P oyec o Fin de Ca e a
Ingenie ía de Telecomunicación
Fo ma o de Publicación de la Escuela Técnica
Supe io de Ingenie ía
Au o : F. Ja ie Payán Some
Tu o : Juan José Mu illo Fuen es
Dep. Teo ía de la Señal y Comunicaciones
Escuela Técnica Supe io de Ingenie ía
Uni e sidad de Se illa
Se illa, 2013
Mas e ´s Thesis
Mas e in Ae onau ical Enginee ing
AUTO-based nume ical s udy o h ee body
p oblem o bi s and ajec o ies wi h applica-
ions o he Luna Deep Space Ga eway
Au ho : José Manuel Mon illa Ga cía
Tu o : Ra ael Vázquez Valenzuela
Ae ospace Enginee ing and Fluid Mechanics Depa men
Highe Technical School o Enginee ing
Uni e si y o Se ille
Se ille, 2018
Mas e ´s Thesis
Mas e in Ae onau ical Enginee ing
AUTO-based nume ical s udy o h ee body
p oblem o bi s and ajec o ies wi h
applica ions o he Luna Deep Space Ga eway
Au o :
José Manuel Mon illa Ga cía
Tu o :
Ra ael Vázquez Valenzuela
Di ec o o he Depa men o Ae ospace Enginee ing
Ae ospace Enginee ing and Fluid Mechanics Depa men
Highe Technical School o Enginee ing
Uni e si y o Se ille
Se illa, 2018
Mas e ´s Thesis:
AUTO-based nume ical s udy o h ee body p oblem o bi s and ajec o-
ies wi h applica ions o he Luna Deep Space Ga eway
Au o : José Manuel Mon illa Ga cía
Tu o : Ra ael Vázquez Valenzuela
El ibunal nomb ado pa a juzga el abajo a iba indicado, compues o po los siguien es p o eso es:
P esiden e:
Vocal/es:
Sec e a io:
acue dan o o ga le la cali icación de:
El Sec e a io del T ibunal
Fecha:

Abs ac
This wo k can se e he eade as an ini ia ion in nume ical con inua ion h ough he AUTO so wa e, which
is specialized in bi u ca ion analysis and has p o ed o be ex emely e sa ile and e icien in calcula ing
o dina y di e en ial sys em solu ions ha would o he wise be much mo e di icul o ob ain. A b ie e iew
is made abou he concep s o non-linea dynamics ha es on he basis o he p oblem unde s udy, he
ci cula es ic ed h ee body p oblem. Then, he bases o he nume ical con inua ion a e p esen ed, as well
as he conc e e s a egies ha will be used o he calcula ion o pe iodic solu ions and hei mani olds. A e
his, he eade inds a guide o he basic concep s ha would allow him o use he AUTO so wa e, as well as
o unde s and he subsequen s udy ha is ca ied ou wi h i . Finally, AUTO is used o gene a e he amilies
o pe iodic o bi s ha a ise om he Lag ange poin s in he ci cula es ic ed h ee body p oblem, in addi ion
o he associa ed s able and uns able mani olds when possible. This analysis is used o commen on he
cha ac e is ics o he o bi s conside ed o be he bes candida es o he Deep Space Ga eway p ojec , which
aims o use nea ec ilinea Halo o bi s.
I
Resumen
Es e abajo puede se i al lec o como iniciación en con inuación numé ica a a és del so wa e AUTO,
especializado en análisis de bi u caciones y que demues a se ex emadamen e e sá il y e icien e en el
cálculo de soluciones a sis emas di e enciales o dina ios que de o a o ma se ían mucho más di íciles de
ob ene . Se hace un b e e epaso po los concep os de dinámica no lineal que descansan en la base del
ema p incipal bajo es udio, el p oblema ci cula es ingido de los es cue pos. Después se plan ean las
bases de la con inuación numé ica, así como las es a egias conc e as que se usa án pa a el cálculo de
soluciones pe iódicas y sus a iedades. T as es o, el lec o se encuen a con una guía de los concep os básicos
que le pe mi i ían u iliza el so wa e AUTO, así como comp ende el pos e io es udio que se ealiza con
es e. Finalmen e, se u iliza AUTO pa a calcula las amilias de ó bi as pe iódicas que nacen de los pun os
de Lag ange en el p oblema ci cula es ingido de los es cue pos, además de las a iedades es ables e
ines ables asociadas cuando p ocedan. Es e análisis se ap o echa pa a comen a las ca ac e ís icas de las
ó bi as conside adas como mejo es candida as pa a el p oyec o Deep Space Ga eway (Po al de espacio
p o undo), que p e ende u iliza ó bi as Halo casi ec ilíneas.
III

1 In oduc ion
I is hough science ha we p o e, bu h ough in ui ion ha we
disco e .
Hen i Poinca é
The h ee-body p oblem is a classical p oblem in physics and mechanics ha , in con as o he Keple ian
wo-body p oblem, doesn’ ha e an analy ical solu ion in gene al. The g a i a ional p oblem da es om 1687,
when New on published his "P incipia" and se ou he p oblem o he mo emen s o h ee masses subjec ed
o he in luence o each o he , he also ied o apply his esul s o s udy he mo ion o he moon unde he
in luence o he Ea h and he Sun. The longi ude p oblem was o g ea impo ance in he 1720s and an
accu a e solu ion o he h ee body p oblem would ha e been an al e na i e answe o he ma ine ch onome e .
