91
Run327(Marker2) 9.29 5.17 4.12
Run329(Marker1) 57.21 10.58 46.63
Run329(Marker2) 57.21 30.08 27.13
Run331(Marker1) 32.71 5.88 26.83
Run331(Marker2) 32.71 23.08 9.63
Run333(Marker1) 53.46 7.42 46.04
Run333(Marker2) 53.46 35.13 18.33
Table13:LeadtimeforneighborhoodlosscorrelationofHullform5514capsizecases
Usingtheoptimistic“marker1”cases,theaveragelead‐timetocapsizeis27.95seconds;themoreconservative“marker2”scenarioshowsa12‐seconddrop,with16.13secondslead‐time.Thoughnotasgoodasthedamagedstabilityresults,itshouldbenotedthattheHullform5514timeseriesweremuchshorter,withasmallerpooloftimehistorytodrawneighborsfrom.Whilethesetimesmaynotseemlikealargeenoughtimeforanyshipcaptaintoreact,18‐24secondsenoughtimetomakeonemaneuver,oracoursecorrectionthatmightmeanthedifferencebetweenalargeamplitudeeventandacapsizeevent.
The“flag”summationmethodwasalsotestedfortheDDG51data;inthiscaseeachrunwasanalyzedforthemaximumflagvalueobtained,andnormalizedwiththetimelengthfortherun.Theaverageresultforthisnormalizedvalueonacapsizerunwas12.12flags/second,whereastheaveragenormalizedvalueforanon‐capsizerunwas5.68flags/second.Thismakesitapparentthatthecapsizerunsarefindingsignificantlylessneighborsthanthenon‐capsizeruns,whichtranslatestomoredangerflagsgoingupinthealgorithm.The
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normalizedcapsizevaluesrangedfrom5.22flag/sto21.27flag/s;thehigherthevalue,themoreoftenthenumberofneighborsisfallingbelowthe50‐neighborthreshold.Thenon‐capsizecaseshadanumberofanomalousruns,withlargenormalizedvalues‐manyoftheserunscameextremelyclosetocapsize,butregainedstabilityatthelastinstant.Thevaluesforthenon‐capsizerunstypicallyrangedfrom0.42flag/sto5.04flag/s,withmostofthevalueslyinginthe0.5‐1.0range.Theanomalousrunsrangedinvaluefrom6.23flag/sto23.40flag/s,whichwaslargeenoughtomarkitasacapsizerun.Whenanalyzingthenon‐capsizeruns,7ofthe31caseswereanomalous,andflaggedmistakenlyasacapsizerun.
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4. ApplicationofNeighborSearchingMethodtoReal‐timeShipMotions
4.1 Motivation
Thenextstepinapplicationofthepredictiveneighbormethodwastoapplythealgorithminareal‐timesetting,withadataacquisitionsysteminplacetomeasurerollandpitch,withtheneighborsbeingcountedateachinstantintime.Theobjectivewastorecordtherollandpitchvaluesforthedamagedshipdatainreal‐time,andreplicatetheneighborcountingmethodasseeninFigures38and39.Motivationforthisistoeventuallyimplementasimilarsystemonnavalorfishingvessels,withneighborsbeingcountedinanattempttopredictlarge‐amplitudemotionsatsea.
4.2 ExperimentalSetup
4.2.1 Data‐collection
Thedatafromthedamagedshipcasewasreplicatedonthemotionplatform(MOOG6DOF2000E)locatedintheVirginiaTechCAVE(AutomaticVirtualEnvironment).TheMOOGisa6D.O.F.hydraulicmotionplatform,withfreedomof20degreesinbothrollandpitch.TheplatformcanbeseeninFigure49:
Figure49:MOOGmotionplatform
TheMOOGiscontrolledbyanSGI/IRIXsystem,withapositionvectorwrittentoaDTKsharedmemorysegment.Thepitchandrollvaluesforthedamagedshipwerefedtothe
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platform,andwerethenreadtoaDellLatitudeD610laptopviaaCrossbowCXTILT02ECtiltsensor,asseeninFigure50:
Figure50:Crossbowtiltsensormountedonmotionplatform
Thetiltsensorisaccuratetowithin0.2degrees,withdigitaloutputviaaRS‐232serialinterface.ThedatafromtheserialportwasfeddirectlyintotheneighborcountingalgorithminMatlab,whichrecordedtherollandpitchvaluesinadditiontothenumberofneighbors.
