The dynamics around the collinear point L3 of the RTBP

XX Cong eso de Ecuaciones Di e enciales y Aplicaciones X Cong eso de Ma em´ a ica Aplicada Se illa, 24-28 sep iemb e 2007 (pp. 1–8) The dynamics a ound he collinea poin L3o he RTBP E. Ba ab´ es1, J.M. Mondelo2, M. Oll´ e3 1Dep . In o m`a ica i Ma em`a ica Aplicada, Uni e si a de Gi ona, 17071, Gi ona, Spain. E-mail: [email p o ec ed]. 2Dep . de Ma em`a iques, Uni e si a Au `onoma de Ba celona, Campus de Bella e a, Ed. C, 08193-Bella e a (Ba celona), Spain. E-mail: [email p o ec ed]. 3Dep . de Ma em`a ica Aplicada I, ETSEIB, Uni e si a Poli `ecnica de Ca alunya, Diagonal 647, 08028 Ba celona, Spain. E-mail: [email p o ec ed]. Palab as cla e: RTBP, collinea poin s, pe iodic o bi s, homoclinic o bi s Resumen We conside he Res ic ed Th ee Body P oblem (RTBP), and we es ic ou a en ion o he equilib ium poin L3. Ou aim is cen e ed in he desc ip ion, as global as possible, o he dynamics a ound his equilib ium poin . In his communica ion, we ini ially conside small alues o µ, o which homoclinic connec ions o he equilib ium poin L3a e ho seshoe-shaped, and hen, o he alues o µa e conside ed. We compu e he objec s in he cen e mani old o L3, including he in a ian mani olds associa ed wi h hem. They a e compu ed by pu ely nume ical p ocedu es, in o de o a oid he con e gence es ic ions o he semi-analy ical ones ( ypically used a ound L1 o L2). We deal wi h homoclinic connec ions o pe iodic o bi s and de elop some nume ical ools in o de o compu e hem. These ools can be ex ended o compu e also homoclinic connec ions o in a ian o i. 1. In oduc ion Le us conside he ci cula es ic ed h ee-body p oblem (RTBP) whe e wo bodies (called p ima ies) desc ibe ci cula o bi s a ound hei common cen e o mass, and a hi d body o in ini esimal mass mo es unde he g a i a ional e ec o he p ima ies bu ha ing negligible e ec on hei mo ion. Wi h sui able uni s, we can assume ha he p ima ies ha e masses 1 −µand µ,µ∈(0,1/2], ha he pe iod o hei mo ions is 2π and ha hei dis ance is he uni . Le = (x, y, z) be he coo dina es o he hi d body and p= (px, py, pz) he co esponding momen a in a o a ing e e ence sys em whe e 1 E. Ba ab´es, J.M. Mondelo, M. Oll´e he p ima ies a e ixed a (µ, 0,0) and (µ−1,0,0). Then, he Hamil onian go e ning he mo ion o he in ini esimal pa icle is gi en by H=1 2(p2 x+p2 y+p2 z)−xpy+ypx−1−µ 1 −µ 2 ,(1) whe e 1=p(x−µ)2+y2+z2and 2=p(x−µ+ 1)2+y2+z2(see, o example [8]). The alue o he Hamil onian on each o bi will be e e ed o as he ene gy o he o bi and i s ela ion wi h he Jacobi in eg al is gi en by C=−2H+µ(1 −µ).Fu he mo e, he equa ions o he p oblem sa is y he symme ies ( , x, y, z, px, py, pz)−→ (− , x, −y, z, −px, py,−pz), ( , x, y, z, px, py, pz)−→ (− , x, −y, −z, −px, py, pz).(2) The RTBP has i e equilib ium poin s: he collinea poin s, L1,L2and L3, and he equila e al ones, L4and L5. We will conside L1loca ed be ween he wo p ima ies, L2 loca ed such ha he small p ima y is be ween L1and L2, and L3such ha he big p ima y is be ween L1and L3. We deno e by Ciand Hi he alue o he Jacobi cons an and he ene gy a he equilib ium poin Li,i= 1, . . . , 5. I is well known ha 3 = C4= C5< C3≤C2< C1,and C3=C2 o µ= 1/2. In his communica ion, we ocus ou a en ion on he dynamics o he RTBP a ound he equilib ium poin L3. The dynamics a ound he collinea poin s ha e been s udied by se e al au ho s using di e en echniques and app oaches (see, o example, [3], [4], [5], [6] and he e e ences he ein). I is well known ha he linea beha iou o he h ee