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Two-dimensionalinvariantmanifoldsinfour-dimensionaldynamicalsystemsHinkeOsingaDepartmentofEngineeringMathematics,UniversityofBristolBristolBS81TR,UKE-mail:H.M.Osinga@bristol.ac.ukDecember22,2003AbstractThispaperexploresthevisualizationoftwo-dimensionalstableandunstablemanifoldsoftheorigin(asaddlepoint)inafour-dimensionalHamiltoniansystemarisingfromcontroltheory.Themanifoldsarecomputedusinganalgorithmthat ndssetsofpointsthatlieatthesamegeodesicdistancefromtheorigin.Bycoloringthemanifoldsaccordingtothisgeodesicdistance,onecangaininsightintothege-ometryofthemanifoldsandhowtheysitinfour-dimensionalspace.Thisiscomparedwiththemoreconventionalmethodofcoloringthefourthcoordinate.Wealsotakeadvantageofthesymmetriespresentinthesystem,whichallowustovisualizethemanifoldsfromdi erentviewpointsatthesametime.Keywords:Hamiltoniansystem,dynamicalsystem,globalunstableman-ifolds,optimalcontroltheory.Correspondingauthor:Tel:+44(0)117928-7600,Fax:+44(0)117954-68331HinkeOsingaTwo-dimensionalmanifoldsinR411IntroductionThecomputationofstableandunstablemanifoldsinvector eldshasrecentlybecomea eldofrenewedactivity;see[1,6,7]andtherecentpublications[2,4,8].Severalnewalgorithmshavebeendevelopedand,eventhoughattentionremainsfocussedonlow-dimensionalsystems,higher-dimensionalproblemsarebeginningtoattractmoreinterestaswell.Themaingoalforcomputingstableandunstablemanifoldsistogaininsightintotheirgeometryandhowtheyareembeddedinphasespace.Asaconsequence,thevisualizationofthemanifoldsisasimportantastheactualcomputationalchallenge.Forhigher-dimensionaldynamicalsystems,itisalreadyhardtovisualizeone-ortwo-dimensionalmanifolds.Inparticular,theprojectionusedinthevisualizationmayresultinself-intersectionofthemanifolds.Thismakesitmuchhardertoassessthestructureofthemanifolds,forexample,whethertwomanifoldsformaheteroclinictangleornot.Inthispaperweexploreaspectsofvisualizingtwo-dimensionalmanifoldsofafour-dimensionaldynamicalsystem.Weconsideramodelarisinginoptimalcontrolandthemanifoldsarestableandunstablemanifoldsoftheorigin(asaddleequilibrium)ofaHamiltoniansystem.The guresareren-deredwiththepackageGeomview[11],whichisalsousedfortheanimationsintheassociatedmultimediasupplement[14].Itisalreadyaseriouschallengetoaccuratelycomputetwo-dimensionalmanifoldsinfourdimensions.Thealgorithmusedtocomputethemanifoldsinthispaperisdescribedindetailin[8].Thisalgorithmproducesadata lethatcanbevisualizedwiththepackageGeomview[11].Geomviewactuallydisplaysthisdatainafour-dimensionalspace,thatis,theautomaticshadowe ectsarecreatedbyalightsourceinfour-dimensionalspace,butitisveryhardtointerprettheresult.Thehumaneyeisverygoodatperceivingdepthinaat-screenpicture,butourintuitionfailswhenwetrytodothisforaprojectionofanobjectthatsitsinafour-dimensionalspace.Onewaytoenhancethevisualizationisthecleveruseofcolor.Inaparticularprojectionontothreeofthefourcoordinates,themanifoldcanbevisualizedusingacolorthatvarieswiththefourth(missing)coordinate.Inparticular,(self-)intersectionsofthemanifoldscanbeidenti edquicklythisway,sinceanintersectionisatrueintersectiononlyifthecolormatches.Thispaperexploresanotherwayofusingcolorinthevisualization,namelycoloringthemanifoldaccordingtoitsgeometry.Thealgorithmin[8]com-putesatwo-dimensionalmanifoldofasaddle(theorigininourexample)HinkeOsingaTwo-dimensionalmanifoldsinR42asasetoftopologicalcirclesthatconsistofallpointsthatlieatthesamegeodesicdistancetothesaddle;thegeodesicdistancebetweentwopointsisthearclengthoftheshortestpathbetweenthesetwopointsthatliesentirelyonthemanifold.Byassigningadi erentcolortoeachofthesegeodesiclevelsetsweobtainavisualizationwherethecolorindicateshowfarthepointsarefromtheoriginalongthemanifold.E ectively,thistechniquealsohelpstoperceivedepthinthedirectionthatismissingintheprojection.TheHamiltoniansysteminourexamplehasspecialsymmetrieswhichim-pliesthatthestableandunstablemanifoldsoftheoriginalsosatisfycertainsymmetryproperties.Bytakingadvantageofthesesymmetries,togetherwiththeabovetwodi erentmethodsofcoloringthemanifolds,wecanem-phasizedi erentpropertiesofthemanifoldsandlearnhowto\lookintofour-dimensionalspace.Thispaperisaccompaniedbyamultimediasupple-ment[14]showinganimationsofthemanifolds,whichprovideanotherusefultoolforvisualizationinfour-dimensionalspace.Thispaperisorganizedasfollows.InthenextsectionweintroducetheHamiltoniandynamicalsystemandexplainhowitisrelatedtooptimalcon-troltheory.Section3introducesthe(global)stableandunstablemanifoldsoftheoriginanddescribesthesymmetriespresentinthedynamicalsystem.ThecomputationandvisualizationofthestableandunstablemanifoldsispresentedinSec.4.WeendwithconclusionsinSec.5.2Four-dimensionalHamiltoniansystemWeconsiderafour-dimensionalHamiltoniansystemthatariseswhenstudy-ingtheoptimalcontrolproblemofbalancinganinvertedpendulumonacartsubjecttoaquadraticcostfuction;adetailedintroductionofthiscontrolproblemcanbefoundin[3].Thefrictionlesspendulumhasatwo-dimensionalphasespaceandthemotioniscontrolledbyapplyingahorizontalforcetothecartintheplaneofmotionofthependulum.IfthemassofthecartisM,the(unif