4.2.2 Algorithm/Datamodification
Therollandpitchvelocitieswerecalculatedusingabackwardsapproximationfromtherollandpitchvalues,similarlytotheneighborcountingmethodsusedinsection3.Unlikethemeasurementsofthatsection,therewasnonormalizingofrollandpitchvalues;thenormalizationsforthatsectionwereperformedusingmeanandstandarddeviations,andtheobjectiveforthissectionweretoobtainneighbormeasurementsinreal‐timewithoutanysortofstandardizing.Futureworkcouldincludesomesortofmethodtonormalizevaluesinreal‐timebasedonpreviouslycollecteddata,buttheresultsofthisstudyweresatisfactorywithoutit.Additionally,furtherworkcouldbedoneinthefollowingareas:
• Normalizevaluesinreal‐time.Ameanandstandarddeviationwouldhavetobecalculatedforeachship,basedonlargesetsofpreviouslycollecteddata;thesewouldbeusedtoperformastandardnormalizationofthedata.
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• Optimizethealgorithmforlongdatasets,knowingthatitwillneedtoperformcalculationsfordays,weeks,orevenmonthsatsea.Notonlywilldatabuffersneedtoberoutinelycleared,butthedataneedstobedatabasedfortheneighborsearches.
• Databaseswouldneedtobesearchedforneighborsefficientlyenoughtoreactinreal‐time,asignificanttaskwhendealingwithtensofthousandsofdatapoints.
• Ahardwaresystemwithsimplewarnings,a“black‐box”sotospeak,wouldneedtobedevelopedforshipoperators.Captainscouldnotbeexpectedtoreadcomplicatedoutputsinacriticalsituation‐thewarningsystemwouldneedtobesimplebuteffective.
Thealgorithmusedinsection3couldnotbeimplementeddirectlyforreal‐timeneighborcalculations.Withthedatabeingcollectedontheorderofonevalueper0.03seconds,anumberofefficiencyandstorageproblemsarise.Withtheoldalgorithmthecomputerhadtosearchthroughtheentirehistoryofdatavaluesateachinstantintimetofindneighbors;whilethereisnoissuewiththiswhenoperatingonaprescribedsetofdata,alaptoplikeonethatwouldeventuallybeusedonashipboardapplicationisnotcomputationallyquickenoughtoperformasearchofallpreviouspointsonthetimescaledescribedabove.
Instead,thealgorithmwasmodifiedtoonlysearchthetime‐historyforneighborswhenanewareaofphasespacewasentered.Thenewalgorithmonlycountsneighborsforarollvaluethathasnotbeenencounteredbefore;iftherollvaluehasbeenloggedinthehistory,itdefaultstotheneighborvaluepreviouslyrecorded.Thisreducedan80,000+steptimeseriestoonly800‐1000actualneighborsearches,greatlyincreasingtheefficiencyofthealgorithm.
However,evenwiththealgorithmrunningmoreefficientlythanbefore,memorylimitationsbecameanissue.Withmultiplematricesover100,000pointsinsize,thealgorithmbegantofailafterabout80,000stepsintothetimeseries.Therefore,theresultsseeninthefollowingsectionwillbemissingtheveryendofeachtimeseries.Luckily,allofthelarge‐amplitudemotionsineveryrunoccurearlierthanthiscutoffpoint,sothedatacanbecompareddirectly.