In 1747 Jean le Rond d’Alembe and Alexis Clai au bo h submi ed hei analyses o he Académie Royale
des Sciences, in which hey used di e en ial equa ions o be sol ed by successi e app oxima ions. The name
" h ee-body p oblem" (P oblème des T ois Co ps) began o be used in he 1740s Pa is due o his esea ch.
I was Lag ange in 1772 who demons a ed analy ical solu ions do exis i ce ain es ic ions a e imposed,
as discused he eina e . Finally, in 1887, Hein ich B uns and Hen i Poinca é showed ha no analy ical
solu ions gi en by algeb aic exp essions and in eg als can exis in gene al, and e en hough he wo ds "chao ic
beha iou " we en’ used in his con ex un il he six ies, in [
1
]Poinca é does desc ibe exac ly his when
conside ing an uns able solu ion in a wo deg ees o eedom Hamil onian sys em:
I on ies o ep esen he igu e o med by hese wo cu es wi h an in ini e numbe o in e sec-
ions whe eas each one co esponds wi h a double asymp o ic solu ion, hese in e sec ions a e
o ming a kind o la ice-wo k, a issue, a ne wo k o in ini e closely packed meshes. Each o he
wo cu es mus no cu i sel bu i mus old on o i sel in a e y complex way o be able o cu
an in ini e numbe o imes h ough each mesh o he ne wo k.
One will be s uck by he complexi y o his pic u e ha I do no e en da e o ske ch. No hing is
mo e app op ia e o gi e us an idea o he in ica eness o he h ee-body p oblem and in gene al
all p oblems o dynamics whe e one has no a uni o m in eg al and whe e he Bohlin se ies a e
di e gen . (Hen i Poinca é)
Fo he gene al case he sys em is 18 h-o de , and 10 analy ical in eg als can be w i en, co esponding o
conse a ion o momen um ( h ee in eg als), ene gy (one in eg al), and mo ion o he sys em’s mass cen e
(six in eg als). Toge he , his in eg als and special case condi ions allow o some solu ions o be analy ically
s udied (Lag ange’s h ee-body Solu ions [
2
]). In one o hose solu ions he masses lie a he e ices o a
o a ing equila e al iangle, wi h a ying angula eloci y and size ( igu e (1.1)). Ano he case is ha o he
h ee masses collinea wi h each o he a any momen ( igu e (1.2)). Sub-solu ions o he p e ious ypes can
be ound i he mo ion o he bodies is es ic ed o be ci cula , ins ead o a gene al conic ( igu es (1.3) and
(1.4)).
O he s special es ic ed o ms o he p oblem ha e been s udied, bu so a he p e ious cases shown a e
he only ones wi h analy ical solu ion. He e, he sub-p oblem o be add essed is ha o one o he masses o be
negligible o he mo ion o he o he wo, he es ic ed h ee-body p oblem. In pa icula , he case o s udy
impose he wo massi e bodies o be in ci cula mo ion a ound hei cen e o mass, and so i is called Ci cula
1
2Chap e 1. In oduc ion
Figu e 1.1 Gene al equila e al iangle solu ion. F om [2].
Figu e 1.2 Gene al in a ian collinea solu ion. F om [2].
Figu e 1.3 Equila e al iangle solu ion wi h ci cula o bi s. F om [2].
1.1 Poinca é’s dynamical sys em heo y 3
Figu e 1.4 Collinea solu ions wi h ci cula o bi s. F om [2].
es ic ed h ee-body p oblem, CR3BP om he e on ou . The p oblem i sel has been ex ensi ely s udied, and
analy ical solu ions o s a iona y poin s can be deduc ed om he equa ions o mo ion (Lag ange Lib a ion
Poin s, a degene a e case o he p e iously shown ci cula solu ions o he gene al p oblem iewed om a
o a ing ame), bu he main in e es is in he pe iodic o bi s a ound wo body sys ems as he Ea h-Moon
o Sun-Ea h sys ems, in o de o design missions ha makes use o such p ac ical o bi s. Because o his,
he main goal o his wo k is o ind said pe iodic o bi s and s udy he in a ian mani olds associa ed, all o
which can be applied o he design o missions, such as he Deep Space Ga eway [
3
]. To do his we ha e o
ely on nume ical me hods, and con inua ion, as we will see, has many ad an ages o e o he app oaches.
Con inua ion echniques ely on much o he wo k o Hen i Poinca é, an so we will e iew i s mo e ele an
elemen s as a mean o his o ical in oduc ion o he ma hema ical aspec s ea ed he e.
1.1 Poinca é’s dynamical sys em heo y
Hen i Poinca é (1854-1912) was a amous ma hema ician ha oday is know as he las polyma h [
4
], because
he made signi ican con ibu ions in mul iple a eas o ma hema ics and he physical sciences. His wo k in
dynamical sys ems and opology is specialy impo an , and i will be he aspec o be commen ed in his
chap e . His mos impo an in en ions [
5
] ega ding dynamical sys ems a e algeb oid unc ions, index heo y
o plane ODEs, he Poinca é-Bendixon heo em, con e gence o se ies solu ions o ODEs, he use o he
implici unc ion heo em,
bi u ca ion heo y
( he Hop bi u ca ion), asymp o ic se ies, ixed poin heo ems
o dynamical sys ems, homoclinic chaos...
