the vrtonentrctes nd orbtl tons for the ner four plsthe tl nd fl prtthe tegrtons showxpected， the chrcterthe vrtonry orbtl elents does not dffer sgnfcntly between the tl nd fl prtech tegrton，lest for venus， erthelentsrcury， especlly ts entrcty， seenbsp;to chn sgnfcnprtly becuse the orbtl tsclethes the shortestll the pls， whch ledsrpd orbtl evoluton thn other pls; the nerstyneresesult ppersbeso greent wthskr&039;s 1994， 1996 expecttons thtr nd rregr vrtons pperthe entrctes nd tonsrcury tscleseverl 109 yr. however， the effectthe possble stbltythe orbtrcurot ftlly ffect the globl stbltythe whole ry systenbsp;owgthe sllo wll nton brefly the longternbsp;orbtl evolutonrcurytersecton 4 usg lowpss fltered orbtl elents.

the orbtl tonthe outer fve pls see rgorously stble nd qute regr over ths tspn see lso secton 5.

3.2 t–frequency ps

lthough the ry ton exhbts very longternbsp;stblty defedthe nonexstenceclose encounter events， the chotc nturery dyncs cn chn the osctory perod nd ltudery orbtl ton grdully over such lon such slght fluctutonsorbtl vrtonthe frequency do， prtcrlythe cseerth， cn potentlly hvgnfcnt effectts surfce clte systenbsp;through sr ston vrton cf. berr 1988.

to gveovervewthe longternbsp;chnperodctyry orbtl ton，perfordfst fourer trnsfortons ffts long thexs， nd superposed the resultg perodgrdrw twodsonl t–frequency ps. the specfc pprochdrwg these t–frequency ths ppervery sle –sler thn the wvelet nlysskr&039;s 1990， 1993 frequency nlyss.

dvde the lowpss fltered orbtl dt tofrgntsthengthech dt segnt should ltple2orderpply the fft.

ech frgntthe dt hr oveppg prt: for exle， when the th dt begs fronbsp;tt nd endsttt， the next dt segnt rns fronbsp;tδttδtt， where δtt.contue ths dvson untlrecert nuer nwhch tnt reches the totl tegrton length.

we pplyfftechthe dt frgnts， nd obtrequency dgr.

ech frequency dgrnbsp;obted bove， the strengthperodcty cnreced greysclecolour chrt.

we perfornbsp;the recent， nd connect ll the greysclecolour chrts to one grph for ec horzontl xsthese new grphs shouldthe t， .e. the strtg tsech frgntdt t， where 1，…， n. the vertcl xs represents the perodfrequencythe osctonorbtl elents.

we hve doptedfft becusets overwhelng speed， sce the untnurcl dtbe dposed to frequency&nbspponentsterrbly hu severl tensgbytes.

typcl exlethe t–frequencretedthe bove proceduresshown greyscle dgrnbsp;， whch shows the vrtonperodctythe entrcty nd tonerthn fg. 5， the drk re shows thtthedctedthe vluethe bscss， the perodcty dctedthe ordtestronr thnthe lghter re round t.cn recognze fronbsp;thht the perodctythe entrcty nd tonerth only chns slghtly over the entre perod coveredtherly regr trendqulttvely the other tegrtons nd for other pls， lthough typcl frequences dfferbynd elentelent.

4.2 longternbsp;exchnorbtl energy nd ngr ntubr>

we clcte very longperodc vrton nd exchnry orbtl energy nd ngr ntunbsp;usg fltered duny elents l， g， h. g nre equvlentthe ry orbtl ngr ntunbsp;nd ts vertcl&nbspponent per unt ss. lrtedthe ry orbtl enerr unt the systenbsp;s&nbsppletely ler， the orbtl energy nd the ngr ntunbsp; ech frequency bbthe ry systenbsp;cn cuseexchnenergy nd ngr ntunbsp; the frequenc ltudethe lowestfrequency oscton should cresethe systenbsp;s unstble nd breks dow， sucytonbsp;of stbltynot pronentour longternbsp;tegrtons.