poin s is o ype cen e ×cen e ×saddle. In spi e o his, he e is a s ong di e ence be ween Li, i= 1,2, and L3: while he i s wo poin s a e s ongly a ec ed by bo h p ima ies, he e ec o he small p ima y on L3is almos negligible. Fu he mo e, in he case o L1and L2, he in a ian mani olds can be compu ed in se ies expansion by semi-analy ical p ocedu es as he ones based in Linds ed -Poinca ´e me hod o educ ion o he cen e mani old (see [7]). These me hods p oduce expansions o he mani olds up o an a bi a y o de , which gi e ini ial condi ions on he mani olds up o a high deg ee o accu acy. Howe e , hese me hods a e no sui able o he neighbou hood o L3due o he small ange o alues o he ene gy o which he unca ed se ies a e alid. We explo e nume ically he exis ence and o ganiza ion o in a ian objec s a ound L3, as well as he exis ence o homoclinic o bi s. We s a dealing wi h he in a ian mani olds o he equilib ium poin L3and hen wi h he amilies o plana Lyapuno pe iodic o bi s (LPO) ha a e bo n a L3. Then, he in a ian mani olds associa ed wi h LPO a e conside ed. We de elop nume ical me hods in o de o compu e, a he same ime, bo h a pe iodic o bi and an homoclinic connec ion o i . The explo a ions a e done o µ∈[10−4,0.03], which con ains he Ea h-Moon alue, µEM = 0.01215058560962404 and he Sun-Jupi e alue, µSJ = 9.53875 ×10−4. Finally, we gene alize he me hods used in o de o ind homoclinic connec ions o in a ian o i. 2. Linea beha iou a ound L3and homoclinic phenomena I is well known ha he linea beha iou o he equilib ium poin s is o ype cen e × cen e ×saddle and ha wo amilies o pe iodic o bi s a e bo n a hem: he plana and 2 The dynamics a ound he collinea poin L3o he RTBP e ical Lyapuno o bi s. In he case o he plana Lyapuno o bi s (see, o ins ance, [4]), and o alues o he ene gy less han he i s e ical bi u ca ion o bi (co esponding o he alue a which he amily o Halo o bi s is bo n), he o bi s ha e cen al and hype bolic pa s. Fo a ixed alue o he ene gy, he co esponding Lyapuno o bi has s able and uns able in a ian mani olds and he e exis s a (can o ian) amily o in a ian o i connec ing he plana o bi wi h he e ical one wi h he same ene gy. Each o hese o i inhe i s he hype bolic beha iou o he backbone pe iodic o bi s. Le Xbe one o he in a ian objec s a ound L3. We deno e by Wu(X) he uns able mani old and Ws(X) he s able one (o simply Wuand Ws). In he case o a collinea equilib ium poin , he in a ian mani olds ha e dimension 1 and con ain plana o bi s ha end (backwa ds o o wa ds in ime) o he equilib ium poin . Fo each mani old, Wu/s +(Li) deno es he b anch co esponding o he eigen ec o ha poin s o he uppe hal plane {y > 0}and by Wu/s −(Li) he b anch co esponding o he eigen ec o poin ing o he lowe hal plane {y < 0}. This no a ion can be ex ended o he b anches o he in a ian mani olds o a pe iodic o bi o an in a ian o us: Wu/s +( espec i ely Wu/s −) deno es he b anch ha en e s in o he uppe space {y > 0}( esp. lowe space {y < 0}) a e lea ing o wa d ( esp. backwa d) in ime a neighbou hood o he in a ian objec . We obse e ha he symme ies gi en by (2) map o bi s on Wu − o Ws +and ice e sa. We a e in e es ed in homoclinic connec ions o an in a ian objec X, which a e so- lu ions o he RTBP such ha end o X o wa d and backwa d in ime. Such solu ions belong o he s able and uns able mani olds associa ed wi h X, his is, o he in e sec ion Wu∩Ws. In o de o compu e homoclinic o bi s, we ix a Poinca ´e sec ion Σ and we look o elemen s o ¡Wu∩Σj¢∩¡Ws∩Σk¢, whe e Wu/s ∩Σmdeno es he m- h in e sec ion o he in