Poinca é was educa ed mainly in geome y and analysis, bu as has been s a ed, he didn’ ocus on one
pa icula discipline. S ill, his mos impo an wo ks a e cha ac e ized by he in e ac ion o analy ical and
geome ical hinking.
The Mémoi e, 1881-82
I is conce ned wi h wo-dimensional p oblems. The Mémoi e ocus on au onomous second o de equa ions,
as many a icles on ODEs a he ime, bu wi h an app oach ha di e s b oadly o m o he s, o such poin ha
oday he philosophy o he s udy in nonlinea dynamics hasn’ change much in i ’s gene ali y and s a egies.
In he e, Poinca é ema ks he impossibili y o in eg a ing ODEs in gene al wi h known unc ions, and so
di ides he s udy o hese sys ems in wo pa s:
1. Quali a i e pa , s udy o he geome y de ined by he unc ion
2. Quan i a i e pa , nume ical calcula ions
He e, Poinca é uses gnomic p ojec ion o analyse sys ems, which leads o he de ini ion o he nowadays
well-known saddle,node, ocus and cen e, he singula i ies o i s ype. This also de elops in he basis o
index heo y, a ool o he s udy o cycles.
Ano he use ul ool is he ’ héo ie o conséquen s’, wha is now called he heo y o Poinca é maps, his
helps o unde s and high-dimensional ODEs and will be use ul la e .
4Chap e 1. In oduc ion
The P ize Essay o Osca II, 1888-89
This essay con ains al eady undamen al heo ems, and impo an esul s in ol e se ies expansions,
pe iodic
solu ions
and
bi u ca ions
. A he ime, se ies expansions wi h espec o a small pa ame e we e he main
ool in celes ial mechanics. Poinca é ga e explici c i e ia o he con e gence and di e gence o such se ies
based on he implici unc ion heo em (IFT) and holomo phic expansion heo ems. He also se -up a ield
o s udy which is cen ic in he p esen wo k, such us
bi u ca ion heo y
. As i will be clea la e on,
bi u ca ions a e he ma hema ical key o he Halo o bi s ha will be s udied in some dep h he e.
Ano he basic esul ha de elops in he e, which in ac was a co ec ion o he i s e sion o he essay,
is he non-in eg abili y o conse a i e sys ems. A he ime, i was hough ha inding an in eg al o , o
ins ance, he h ee-body p oblem, was a ma e o analy ical skill, and ha all Hamil onian sys ems we e
always in eg able. In he i s e sion he p o ed in eg abili y o he CR3BP, iden i ied an uns able pe iodic
solu ion and app oxima ed i s s able and uns able mani olds by se ies expansions. Poinca é called hese
in a ian mani olds asymp o ic su aces. He inco ec ly concluded ha he con inua ion o s able and uns able
mani olds could be glued oge he o o m a second i s in eg al o he sys em. A e inding abou his
mis ake, he ound ou ha his gluing was no possible in his pa icula case, and ha he e was in ac an
in ini e numbe o in e sec ion be ween he mani olds (ins ead o me ging).
Les Mé hodes Nou elles de la Mécanique Céles e, 1892-1899
This wo k con ains he i s gene al heo y o dynamical sys ems desc ibing bo h conse a i e and dissipa i e
sys ems by analy ical and geome ical me hods. Despi e he i le, celes ial mechanics is only used as a mean
o illus a e he me hods, and i is no he subjec o s udy.
Poinca é used he IFT o demons a e condi ions o he con e gence o se ies expansions in ODEs, wi h
consequences o he bi u ca ion solu ions, all o which was a new use o he IFT. He also in oduces he
no ion o bi u ca ion se , which in Chap. 3 leads o a gene al discussion o wha is now called he Hop
bi u ca ion. In his same chap e he implici ly poin s ou he "chao ic na u e" o he beha iou o an uns able
pe iodic solu ion in a wo deg ees o eedom Hamil onian sys em, as was p e iously shown.
1.2 Halo o bi s and he Deep Space Ga eway
Leaded by NASA and suppo ed by he main space agencies in he wo ld (ESA, JAXA, Roscosmos and
CSA), he nex s ep ha will be aken in o de o ad ance in he explo a ion o space s a s by building a new
space s a ion in cisluna space
1
, he Deep Space Ga eway (DSG). The p incipal goal o his s a ion is o he
as onau s and he agencies o es he sys ems needed o missions in deep space (Ma s, as e oids...), and his
a ea o he Ea h-Moon sys em o e s a g ea oppo uni y o gain expe ience. I could also se e as a middle
poin be ween Ea h and a Moon base. Fo his space s a ion o be o p ac ical use i needs o be placed in an
o bi such ha i ’s easily accessible and ela i ely cheap o main ain, and a he same ime i has o se e as
he ga eway o he u u e missions o explo a ion in he Sola Sys em. The s udy o pe iodical o bi s in he
CR3BP may shed some ligh on his issue.
