ApplicationofLyapunovExponentstoStrangeAttractorsandIntact&Damaged

ShipStability

WilliamR.Story

ThesissubmittedtotheFacultyoftheVirginiaPolytechnicInstituteandStateUniversityinpartialfulfillmentofthe

requirementsforthedegreeof

MasterofScienceIn

OceanEngineering

LeighMcCue,ChairAlanBrownWayneNeu

April29,2009

Blacksburg,VirginiaTech

Keywords:Stability,Capsize,Lyapunov,Attractor,Lorenz

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.

iii

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.

v

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

vi

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

vii

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

viii

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

ix

�

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

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).

2

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).

3

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).

4

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

6

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.

7

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%.

8

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

9

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).

10

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.

11

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.

12

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 

13

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.

14

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:

15

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.

16

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:

17

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

18

Figure11:LorenzsecondLyapunovExponentchangeforlarge‐scalechangesin

�

σ

Bothexponentsshowarapidincreaseforearlysigmachanges,transitioningtoarelativeplateauofstabilitywithlittletonosignificantchange,andfinallyadecreaseatincreasingvaluesofsigma;theinitialvaluesforbothvariationsrestinthestableareaoftheparameterrange.Theresultsindicatethataparametertransitionboundarydoesexistatbothlowandhighendsofthesigmaspectrum,withhighinstabilityinLyapunovvaluesforsmallvaluesofsigma.TheHopfbifurcationsmarktheboundariesinmostcases,thoughforthefirstexponentitoccursearlierthantheLyapunovtransitionindicates.

19

Thethirdexponentshowsalineardecreaseas

�

σ increases:

Figure12:LorenzthirdLyapunovexponentchangesforsmall‐scalevariationinSigma

Thereisonlyaslighttransitionaryperiodforthethirdexponent,withbothvariationsstabilizingforthemiddlerangeofsigmavalues;theClassicvariationappearstotransitionagainattheendofthesigmarange.Againthebifurcationsindicatethetransitionperiods,thoughmoresubtlethanthoseindicatedbythefirstandsecondexponents.

20

Figure13:LorenzfirstLyapunovexponentchangeandphase‐spaceforlarge–scalechangein

�

σ fortheClassicvariation

TheLorenzphasespacetransitionsthroughthreedistinctformsasthe

�

σ parameterischangedintheClassicalvariation.Thefirstphase,seeninFigure14,isaunitcycle;inthisregionof

�

σ theequationsofmotionhaveasteadystatesolution.

21

Figure14:LorenzClassicvariationsteady‐statephase‐spacefor

�

σ =2.0

Afterthetransitiontochaoticbehavior,theoscillatorphasespacelookslikeFigure2,withtwodistinctlobesandclassicattractorbehavior.Afterthesecondtransitionthephasespacereturnstoastablefixed‐pointattractor,andtheorbitsallconvergetoonepoint,asseeninFigure15:

22

Figure15:LorenzClassicvariationunstructuredphase‐spacefor

�

σ =19.0

23

2.6.1.2 RVariation

ChangesintheRparameteraffectthesystemfarlesssignificantlythanthoseofsigma,bothinoverallLyapunovvariationandinrecognitionofclearparameterboundarytransitionswithbifurcationanalysis.ThechangeinRforbothvariationscanbeseeninFigures16and17:

Figure16:LorenzLyapunovspectrumchangesforRintheWolfVariation(

�

σ = 16, R = 45.92, b = 4.0)

24

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.

25

2.6.1.3 b(large‐scale)

Forvariationsinthebparameter,theClassicvariationshowedsignificantdifferencesinbothparametertransitionandoverallexponentbehavior.Thethreeexponentvariationfiguresandstandarddeviationtablearedetailedbelow:

Figure18:LorenzfirstLyapunovexponentchangesforlarge‐scalevariationsinb 

26

Figure19:LorenzsecondLyapunovexponentchangesforlarge‐scalevariationsinb 

27

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:

28

Figure21:LorenzClassicalvariationtransitionbetweenchaoticandstablebehaviorforchangesinb 

29

2.6.2 RosslerSystem

2.6.2.1 a (large‐scale)

LiketheLorenzsystem,theRosslerattractorisathree‐dimensionalsystemthatcanbreakdownfromitstypicalstructurewhenparameterchangesbecometoopronounced.ItshouldbenotedthatAUTOhaddifficultycalculatingthebifurcationpointsforthissystem.Thismayhavebeenduetotheneedforanegativeaparametertofindthesteadystatesolution,fromwhichthechaoticbifurcationsmaybecalculated.ChangesinLyapunovspectrafromvariationina canbeseeninFigures22‐24:

Figure22:RosslerfirstLyapunovexponentchangesforlargescalechangesina 

30

Figure23:RosslerfirstLyapunovexponentchangesforlargescalechangesina 

31

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,

32

onlytoincreaseafterthe0.2mark.Thefirstexponentistheleastdefinedofthethree,withnoapparenttransitionareasatanypointintherangeofparametervalues.