a ian mani old wi h Σ. Depending on he b anches ha a e in ol ed we look o he ollowing kinds o homoclinic connec ions: connec ion o ype (−j, −k), which akes place when ¡Wu −∩Σj¢∩¡Ws −∩Σk¢6=∅. Simila ly, a connec ion o ype (+j, +k) can be de ined. Obse e ha , i he e exis s a connec ion o ype (−j, −k), hen i is a non-symme ic o bi and he symme ic o bi is a connec ion o ype (+j, +k). connec ion o ype (−j, +k), which akes place when ¡Wu −∩Σj¢∩¡Ws +∩Σk¢6=∅. Simila ly, a connec ion o ype (+j, −k) can be de ined. I he e exis symme ic connec ions, hey mus be o one o hese ypes. 3. Homoclinic connec ions o L3 In his Sec ion, we look o alues o µ o which he e exis s an homoclinic connec ion o L3. Fon , in [2], p o ed ha he e exis s an in ini e sequence o µ ending o ze o, such ha he e exis s an homoclinic connec ion o L3 o each one o hese alues. These connec ions a e all symme ic. In [1], he beha iou o he in a ian mani olds o L3as µ a ies and i s ela ion wi h ho seshoe o bi s a e desc ibed and a p ocedu e o compu e symme ic homoclinic connec ions o L3using he symme y o he o bi s is gi en. He e we wan o gene alize ha p ocedu e in o de , o ind, i hey exis , non symme ic connec ions as well. 3 E. Ba ab´es, J.M. Mondelo, M. Oll´e We conside in his sec ion Σ = {x=µ−1/2}. We s a wi h an ini ial condi ion on he linea app oxima ion o he in a ian mani old, and we ollow he low o he RTBP un il he co esponding in e sec ion wi h Σ. As he in a ian mani olds a e 1-dimensional, Wu/s ±∩Σmconsis s o one poin zu/s(µ), so we look o alues o µsuch ha zu(µ) = zs(µ). We conside he alues o µ∈[10−4,0.03], which con ains he cases o Sun-Jupi e and Ea h-Moon p oblems. Conce ning homoclinic connec ions o ype (−j, −k), we explo e he cases (−1,−2), (−2,−3) and (−3,−4) and we do no obse e nume ical e idence o non-symme ic ho- moclinic connec ions. In he case o an equilib ium poin , all homoclinic connec ions o ype (−j, +k) o (+j, −k) a e symme ic (which is no ue in he case o pe iodic o bi s o in a ian o i). We explo e he cases j= 2, k = 3 and j= 2, k = 5 and j= 4, k = 5. We ind homoclinic connec ions in all o hem, in pa icula , all he symme ic homoclinic o bi s desc ibed in [1]. In Figu e 1, wo di e en homoclinic o bi s o L3 o di e en alues o µa e shown. -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 Figu a 1: Homoclinic o bi s o L3o ype (−3,2) o µ= 0.0010015432 (le ) and (+2,−5) o µ= 0.012143988024852 ( igh ). (Wuin con inuous ed line, Wsin dashed blue line) 4. Homoclinic connec ions o Lyapuno o bi s In his sec ion we deal wi h he amily o plana Lyapuno pe iodic o bi s (LPO) a ound L3and hei homoclinic o bi s. Fo alues o he ene gy Hclose o H3, he LPO inhe i he beha iou o he equilib ium poin , so he associa ed in a ian mani olds ha- e a shape simila as he in a ian mani olds o L3. Fixed a plana Lyapuno o bi X, we conside he di e en b anches o each in a ian mani old Wu/s(X), and we look o hei in e sec ions wi h he sec ion Σ = {x=µ−1/2}. The in a ian mani olds a e 2- dimensional objec s ha can be iewed as ubes in he phase space. I can be expec ed ha he i s c ossings o each Wu/s ±(X) wi h Σ will be like S1cu es, so one way o compu e he homoclinic connec ions is o look o he in e sec ions o hese cu es (see Figu e 2). Howe e , his p ocedu e p esen s some p oblems when he ene gy o µinc eases due o he p esence o mul iple loops ha make di icul o compu e ‘exac ly’ he m- h c ossing wi h a sec ion (see Figu e 5). In his case, we use a di e en app oach in o de o ind a speci ic homoclinic o bi wi hou