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4.3 Real‐timeNeighborCountingResults
Aside‐effectoftheefficientalgorithmwasthatonlysignificantneighborlosseswererecorded.Forexample,compareFigure38insection3tothefollowingfigure:
Figure51:DamagedStabilityRun101,datarecordedfromMOOGplatform
AsseeninFigure51,themodifiedalgorithmismuchmoreefficientatrecordinglossesofneighborsthanbefore,whilestillcapturingthemajorneighborlossesthatoccuratsignificantmotioneventsinthetime‐series.Theseneighbordropoutsfunctionthesamewayassection3.2.2,justinreal‐time;thoughatfirstglanceitappearsthatthedropoutsarepurelyreactive,furtherexaminationoftimeseriesmakesitapparentthattheyareidentifyingsignificantchangesinshipbehavior,notjustlargeamplitudemotions.Thetwobeginningneighborlossessignifyanewregionsofunstablebehaviorfortheship,andthenumbersreflectthesefuturemotions.
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Knowingthatthedataqualitativelysatisfactorytotheresultsofsection3,thenextstepwastocomparelead‐timestothatofTables11and12.Itshouldbenotedthatthewarning‐flagmethodwasattempted,butabandonedwhennumerousrunsproducedgarbagedata.Thisisaresultofthecomputerattemptingtoiteratenestedfor‐loopsevery0.03seconds,atwhichpointitfailedtoevenrecordthecorrectrollandpitchvalues.Therefore,thelead‐timerepresentseverymajorlossofneighbors,whereawarningflagwouldhavecertainlyoccurredinthepreviousalgorithm;aconservativeestimatewasusedineverycase,andevenatthatthereal‐timemethodproducedsomestartlingresults,asseeninTable14:
RunID Max.RollAmplitude(Degrees)
TimeofMax.Roll(Seconds)
TimeofNeighborDropout(Seconds)
Lead‐Time
Run101 ‐10.368 575.44 514.11 61.33
Run101 ‐15.983 1083.19 1009.79 73.40
Run102 ‐13.98 1144.13 1102.91 41.22
Run366 ‐11.613 1050.93 975.48 75.45
Run398 ‐18.296 715.87 642.36 73.51
Run399 ‐16.49 1122.41 1008.83 113.58
Run400* ‐29.345 1993.90 1924.80 69.10
Run401 ‐12.52 502.40 412.11 90.29
Run402 ‐17.46 500.50 435.20 65.30
Table14:Leadtimeforreal‐timeneighborhoodlosscorrelationofDamagedStabilitycases
Forthereal‐timecasestheaveragelead‐timeis73.69seconds,withastandarddeviationof19.90secondsandavarianceof396.1;thelattertwovaluesarelowerforthiscasethanineitheroftheotherpreviousmethodsoutlinedforTables11and12.Thoughthisresultwasinitiallysurprising,thechangesmadetothealgorithmandmethodinwhichthedataiscollectedpointtothesignificantimprovementsinlead‐timevalues.Thedataisbeingcollectedataratemuchhigherthantheoriginaltime‐series,andthusthealgorithmiscollectingneighborsatamuchhigherrate.This,combinedwiththechangesoutlinedearlier,
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allowthealgorithmtoreactmorequicklytotheshipenteringapreviouslyunseenareaofphasespace.Itisapromisingsetofresultsforfutureapplications.
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5. Conclusions
5.1 VerificationandValidation
Chaoticattractorscanbeextremelysensitivetoinputsbynature.TheLorenzandRosslersystems,boththree‐dimensionalchaoticattractors,canundergoverylargechangesinparameterswithoutlosingtheirstandingasastrangeattractor.Forcertaincombinationsofparametervaluesbothsystemshavethepotentialtoshiftfromattractorstofixedpoints,ortoshowvaryinglevelsofperiodicity.