La e i will be shown ha many amilies o pe iodic o bi s exis in he CR3BP, and one o hese amilies is
known o ha ing a peculia shape as seen om Ea h. They a e he
Halo o bi s
, and as hei name sugges s
hey a e seen as a halo a ound he Moon. The e is also a g ea a ie y o halo o bi s hemsel es, and hey
ha e been ex ensi ely s udied in he con ex o space explo a ion [
6
]. Inside his g oup o o bi s he e is a
subse ha s ands ou o he pu pose o ha ing an inhabi ed s a ion wi h deep space access [
7
], which a e
he Nea Rec ilinea Halo O bi s, o
NRHOs
. These o bi s a e a he edge o Ea h’s g a i y well om an
ene ge ic poin o iew, which makes hem ideal as s epping s one o deep space missions, and hei s abili y
p ope ies a e qui e p omising.
1.3 Objec i es and scope o wo k
Con inua ion will se e he e as a ool o s udying hese o bi s and hei dynamical p ope ies in he CR3BP
amewo k, in o de o s ablish he ounda ions o la e s udies wi h mo e de ailed models. Al hough a
comple e ma hema ical jus i ica ion o con inua ion isn’ he scope o his wo k, some o he bases i elies on
ha e been plainly es ablished. This wo k ies o se e as a how- o basic guide in he ways he AUTO so wa e
could be used in he con ex o compu ing he dynamical s uc u es a ound he CR3BP wi h con inua ion. I
1I is he olume wi hin he Moon’s o bi , beyond cisluna space lies ansluna space
1.4 S uc u e o he documen 5
shows he basic u ilisa ion o AUTO wi h he Py hon in e ace, he calcula ion o ixed poin s, cycles and
inally he mani olds, as well as how o ex ac he ele an in o ma ion o hem om he ou pu iles ha
AUTO p o ides.
1.4 S uc u e o he documen
Aside om his in oduc ion, his wo k is composed o 4 mo e di e en chap e s ha add ess ele an aspec s
o he opics ega ding he CR3BP. Chap e 2 se es as a e ision o nonlinea dynamics heo y ha helps
o unde s and concep s used la e on, i also explains e y ew impo an concep s ega ding bi u ca ion
heo y. Chap e 3 se es as a ma hema ical o e iew o con inua ion as a ool, i also explains some o he
in e nal wo kings o he so wa e AUTO, in o de o help unde s and wha happens inside he black box ha
his so wa e will ul ima ely be o us. This chap e also exempli ies he s a egies ha will be used la e o
compu e cycles, and specially mani olds, om a nume ical poin o iew. Chap e 4 se es as a s a ing poin
o he eade o amilia ise himsel wi h he AUTO en i onmen and capabili ies, i ies o be he minimum
amoun o in o ma ion needed o someone ha wan s o use he so wa e in he con ex o his wo k and
doesn’ wan o deal wi h he comple e o icial manual. Chap e 5 is he s udy conce ning he CR3BP using
AUTO as main compu a ional ool, i is ai ly de ailed in how he calcula ions ha e been done and i also
analyses he iabili y o he NRHO as he u u e des ina ion o he DSG, only in he con ex o he idealised
ci cula es ic ed p oblem a hand.

2 Nonlinea dynamics and bi u ca ion
heo y
This chap e ies o s ablish he main ideas ha a e used in he s udy o non-linea dynamical sys ems, and
ha will be use ul in he la e con inua ion s udy o he CR3BP. I will ollow some o he opics discussed in
[8] and [9].
2.1 Dynamical sys ems
The idea o dynamical sys ems a ise om he ma hema ical concep ualiza ion o a de e minis ic p ocess.
This means ha gi en an ini ial s a e and he laws go e ning i s e olu ion, he u u e s a es a e comple ely
de ined. Because o his, he concep o dynamical sys em includes a se o i s possible s a es (s a e space)
and a law o he e olu ion in ime. This way, i a poin x ha is pa o he s a e space (o phase space)Xis
gi en, i is enough o desc ibe he sys em in i s ac ual "posi ion" and he ones ollowing i .
In a gene al way, he e olu ion o he sys em comes om an e olu ion ope a o , which o a gi en
∈T
can be de ined as a map ϕ in he phase space X ha :
ϕ :X−→X
I ans o ms an ini ial s a e x0∈Xin some s a e x ∈Xa ime :
x =ϕ x0
The map
ϕ
migh be known explici ly, bu in mos cases i is de ined only indi ec ly. When he go e ning
beha iou o he sys em does no change in ime i is said ha he sys em is au onomous, which can be
exp essed as:
ϕ +s=ϕ (ϕsx)
2.1.1 O bi s and phase po ai s
O bi s a e geome ical objec s associa ed wi h a dynamical sys em, and phase po ai s a e a composi ion o
hese o bi s. Fo a gi en x0, he associa ed o bi (o ajec o y) is:
O (x0) = {x∈X:x=ϕ x0, o all ∈Tsuch ha ϕ x0is de ined }
A poin
x0∈X
is called an equilib ium ( ixed poin ) i
ϕ x0=x0
o all
∈T
, hus, an e olu ion ope a o
maps an equilib ium on o i sel . The e m equilib ium is usually ese ed o con inuous- ime dynamical
sys ems, while ixed poin is mo e commonly used in disc e e- ime sys ems.
