Two-dimensional invariant manifolds in four-dimens

Two-dimensionalinvariantmanifoldsinfour-dimensionaldynamicalsystemsHinkeOsingaDepartmentofEngineeringMathematics,UniversityofBristolBristolBS81TR,UKE-mail:[e-mail]22,2003AbstractThispaperexploresthevisualizationoftwo-dimensionalstableandunstablemanifoldsoftheorigin(asaddlepoint)inafour-dimensionalHamiltoniansystemarisingfromcontroltheory.Themanifoldsarecomputedusinganalgorithmthatndssetsofpointsthatlieatthesamegeodesicdistancefromtheorigin.Bycoloringthemanifoldsaccordingtothisgeodesicdistance,onecangaininsightintothege-ometryofthemanifoldsandhowtheysitinfour-dimensionalspace.Thisiscomparedwiththemoreconventionalmethodofcoloringthefourthcoordinate.Wealsotakeadvantageofthesymmetriespresentinthesystem,whichallowustovisualizethemanifoldsfromdierentviewpointsatthesametime.Keywords:Hamiltoniansystem,dynamicalsystem,globalunstableman-ifolds,optimalcontroltheory.Correspondingauthor:Tel:[phone],Fax:[phone]HinkeOsingaTwo-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,theautomaticshadoweectsarecreatedbyalightsourceinfour-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.Byassigningadierentcolortoeachofthesegeodesiclevelsetsweobtainavisualizationwherethecolorindicateshowfarthepointsarefromtheoriginalongthemanifold.Eectively,thistechniquealsohelpstoperceivedepthinthedirectionthatismissingintheprojection.TheHamiltoniansysteminourexamplehasspecialsymmetrieswhichim-pliesthatthestableandunstablemanifoldsoftheoriginalsosatisfycertainsymmetryproperties.Bytakingadvantageofthesesymmetries,togetherwiththeabovetwodierentmethodsofcoloringthemanifolds,wecanem-phasizedierentpropertiesofthemanifoldsandlearnhowto\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