compu ing he whole cu e Wu/s ±(X)∩Σm. Nex , we desc ibe sligh ly he me hods used. 4 The dynamics a ound he collinea poin L3o he RTBP 0.034 0.035 0.036 0.037 0.038 -0.927 -0.9265 -0.926 -0.9255 -0.925 0.034 0.035 0.036 0.037 0.038 -0.927 -0.9265 -0.926 -0.9255 -0.925 Figu a 2: Wu −∩Σ1and Ws −∩Σ2cu es in he (y, y0) plane (y0=py−x) o wo plana LPO a ound L3o ene gy H=−1.500476742438758 (le ) and H=−1.500476517438758 ( igh ). (µ=µSJ and Σ = {x=µ−1/2}) 1. In e sec ion o he in a ian mani olds wi h a de ined sec ion. Fixed Xa pe iodic o bi , a simple me hod consis s o aking, o each poin o he o bi , he linea app oxima ion o he in a ian mani old, and ollow he low ( o wa d o backwa d, depending on he mani old) by nume ical in eg a ion un il he desi ed in e sec ion wi h Σ. This me hod has p oblems when he o bi on he mani old has loops. In his case, we use a global pa ame iza ion o an in a ian mani old in o de o ob ain a pa ame iza ion o he cu e Wu/s ±(X)∩Σm. In Figu es 2 and 5 he in e sec ions o he in a ian mani olds wi h di e en sec ions (Σ = {x=µ−1/2} and Σ0={y= 0}) o di e en LPO a e shown. Once we ha e he wo cu es, ob ained om he in e sec ion o each in a ian ma- ni old and Σ, he homoclinic o bi s can be compu ed as he in e sec ion o bo h cu es. As we ha e hem as a union o polygons, we can check o in e sec ions be ween segmen s and hen e ine he p ocess. 2. Compu a ion o a homoclinic o a pe iodic o bi : ma ching o mani olds on a sec ion. The idea is o compu e di ec ly an homoclinic o bi wi hou compu ing he whole in e sec ion o an in a ian mani old wi h a sec ion. In o de o do his, we look o wo o bi s, each one in a di e en in a ian mani old, such ha ma ch a hei co esponding in e sec ion wi h Σ. We ema k ha in o de o ob ain obus esul s his me hod is implemen ed by using a mul iple shoo ing s a egy. As we ha e said be o e, as he ene gy inc eases, so do he loops, which is a p oblem i he numbe o in e sec ions wi h Σ is ixed. In o de o a oid coun ing he numbe o in e sec ions when compu ing an homoclinic o bi , we in oduce in he equa ions as an unknown quan i y he ime o each Σ. Finally, i we wan o implemen a con inua ion me hod i is necessa y o a y he pe iodic o bi . Thus we compu e, a each s ep, a pe iodic o bi and an homoclinic connec ion o i . Le us show some o he esul s ob ained. We conside he mass pa ame e µSJ = 9.53875×10−4. We s a looking o homoclinic o bi s o ype (−1,−2). We conside he sec ion Σ = {x=µ−1/2}and compu e he cu es 5 E. Ba ab´es, J.M. Mondelo, M. Oll´e -1.08 -1.06 -1.04 -1.02 -1 -0.98 -0.96 -0.94 -0.92 -0.9 -1.501 -1.499 -1.497 -1.495 -1.493 Figu a 3: Cha ac e is ic cu e (H, y) o a amily o homoclinic o bi s o Lyapuno o bi s o µ=µSJ . -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 -1 -0.5 0 0.5 1 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 -1 -0.5 0 0.5 1 Figu a 4: Non symme ic homoclinic o bi s o µ=µSJ H=−1.5004762674387578125 (le ) and H=−1.4935767674387578125 ( igh ). Wu −∩Σ1and Ws −∩Σ2, which can be ep esen ed in he (y, y0) plane (y0=py−x). Recall ha he e is no homoclinic connec ion o L3o his ype. Then, o alues o Hsligh ly bigge han H3, no homoclinic o bi s o his ype a e expec ed, and he cu es do no in e sec (see Figu e 2 le ). Bu as we inc ease H, i s he cu es become angen a one poin –so he e is one homoclinic o bi –, and hen in e sec a wo poin s, gi ing ise o wo homoclinic o bi s (see Figu e 2 igh ). Then, gi en a homoclinic o bi , we ollow he amily i belongs o, p edic ing a new o bi using he angen ec o o he cu e ha ep esen s