A cycle is a pe iodic o bi , so i
L0
is he o bi , hen any poin
x0∈L0
will sa is y
ϕ +T0x0=ϕ x0
o some
T0>0
a any
∈T
. I he e a e no o he cycles in he neighbou hood hen i is called a limi cycle. Finally,
a phase po ai is a pa i ioning o he s a e space in o o bi s, e ealing he beha iou o he sys em. An
example o i can be seen in igu e (2.1) o a con inuous- ime dynamical sys em.
7
8Chap e 2. Nonlinea dynamics and bi u ca ion heo y
Figu e 2.1
This image om [
9
] shows some o he mos impo an ea u es o a phase po ai : ixed poin s
(A, B, C), cycles (D) and he di e en ypes o beha iou nea ixed poin s.
2.1.2 In a ian se s
An in a ian se o a dynamical sys em
{T,X,ϕ }
is a subse
S⊂X
such ha
x0∈S
implies
ϕ x0∈S
o all
∈T
. O bi s
O (x0)
a e in a ian se s, and we could conside closed in a ian se s in X, such as equilib ia
and cycles. The nex mo e complex in a ian se s a e in a ian mani olds, ini e-dimensional hype su aces
in some space RK, like he in a ian o us o igu e (2.2).
Figu e 2.2
This image om [
8
] shows an in a ian wo-dimensional o us
T2
o a con inuous- ime dynamical
sys em in R3.
An in a ian se
S0
is called s able i o any su icien ly small neighbou hood
U⊃S0
he e exis s a
neighbou hood
V⊃S0
such ha
ϕ x∈U
o all
x∈V
and all
>0
; o i he e exis a neighbou hood
U0⊂S0
such ha
ϕ x→S0
o all
x∈U0
, as
→+∞
. The i s ype o s abili y is called Lyapuno s abili y, and he
second asymp o ic s abili y ( igu e (2.3) (a) and (b) espec i ely).
Figu e 2.3 (a) Lyapuno s abili y e sus (b) asymp o ic s abili y. F om [8].
2.2 Poinca é maps 9
2.1.3 Di e en ial equa ions
Di e en ial equa ions a e he mos common way o de ine a con inuous- ime dynamical sys em. Fo a s a e
space
X=Rn
wi h coo dina es
(x1,x2,...,xn)
he laws o e olu ion a e gi en implici ly in e ms o eloci ies
˙xias unc ions o he coo dina es:
˙xi= i(x1,x2, ...,xn),i=1,2,..., n
o in he ec o o m
˙x= (x)(2.1)
whe e he ec o - alued unc ion
:Rn→Rn
is su icien ly di e en iable (smoo h). Equa ion (2.1) is a
sys em o n au onomous o dina y di e en ial equa ions, ODEs o sho . Fo an isola ed ene gy-conse ing
mechanical sys em wi h s deg ees o eedom, he equa ions o mo ion can be de e mined by 2s Hamil onian
equa ions:
˙qi=∂H
∂pi
,˙pi=−∂H
∂qi
o i=1,2,...,s. The scala unc ion H=H(q,p)is he Hamil on unc ion.
The condi ions o exis ence, uniqueness and smoo h dependence o he unc ion
x=x( ,x0),x:R1×Rn→
Rn
ha is solu ion o equa ion (2.1) a e exp essed wi h de ail in [
8
]. The unc ion o ime
x=x( ,x0)
is
called solu ion s a ing a
x0
. I de ines a solu ion cu e
C (x0)
( ime dependen ), and an o bi , which is he
p ojec ion on o he s a e space o he solu ion cu e. He e, he e olu ion ope a o ϕ can be de ined as
ϕ x0=x( ,x0)
in an in e al o
. Dynamical sys ems heo y ies o analyse he beha iou o a dynamical sys em de ined
by ODEs. This can be done by simply compu ing many o bi s nume ically, howe e , his app oach isn’ e y
p ac ical, and i is possible o p edic some ea u es o he phase po ai wi hou ac ually ha ing o sol e
he sys em. The i s hing ha can be done is o s udy he numbe and posi ions o equilib ia by inding he
solu ions o
(x) = 0(2.2)
I is also possible o s udy he s abili y o an equilib ium, and su icien condi ions a e gi en by he ollowing
heo em (Lyapuno [1892]):
Conside a dynamical sys em de ined by
˙x= (x),x∈Rn,
whe e is smoo h. Suppose ha i has an equilib ium
x0
(i.e.,
(x0) = 0
), and deno e by A he Jacobian
ma ix o (x) e alua ed a he equilib ium,
A= x(x0)
. Then
x0
is s able i all eigen alues
λ1,λ2,...,λn
o
A
sa is y Re λ<0.
Whe e he eigen alues a e he oo s o he cha ac e is ic equa ion (
de (A−λI) = 0
). Mo e complex
p ope ies such as cycles a en’ so easy o s udy by jus sol ing algeb aic equa ions, bu as we’ll see la e ,
he e is a way wi h con inua ion.
2.2 Poinca é maps
I is no uncommon ha disc e e- ime dynamical sys ems (maps) appea in he s udy o con inuous- ime
sys ems de ined by ODEs [10]. These maps a ising om ODEs a e called Poinca é maps.