Thesefactorsbecomeimportantinvalidatingcodeforbothnumericalandexperimentalresearch.Whilebifurcationanalysisisausefultoolfordeterminingregionsofchaoticbehaviorfromanumericalapproach,itislimitedinitsapplicationtoexperimentaltimeseries.TheLyapunovexponentprovestobeaveryrobusttoolinthisregard;multiplemethodsofcalculatingtheexponentexist,bothfornumericalandexperimentaldata.ThisresearchhasshownthattheLyapunovapproachaccuratelycaptureschangesinphasespaceforchaoticbehavior,andcandosowithsimilaraccuracytoothermethodslikebifurcationanalysis.Bycomparingexponentspectrums,theresearchercaneffectivelyvalidatetheunderlyingphysicsofthedevelopedtheoreticalmodel,makingtheLyapunovapproachaveryimportantpieceoftheverificationandvalidationframework.
5.2 ApplicationtoShipCapsize
ThedatapresentedshowsthattheLyapunov/neighborcountingmethodprovestobeavalidwaytopredictcapsizeandlargeamplitudemotionsforagiventimeseriesofexperimentaldata.Thedamagedshipdatashowsthatthealgorithmsproveusefulforlarge‐motionanalysis,butitappearsthatthemethodismuchmoreusefulforcapsizecasesliketheonespresentedbythe5514data.Whilethelead‐timesgivenbythemethodsmaynotbeonascaleofminutes,butratherseconds,itmayoftenbethecasethatifashipcaptainknowsacapsizeeventisabouttooccur,asingledrasticcoursecorrectionormaneuvercouldbeundertaken.
5.3 FutureWork
WhiletheapplicationofLyapunovExponentstoshipcapsizeinanumericalandcontrolledexperimentalenvironmentisagoodstart,thereisstillmuchworktobedoneinordertoprovideausefulandreliabletoolinthefieldtoassistshipcaptainsinextremeseastates.RealisticallytheLyapunovmethodisbutoneapproachbeingtakeninregardstopredictingcapsizeorlarge‐amplitudemotions,andcanbeviewedasanothertoolinthenonlineardynamicanalysistoolbox.Thenextstepforthisresearchistoimplementasystemonatestvesselatsea,andbeingtoacquiresetsofnumericaldatafromwhichtodrawneighbors.Thispresentsafewnumericalproblems,includingdatastorageandalgorithmefficiency.Thetaskofpredictingnonlinearshipdynamicsisacomplicatedone,buthopefullyworksuchasthisinacademicandcommercialinstitutionsaroundtheworldwilleventuallyleadtosaferenvironmentsforbothpassengersandcrewatsea.
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AppendixA1. Figures
Figure52:Rosslerhyperchaoticattractorfor
�
a = 0.25,b = 3.0,c = 0.05,d = 0.5
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Figure53:LorenzLyapunovspectrumforsmall‐scalechangesinR
Figure54:LorenzLyapunovspectrumforsmall‐scalechangesinb
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Figure55:RosslerHyperchaosLyapunovspectrumchangesforsmall‐scalechangesinbforWolfsystem
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Figure56:RosslerHyperchaosLyapunovspectrumchangesforsmall‐scalechangesincforWolfsystem
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Figure57:RosslerHyperchaosLyapunovspectrumchangesforsmall‐scalechangesind forWolfsystem
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2. Tables
Variation Exponent1
�
σ Exponent2
�
σ Exponent3
�
σ Wolf 0.03228 0.0003941 0.2804Classic 0.04187 0.001866 0.04219
Table14:StandarddeviationsforLorenzLyapunovspectrumforsmall‐scalechangesinsigma
Variation Exponent1
�
σ Exponent2
�
σ Exponent3
�
σ Wolf 0.01722 0.0009249 0.1817Classic 0.03228 0.0093941 0.2804
Table15:StandarddeviationsforLorenzLyapunovspectrumforsmall‐scalechangesinb
Variation Exponent1
�
σ Exponent2
�
σ Exponent3
�
σ Wolf 0.003200 0.0005432 0.005284
Table16:StandarddeviationsforRosslerLyapunovspectrumforsmall‐scalechangesinb
Variation Exponent1
�
σ Exponent2
�
σ Exponent3
�
σ Exponent4
�
σ Wolf 0.004902 0.002564 0.000474 1.380
Table17:StandarddeviationsforRosslerHyperchaosLyapunovspectrumforsmall‐scalechangesinb
Variation Exponent1
�
σ Exponent2
�
σ Exponent3
�
σ Exponent4
�
σ Wolf 0.005657 0.003972 0.0007657 3.77
Table18:StandarddeviationforRosslerHyperchaosLyapunovspectrumforsmall‐scalechangesind
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AppendixB:ChoosingD.O.F.ParametersforbestNeighbors/FTLEresults
TheauthorconductedaparametersearchintotheinfluenceofchosenDOFforcalculatingtheFTLEvalues.Themostpronouncedshipmovementwasinroll,andsubsequentlyitprovedtobethemostrobustdegree‐of‐freedomforgeneratingFTLEvalues;furtherrunsdeterminedthatincludingroll‐velocityvs.rollprovidedevenbetterpredictiveresultsthanmeasuringthevaluesbasedonrollalone.ThetwofiguresbelowshowtheFTLEmeasurementsforthesamedatarun,thefirstonlygeneratingfortherolldegree‐of‐freedombyitself,andthesecondforrollvs.rollvelocity.
Figure58:DamagedStabilityRun101RollvsTimeandnon‐dimensionalizedFTLEperiodmeasurement
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Figure59:DamagedStabilityRun101Roll/RollVelocityvs.Timeandnon‐dimensionalizedFTLEperiod
Thegraphsabovebothshowthesamerolltimeseries,butwithdifferentembeddedparameters.Thetopsetofdatainblueistherollamplitude,whilethebottomsetofdataingreenistheperiodmeasurementoftheFTLEpointsthatwerecomputedforthetimeseries;eachseriesisrepresentedbyitsowny‐axis.Figure12showstheperiodmeasurementsforRollvs.Time,withoutrollvelocitybeingembeddedinthesolutionfortheFTLE’s,whileFigure13showsthesameperiodmeasurementwhenrollvelocityisembedded.Figure13showsamuchbettercorrelationbetweenlargerollmotionsandmarkedincreasesinperiodmeasurementintheFTLEvaluesascomparedtotheerraticismofthedataintheonestate‐spacevariablecaseseeninFigure12.
FTLEandperiodvalueswerealsocalculatedforpitchandpitchvelocity.Figure14showstheperiodcalcuationsforapurepitchcase:
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Figure60:DamagedStabilityRun101Pitchvs.Timeandnon‐dimensionalizedFTLEperiod
Thefigureshowstheerraticnatureofthepitchmeasurementsforthedata;thepitchmotionsshownoneoftheextrememotionsoftherolldata,andthusisnotasrobustforpredictinglargeamplitudemotions.Thecalculationofperiodvaluesforpitch‐pitchvelocitywasverysimilarinregardstoperiodmeasurement:
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Figure61:DamagedStabilityRun101Pitch/PitchVelocityvs.Timeandnon‐dimensionalizedFTLEperiod
Thepitch/pitchvelocitycaseseeninFigure15isjustaninconclusiveanindicatorasthesinglevariablepitchcase.Othercombinationsofthesedegreesoffreedomwereexplored,includingintegratingthefourDOFcaseofroll,roll‐velocity,pitch,andpitch‐velocity,asseeninFigure16:
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Figure62:DamagedStabilityRun101Pitch/PitchVelocity&Roll/RollVelocityvs.Timeandnon‐dimensionalizedFTLEperiod
Otherdegreesoffreedomwereconsidered,butafterscrutinyitappearedthatthe2D.O.F.caseofpitchandroll,extendedtofourstate‐spacevariableswiththeroll/pitchvelocitycalculations,wasmorethansufficienttocapturethemajorchangesinphase‐space.
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