he amily o homoclinic o bi s. As he ene gy inc eases, he numbe o loops inc eases as well, so al hough we s a he amily wi h a homoclinic o bi o ype (−1,−2), i is possible o ind o he homoclinic o bi s wi h a di e en numbe o in e sec ions wi h Σ in he same amily. In Figu e 3 he cha ac e is ic cu e o a amily o homoclinic o bi s in he (H, y) plane is shown, being y he second coo dina e o he ma ching poin o he in a ian mani olds a Σ. In Figu e 4 wo non symme ic homoclinic o bi s a e shown. All o he abo e homoclinic connec ions a e non-symme ic o bi s. In o de o ind symme ic homoclinic o bi s, we conside he (−1,+4) case. We s a conside ing a LPO wi h H=−1.500476742438758, o which he e no exis s connec ions o ype (−1,−2), and 6 The dynamics a ound he collinea poin L3o he RTBP 0.033 0.0345 0.036 0.0375 0.039 -0.929 -0.928 -0.927 -0.926 -0.925 -0.924 -0.923 -0.002 -0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015 0.002 0.995 0.996 0.997 0.998 0.999 1 1.001 Figu a 5: In e sec ion o he in a ian mani olds o a pe iodic o bi o µ=µSJ and H=−1.500476742438758 wi h wo di e en sec ions: Wu −∩Σ1and Ws +∩Σ4cu es in he (y, y0) plane (le ), Wu −∩Σ0and Ws +∩Σ0cu es in he (x, x0) plane ( igh ). we compu e Wu −∩Σ1and Ws +∩Σ4. The cu es ob ained, ep esen ed in he (y, y0)-plane, a e shown in Figu e 5, le , whe e he e ec o he loops can be obse ed. In his case we ob ain 28 in e sec ion poin s, ha is, 28 homoclinic o bi s which con ain symme ic and nonsymme ic homoclinic o bi s. In o de o iden i y he symme ic o bi s, some imes i is mo e con enien o deal wi h he sec ion Σ0={y= 0}, because he symme ic o bi s co espond o he poin s on he cu es wi h x0= 0. In Figu e 5, igh , Wu −∩Σ0and Ws +∩Σ0, o he same pe iodic o bi as be o e, a e shown, whe e, 14 homoclinic symme ic o bi s can be iden i ied. 5. Homoclinic connec ions o in a ian o i Conside a le el o ene gy smalle han he i s bi u ca ion o he amily o plana Lyapuno o bi s. The amily o in a ian o i o his ene gy le el connec ing he plana and he e ical Lyapuno o bi s inhe i he hype bolic beha iou o he pe iodic o bi s. Thus, o each in a ian o us, we can conside he 3-dimensional in a ian mani olds associa ed o hem and we can look o homoclinic connec ions conside ing again he in e sec ion o each b anch o he in a ian mani olds wi h a gi en sec ion. The me hod o compu ing homoclinic o bi s looking o wo speci ic o bi s in each in a ian mani old and doing sec ion ma ching can be gene alized o in a ian o i. In Figu e 6 wo di e en homoclinic o bi s o he same in a ian o us a e shown. Acknowledgmen s E. Ba ab´es and J.M. Mondelo a e pa ially suppo ed by he MCyT/FEDER g an s BFM2003-09504-C02-01 and MTM2006-05849/Consolide . J.M. Mondelo is also suppo ed by he MCyT/FEDER g an MTM2005-02139. M. Oll´e is pa ially suppo ed by he MCyT/FEDER g an MTM2006-00478. 7 E. Ba ab´es, J.M. Mondelo, M. Oll´e -1.5 -1 -0.5 0 0.5 1 1.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 -0.004 -0.002 0 0.002 0.004 -1.5 -1 -0.5 0 0.5 1 1.5-1.5 -1-0.5 0 0.5 1 1.5 -0.01 -0.006 -0.002 0.002 0.006 0.01 Figu a 6: Homoclinic connec ions o an in a ian o i o µ=µSJ and H=−1.46 (le ) and H=−1.5 ( igh ). Re e encias [1] Es he Ba ab´es and Me c`e Oll´e. In a ian mani olds o l3and ho seshoe mo ion in he es ic ed h ee-body p oblem. Nonlinea i y, 19:2065–2089, 2006. [2] J. Fon . The ole o homoclinic and he e oclinic o bi s in wo-deg ees o eedom Hamil onian sys ems. PhD hesis, Uni e si y o Ba celona, 1999. [3] G. G´omez, A. 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