2.2.1 Poinca é maps and s abili y o cycles
Conside a con inuous dynamical sys em de ined by
16 Chap e 3. Nume ical con inua ion me hods
(G1
u)(ν)(G1
λ)(ν)
˙
u0∗˙
λ0 ∆u(ν)
1
∆λ(ν)
1!= G(u(ν)
1,λν
1)
(u(ν)
1−u0)∗˙
u0+(λ(ν)
1−λ0)˙
λ0−∆s!,
Wi h he di ec ion ec o de ined as:
G1
uG1
λ
˙
u∗
0˙
λ0˙
u1
˙
λ1=0
1,
A e i s compu a ion he di ec ion ec o mus be escaled, so ha i s no m con inues o be
1
. This me hod
can be e icien ly sol ed i co ec ly implemen ed as shown in [12].
Figu e 3.3 G aphical in e p e a ion o pseudo-a cleng h me hod. F om [12].
3.3 Bounda y alue p oblem
In a gene al way, AUTO sol es he BVP o he ollowing shape:
u0( )− (u( ),µ
µ
µ,λ) = 0, ∈[0,1],
whe e "0" is a de i a i e wi h espec o ime, and
u(·), (·)∈Rn,λ∈R,µ
µ
µ∈Rnµ
and subjec o he nex bounda y condi ions
b(u(0),u(1),µ
µ
µ,λ) = 0,b(·)∈Rnb,
and in eg al condi ions i necessa y o he p oblem
Z1
0
q(u(s),µ
µ
µ,λ)ds =0,q(·)∈Rnq
The goal is o sol e he BVP o
u(·)
and
µ
µ
µ
. Fo he p oblem o be well posed he nex condi ion has o be
ue:
nµ=nb+nq−n≥0.
In his case he pa ame e
λ
is he con inua ion pa ame e in which he solu ion
(u,µ
µ
µ)
is con inued. AUTO
sol es BVP using he me hod o o hogonal colloca ion wi h piecewise polynomials, as i is e y accu a e
and allows adap i e mesh-selec ion. He e he gene al idea behind his me hod will be shown, o a mo e
de ailed explana ion see [13] and [14].
Fi s , a mesh is in oduced be ween
=0
and
=1
(AUTO can change his mesh h ough he i e a ions i
necessa y):

3.3 Bounda y alue p oblem 17
{0= 0< 1···< N=1},
wi h
hj= j− j−1,(1≤j≤N).
Nex , he space o ( ec o - alued) piecewise polynomials Pm
his de ined as:
Pm
h={ph∈C[0,1]|ph|[ j−1, j]∈Pm},
whe e
Pm
is he space o ( ec o - alued) polynomials o deg ee
m≤3
. The colloca ion me hod consis s
in inding ph∈Pm
hand µ
µ
µ∈Rnµ, such ha he colloca ion equa ions a e sa is ied:
p0
h(zj,i) = (ph(zj,i),µ
µ
µ,λ),j=1,···,N,i=1,···,m,
and such ha
ph
sa is ies he bounda y and in eg al condi ions. The colloca ion poin s
zj,1
in each
subin e al
[ j−1, j]
a e he oo s o he m h-deg ee o hogonal polynomial (Gauss poin s). Fo a g aphical
in e p e a ion o i see he igu e (3.4).
Figu e 3.4
He e a e shown he colloca ion poin s and he ’ex ended-mesh poin s’ o he case m=3. Also,
wo o he local Lag ange basis polynomials a e shown. F om [12].
The implemen a ion is done wi h Lag ange basis polynomials o each subin e al [ j−1, j]. We de ine
{`j,i( )},j=1,...,N,i=0,1,...,m,
by
`j,i( ) =
m
∏
k=0,k6=i
− j−k
m
j−i
m− j−k
m
whe e
j−i
m= j−i
mhj
Then, he local polynomials can be w i en as
pj( ) =
m
∑
i=0
`j,i( )uj−i
m.
Then he colloca ion equa ions a e
18 Chap e 3. Nume ical con inua ion me hods
p0j(zj,i) = (pj(zj,i),µ
µ
µ,λ),i=1,...,m,j=1,...,N,
he disc e e bounda y condi ions a e
bi(u0,uN,µ
µ
µ,λ) = 0,i=1,...,nb,
and he in eg als cons ain s can be disc e ized as
N
∑
j=1
m
∑
i=0
wj,iqk(uj−i
m,µ
µ
µ,λ) = 0,k=1,...,nq,
whe e he
wj,i
a e he Lag ange quad a u e coe icien s. Gi en he solu ion o he p e iously compu ed
poin on he solu ion b anch,
(u0,µ
µ
µ0,λ0)
and he di ec ion ec o
(˙
u0,˙
µ
µ
µ0,˙
λ0)
, he disc e ized pseudo-a cleng h
equa ion is
N
∑
j=1
m
∑
i=0
wj,i[uj−i
m−(u0)j−i
m]∗(˙
u0)j−i
m+(µ
µ
µ−µ
µ
µ0)∗˙
µ
µ
µ0+(λ−λ0)˙
λ0−∆s=0.
The shape o he sys em is illus a ed in igu e (3.5), whe e he numbe o colloca ion equa ions,
mnN
,
con inui y equa ions,
(N−1)n
and cons ain s,
nb+nq(= n+nµ)
, ma ch he o al numbe o deg ees o
eedom (m+1)nN +nµ(wi h λ ixed).