， the totl orbtl energy nd ngr ntunbsp;of the four ner pls nd ll ne pls re shown for tegrton n2. the upper three pnels show the longperodc vrtontotl energy denoted se e0， totl ngr ntunbsp; g g0， nd the vertcl&nbspponenthof the ner four pls clcted fronbsp;the lowpss fltered dun， g0，denote the tl vluesec bsolute dfference fronbsp;the tl vluesplottedth lower three pnelsech fgure showee0，gg0 ndhh0the totln fluctuton shownthe lower pnelsvrtully entrelesultthe ssve jovn pls.

corg the vrtonsenergy nd ngr ntunbsp;of the ner four pls nd ll ne pls，s pprent tht the ltudesthosethe ner pls resller thn thosell ne pls: the ltudesthe outer fve pls rrr thn thosethe ne does nottht the ner terrestrl ry subsystenbsp;sstble thn the outer one: thsslesultthe rtve sllnessthe ssesthe four terrestrl pls&nbsppred wth thosethe outer jov thgnotcetht the ner ry subsystenbsp;yunstblerpdly thn the outer one becusets shorter orbt cnseenthe pnels denoted sner 4 the lonrperodc nd rregr osctons repprent thnthe pnels denoted stotl 9. ctully， the fluctutonsthener 4 pnels rer extent resultthe orbtl vrtonth，cnnot neglect the contrbuton fronbsp;other terrestrl pls，we wll seesubsequent sectons.

4.4 longternbsp;couplgseverl neghbourgprs

letseedvdul vrtonsry orbtl energy nd ngr ntunbsp;expressedthe lowpss fltered dunndshow longternbsp;evolutonthe orbtl energyechnd the ngr ntunbsp;notce thtpls fornbsp;pprent prsterorbtl energy nd ngr ntunbsp; prtcr， venus nd erth typc the fgures， they show negtve corrtonsexchnenergy nd postve corrtonsexchnngr ntu the negtve corrtonexchnorbtl energy ns tht the two pls fornbsp; closed dyncl systenbsp; terthe orbt postve corrtonexchnngr ntunbsp;ns tht the two pls re sltneously under cert longternbsp; for perturbers re jupter，cn see tht mrs showostve corrtonthe ngr ntunbsp;vrtonthe venus–erth syste rcury exhbts cert negtve corrtonsthe ngr ntunbsp;versus the venus–erth syste whch seebecton cusedthe conservtonngr ntunbsp; the terrestrl ry subsyste

tnot clerthewhy the venus–erth pr exhbtegtve corrtonenergy exchn nostve corrtonngr ntunbsp; y possbly ex ths through observg the nerl fct tht there resecr terry sejor xesto secondorder perturbton theores cf. brouwer & clence 1961; blett & puc ns tht the ry orbtl energy whchdrectly rtedthe sejor xhtch less ffectedperturbg pls thnthe ngr ntunbsp;exchn whch rtese. hence， the entrctesvenus nd erth cndsturbed eslyjupter nd sturn， whch results postve corrtonthe ngr ntunbsp; the other hnd， the sejor xesvenus nd erth re less lkelybe dsturbedthe jov the energy exchn ylted only wth the venus–erth pr， whch results negtve corrtonthe exchnorbtl energythe pr.

s for the outer jovn ry subsyste jupter–sturn nd urnus–neptune seenbsp;todync， the strengthther couplgnotstrong&nbsppred wth thtthe venus–erth pr.

551010yr tegrtonsouter ry orbts

sce the jovn ry sses rrr thn the terrestrl ry sses，tret the jovn ry systenbsp;sdependent ry systenbsp; terthe studyts dync，ddeoupletrl tegrtons tht spn51010 yr， cludg only the outer fve pls the four jovn pls plu results exhbt the rgorous stbltythe outer ry systenbsp;over ths lon confgurton， nd vrtonentrctes nd ton show ths very longternbsp;stbltythe outer fve plsboth thend the frequencdo not showhere， the typcl frequencythe orbtl osctonpluto nd the other outer plslst constnt durg these very longternbsp;tegrton perods， whchdenstrtedthe t–frequencyon our webp.

these two tegrtons， the rtve nurcl errorthe totl energy ws 106 nd thtthe totl ngr ntunbsp;ws 1010.

