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 ).
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[5] `
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