Figu e 3.5
S uc u e o he sys em o he case o
n=2
di e en ial equa ions wi h numbe o mesh in e als
N=3
, numbe o colloca ion poin s pe mesh in e al
m=3
, he numbe o bounda y condi ions
nb=2
, and he numbe o in eg al cons ain s
nq=1
, being he las ow he pseudo-a chleng h
equa ion. F om [12].
AUTO sol es hese sys ems wi h an e icien me hod ha includes he condensa ion o pa ame e s by
Gauss elimina ion done in pa allel.
3.4 Compu ing cycles
Compu ing pe iodic solu ions can be done using a bounda y alue app oach. Conside he i s o de sys em
u0( ) = (u( ),λ),u(·), (·)∈Rn,λ∈R.
In o de o make he pe iod o he solu ion
1
we apply he nex ans o ma ion
→
T
, so he equa ion
becomes
u0( ) = T (u( ),λ),u(·), (·)∈Rn,T,λ∈R.(3.2)
3.4 Compu ing cycles 19
This way, he solu ions we seek a e hose in which
u(0) = u(1).(3.3)
The pe iod T is one o he unknowns o he sys em. Suppose ha we ha e al eady calcula ed a solu ion
(uk−1(·),Tk−1,λk−1)
and we wan o compu e he nex
(uk(·),Tk,λk)
. Then, i i we e only o he las wo
equa ions, a ansla ed solu ion om he las one is also a solu ion:
uk( ) = uk−1( +σ)
. Thus, a phase
condi ion is needed o a i e a he nex solu ion. One op ion is o impose a displacemen using he Poinca é
o hogonali y condi ion (see igu e (3.6)):
(uk(0)−uk−1(0))∗u0
k−1(0) = 0,
hough a nume ically mo e sui able phase condi ion is
Z1
0
uk( )∗u0
k−1( )d =0.(3.4)
This is deduc ed in [
12
] and i ’s called he in eg al phase condi ion, which is wha AUTO has implemen ed.
Figu e 3.6 G aphical in e p e a ion o he Poinca é phase condi ion. F om [12].
The con inua ion o a amily o pe iodic solu ions is also done wi h he pseudo-a cleng h me hod, as his
allows calcula ion pas olds. I also has impo an applica ions o he compu a ion o pe iodic solu ions
o conse a i e sys ems, as i allows calcula ions o a ’ e ical amily’ o pe iodic solu ions. Fo pe iodic
solu ions he pseudo-a cleng h equa ion is:
Z1
0
(uk( )−uk−1( ))∗˙
uk−1( )d +(Tk−Tk−1)∗˙
Tk−1+(λk−λk−1)˙
λk−1=∆s.(3.5)
Equa ions (3.2)-(3.5) a e he ones AUTO sol es o he con inua ion o pe iodic solu ions in gene al. In
addi ion, du ing he con inua ion, he
Flouque mul iplie s
o he cycles a e moni o ed by compu ing a
special decomposi ion o he monod omy ma ix ha a ises as a by-p oduc o he decomposi ion o he
Jacobian o he colloca ion sys em.
3.4.1 Cycles in conse a i e sys ems, un olding pa ame e
The app oach shown un il now o he con inua ion o pe iodic o bi s in dynamical sys ems is alid in gene al,
bu in he case o conse a i e sys ems wi h one conse ed quan i y he s a egy ollowed uses a pa ame e
wi h a special meaning.
Ra he han pe o m he con inua ion using he conse ed quan i y, which would be a educ ion me hod,
p oduc o he cylinde heo em, he al e na i e is abou inc easing he dimension o he sys em, adding an
un olding pa ame e . The addi ional e m is de ined in a way ha he pe iodici y o he solu ion can only
exis when he pa ame e is
0
, so ha looking o he cycle is equi alen o looking o a " e ical" Hop
bi u ca ion.
The sys em wi h he un olding e m becomes:
u0( ) = T (u( ),λ)+αd(u( )),α∈R,
20 Chap e 3. Nume ical con inua ion me hods
whe e
α
is he un olding pa ame e , which ac s as a dissipa ion (posi i e o nega i e), and ha will be
α=0in he pe iodic solu ion. The sys em can now be exp essed as:
u0( ) = T (u( ),α).
All he p e ious addi ional equa ions emains he same. Wi h his se -up, he amily o pe iodic solu ions
will be ha in which
α=0
( o nume ical p ecision), so ha he bi u ca ion diag am in
α
looks like he igu e
(3.7).
Figu e 3.7 Bi u ca ion diag am in αwi h he e ical b anch o pe iodic solu ions. F om [12].
3.5 Compu ing mani olds
The me hod ollowed he e is he one we will use in AUTO o he compu a ion o mani olds in h ee
dimensional sys ems (as he CR3BP) e en hough i could be done o any kind o sys em wi h pe iodic
solu ions.
We will i s assume ha he pe iodic o bi has a single, eal Floque mul iplie ou side he uni ci cle,
so ha i gi es ise o a wo-dimensional uns able mani old o he cycle in phase space. In his case, he
eigen unc ion associa ed o his mul iplie gi es a linea app oxima ion o he mani old close o he pe iodic
o bi . As s a ed in chap e 2 his eigen unc ion can be compu ed as a solu ion o he BVP





w0( ) = T u(u( ),0)w( )−λw( ),
w(0) = sw(1),
|w(0)|=√ρ,
whe e
s= +1
i he mul iplie is posi i e and
s=−1
o a nega i e one. The Floque mul iplie is hen
seλ
. We ha e impose he no m on he no malized eigen unc ion a
=0
o be
√ρ
, whe e ypically
ρ=1
.