5.1 resonncesthe neptune–pluto systebr>

kosht & nk 1996 tegrted the outer fve ry orbts they found tht four jor resonnces between neptune nd pluto re ted durg the whole tegrton perod， nd tht the resonnces ythecusesthe stbltythe orbor four resonnces foundprevous reserch rhe followg descrpton，λ denotes thelongtude，the longtudethe scendg node nds the longtud nenote pluto nd neptune.

n ton resonnce between neptune nd pluto 3:2. the crtcl rgunt32 λnp lbrtes round 180 wthltudeboutnbrton perodbout 2104 yr.

the rguntperhelonpluto pθ2pp lbrtes roundwterodbout 3.8106 yr. the donnt perodc vrtonsthe entrcty nd tonpluto re synchronzed wth the lbrtonts rgunt ontcptedthe secr perturbton theory constructedkoz 1962.

the longtudethe nodepluto referredthe longtudethe nodeneptune，θ3pn， crctes nd the perodths crctonequlthe perodθbes zero， .e. the longtudesscendg nodesneptune nd pluto ovep， the tonpluto besthe entrcty bes nbsp;nd the rguntperhelon bes 90. whenbes 180， the tonpluto besthe entrcty bes xbsp;nd the rguntperhelon bes 9 & benson 1971 ntcpted ths typeresonnce，ter confrdmn， nobl & crpo 1989.

n rgunt θ4pn 3lbrtes round 180 wtong perod， 5.7108 yr.

our nurcl tegrtons， the resonnces – re well ted， nd vrtonthe crtcl rgunts θ1，θ2，θ3 re sr durg the whole tegrton perod fgs 14–16 . however， the fourth resonnceppersbe dfferent: the crtcl rguntlterntes lbrton nd crcton over 1010yr tscn terestg fct tht kosht & nk&039;s 1995， 1996 shorter tegrtons were not bledsclose.

6 dscusson

wht kddyncl chnsnbsp;ts ths longternbsp;stbltythe ry systenbsp;we cn edtely thktwo jor fetures tht yresponsble for the longternbsp;， there seenbsp;tono sgnfcnt lowerorder resonnceston nd secr between ny pr ng the n nd sturn re close 5:2ton resonnce the fus gret equlty， but not justthe resonnc resonnceuse the chotc nturethe ry dyncl ton， but they re notstrongto destroy the stble ry ton wth the tthe rel sr syste the second feture， whchthkre ortnt for the longternbsp;stbltr ry systethe dfferencedyncl dstnce between terrestrl nd jovn ry subsyste to & tnkw 1999， sure ry seprtonsthe tul hll rd r， seprtons ng terrestrl pls re greter thn 26rh， wheres those ng jovn pls re less th dfferencedrectly rtedthe dfference between dyncl feturesterrestrl nd jov pls hve sller sses， shorter orbtl perods nd wder dync re strongly perturbedjovn pls tht hverr sses， lonr orbtl perods nd nrrower dync pls re not perturbedny other ssve bodes.

the present terrestrl ry systenbsp;s stll beg dsturbedthe ssve jov， the wde seprton nd tul tercton ng the terrestrl pls renders the dsturbnce effectve; the degreedsturbncejovn plsoejordergntudethe entrctyjupter， sce the dsturbnce cusedjovn pls forced oscton hvgltude oentrcty， for exl，fr fronbsp;suffcentprovoke stbltythe terrestrl pls hvg sucde seprton ssu tht the present wde dyncl seprton ng terrestrl pls > 26rhprobbly ohesgnfcnt condtons for tg the stbltythe ry systenbsp;over 109y detled nlyssthe rtonshp between dyncl dstnce between pls nd the stblty tsclesr systenbsp;ry tonnow ongog.

lthough our nurcl tegrtons spn the tthe sr syste the nuertegrtonsfr fronbsp;suffcentfll the tl phsnecessryperfornbsp;re ndnurcl tegrtonsconfrnbsp;nd exnedetl the longternbsp;stbltr ry dyncs.

以上文段引自 to， t.& tnkw， k. longternbsp;tegrtons nd stbltyry orbtsour sr syst， 483–500 2002

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