remaine1992，1994).

    Thestepsizeforthenumericalintegrationsis8dthroughoutallintegrationsofthenineplanets(N±1，2，3)，whichisabout1/11oftheorbitalperiodoftheinnermostplanet(Mercury).Asforthedeterminationofstepsize，wepartlyfollowthepreviousnumericalintegrationofallnineplanetsinSussman&Wisdom(1988，7.2d)andSaha&Tremaine(1994，225/32d).Weroundedthedecimalpartofthetheirstepsizesto8tomakethestepsizeamultipleof2inordertoreducetheaccumulationofround-offerrorinthecomputationprocesses.Inrelationtothis，Wisdom&Holman(1991)performednumericalintegrationsoftheouterfiveplanetaryorbitsusingthesymplecticmapwithastepsizeof400d，1/10.83oftheorbitalperiodofJupiter.Theirresultseemstobeaccurateenough，whichpartlyjustifiesourmethodofdeterminingthestepsize.However，sincetheeccentricityofJupiter(～0.05)ismuchsmallerthanthatofMercury(～0.2)，weneedsomecarewhenwecomparetheseintegrationssimplyintermsofstepsizes.

    Intheintegrationoftheouterfiveplanets(F±)，wefixedthestepsizeat400d.

    WeadoptGauss'fandgfunctionsinthesymplecticmaptogetherwiththethird-orderHalleymethod(Danby1992)asasolverforKeplerequations.ThenumberofmaximumiterationswesetinHalley'smethodis15，buttheyneverreachedthemaximuminanyofourintegrations.

    Theintervalofthedataoutputis200000d(～547yr)forthecalculationsofallnineplanets(N±1，2，3)，andabout8000000d(～21903yr)fortheintegrationoftheouterfiveplanets(F±).

    Althoughnooutputfilteringwasdonewhenthenumericalintegrationswereinprocess，weappliedalow-passfiltertothe

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