Figure25:Rossleroscillatorfor

�

a = 0.05,b = 0.2,c = 10

2.6.2.2 b (large‐scale)

VariationsinbfortheRosslersystemwereatorbelowtheorderofaccuracyfortheLyapunovalgorithm.AsseeninFigure26,thevaluesforallthreeexponentswererelativelystableacrosstheentirerangeofchangesinb:

33

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

34

2.6.2.3 c(large‐scale)

Forthisparameter,thesystemevolvedsimilartothatoftheLorenzoscillator.Thethirdexponentunderwentafamiliarlineardecreaseasc increased,thefirstexponentstayedpositiveforalmosttheentirespectrum,andthesecondexponentoscillatedaround0formostoftheduration,seeninFigure27:

Figure27:FirstLyapunovexponentchangesforsmall‐scalevariationsinc 

35

 

Figure28:SecondLyapunovexponentchangesforsmall‐scalevariationsinc 

36

 

Figure29:ThirdLyapunovexponentchangesforsmall‐scalevariationsinc 

Thefirstexponentshowsoscillatorybehavior,withnoapparenttransitionboundaries;itappearsthattheexponentslightlystabilizesbetweenc=10andc=15,butthenbeginstooscillateagain.Thesecondexponentshowsmoredelineatedbehavior,withparametertransitionboundariesatbothlowandhighendsofthespectrumofc values.Therearenoapparenttransitionsforthethirdexponent,whichshowslinearbehaviorthroughoutitslength.

Notably,bothvariationsundergobriefperiodsofperiodicitythroughouttheextentofparametervariation,returningtoachaoticstateaninstantlater.Figure30showsthephasespacebehaviorfordifferingexponentvalues:

37

Figure30:RosslerfirstLyapunovexponentchangeandphase‐spaceforlarge–scalechangeinc 

Forsmallvaluesofc, thesystemexhibitsperiodicity.Figure31wasrunfor500seconds,withan0.001stepsizeforthetimeinterval:

38

Figure31:Rossleroscillatorperiodicityfor

�

a = 0.2,b = 0.2,c = 4

ComparethepreviousfiguretoFigure32,whichruninthesamemannerforac valueof20:

39

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 

 

 

 

 

40

2.6.3 RosslerHyperchaosSystem

Thehyperchaossystemisextremelysensitivetoparameterinputs,andassuchisimpossibletochangeonalargescaleliketheothertwosystems.Asseeninsection2,thereisaverysmallrangeforwhichthesystemcanbesolvedviaanODEsolverinMATLAB.Itappearsthatthesystemhasnotransitionboundaries,andonlyexistsforaverysmallrangeofparametervalues.

41

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.

42

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:

43

Figure34:DamagedStabilityRun101.CloseupofFTLEvaluesandperiodmeasurement

ThoughitappearsfromthescaleofFigure33thatthereisaperiodmeasurementateverytimestep,periodmeasurementsonlyoccurateachmaxima;thegreaterthespacingbetweenmaximapoint,thegreatertheperiod.TheperiodmeasurementoftheFTLE’siscloselylinkedtothedrop‐outpointsintheFTLEmeasurements;thesedrop‐outsoccurwherethecodecannotfindenoughneighboringpointstofillthe yi and zi vectors,andthealgorithmautomaticallyappliesanarbitrarilyhighvaluetotheFTLE,asseeninFigure35:

44

Figure35:DamagedStabilityRun101.FTLEandPeriodmeasurementsvs.Time

Figure35makesitapparentthatthelargestperiodmeasurementsaredirectlytiedtolackofneighbors,ratherthanany“stretching”oftheFTLEvalues;thoughdirectlymeasuringthenumberofneighborsprovedtobeabettersolution,theperiodmeasurementsprovidesimplevisualcuesforextrememotion,andwereadequateadvanceindicatorsforlargerollamplitudes,asillustratedinTable11.Themaximumrollamplitudesforeachtimeserieswererecorded,alongwiththetimeatthatpointandthetimeoftheprecedingperiodspike.

45

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

73

Figure56:RosslerHyperchaosLyapunovspectrumchangesforsmall‐scalechangesincforWolfsystem

74

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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