Only when he e is one eal and g ea e han one in absolu e alue Floque mul iplie hen his p oblems
esul s in an unique uns able eigen unc ion
w( )
. This algo i hm applies equally well o s able mani olds,
in which case he e would only be one eal Floque mul iplie wi h absolu e alue less han one. We will
also es ic o he case
s=1
, so ha he Floque mul iplie is posi i e, and he co esponding mani old is
o ien able a he han wis ed. A line app oxima ion o he uns able mani old a ime ze o is gi en by
(0) = u(0)+εw(0)(3.6)
o
ε
small. Al hough his p ocedu e would indeed esul in he needed eigen unc ion, he ac ual s a egy
ollowed is a bi di e en . Gi en he pe iodic o bi
u0( )
, om which we wan o compu e he uns able
mani old, he sys em ha is sol ed is he one gi en by
3.5 Compu ing mani olds 21
u0( ) = T (u( ),α),
u(0) = u(1),
Z1
0
u( )∗u0
0( )d =0.
w0( ) = T u(u( ),0)w( )−λw( ),
w(0) = w(1),
|w(0)|=√ρ,
Z1
0
(ζ
ζ
ζ( )−ζ
ζ
ζ0( ))∗˙
ζ
ζ
ζ0( )d +(p−p0)∗˙
p0=∆s,ζ
ζ
ζ( ) = (u( ),w( )),p= (α,λ,ρ),
whe e
w( )
and
ρ
a e ini ialized o ze o. This way, he cycle co esponds o a bi u ca ion, whe e b anch
swi ching gi es he non-ze o eigen unc ion
w( )
. This me hod, al hough elabo a e, allows o he calcula ion
o he eigen unc ion in sensi i e cases, as o example when he Floque mul iplie s a e ei he e y la ge o
small. Now ha we ha e he eigen unc ion, he compu a ion o he mani old is as p oceed [15].
The me hod consis s basically in in eg a ing o bi s ha lie in he mani old un il a ce ain condi ion is
ma ched. We s a wi h he condi ions in he linea app oxima ion o he mani old (see equa ion (3.6)), and
pe o m con inua ion using he in eg a ion ime as a ee pa ame e , hen s op he con inua ion when he
desi ed condi ion is achie ed. To go down all he mani old i is impo an ha he ange o he pa ame e
ε
co esponds o a undamen al domain
[ε1,ε2)
, meaning ha he o bi ha s a s a
(0) = u(0)+ε1w(0)
closely passes he line gi en by
(0) = u(0)+ε2w(0)
o he i s ime. Gi en a ce ain
ε1
, ha has o be
su icien ly small so ha he mani old is co ec ly app oxima ed, we can compu e he co esponding linea
app oxima ion o
ε2
using
ε2=seλε1
. A possible s opping c i e ia could be ha he o bi ends in a chosen
sec ion Σ. The pa o he mani old o be app oxima ed is hen gi en by he se o o bi s:
{ ( )|u(0)+εw(0)and (1)∈Σ, o ε1⩽ε⩽ε2}
Whe e ime has been escaled so ha he en i e ini e in eg a ion in e al becomes he uni in e al. The
bounda y alue p oblem o ( )is gi en by:
0( ) = T ( ( ),0),
(0) = u(0)+εw(0),
(1)x=xΣ,
Z1
0
( ( )− 0( ))∗˙
0( )d +(ε−ε0)∗˙
ε0+(T −T 0)˙
T 0=∆s,
whe e
α=0
and
Σ
deno es he sec ion
x=xΣ
. The las equi es a s a ing o bi o be compu ed using ime
in eg a ion.

4 AUTO
This chap e ies o exempli y he mos impo an ea u es o AUTO ega ding non-linea dynamics and
shows how o use i wi h he Py hon in e ace, so ha using his minimun in o ma ion he eade should be
able o wo k wi h he so wa e. I is a selec ion o he mos ele an pa s o he AUTO manual [
16
] ega ding
he subjec o his wo k, so a mo e de ailed desc ip ion o he so wa e capabili ies (qui e ex ensi e) can be
ound in he e i necessa y.
4.1 Capabili ies o he so wa e
AUTO can pe o m bi u ca ion analysis o algeb aic sys ems o he o m
(u,p) = 0, (·,·),u∈Rn.
Al hough he main algo i hms in AUTO a e aimed a he con inua ion o solu ions o ODE’s o he o m
u0( ) = (u( ),p), (·,·),u(·)∈Rn,
subjec o bounda y and in eg al cons ains as i has al eady be explained. Rega ding algeb aic sys ems
AUTO can:
•Compu e solu ion o amilies.
•Loca e b anch poin s and au oma ically compu e bi u ca ing amilies.
•Loca e Hop bi u ca ion poin s and de ec i s p ope ies.
•Loca e olds.
•Find ex ema in an objec i e unc ion and con inue i in mo e pa ame e s.
Rega ding he s udy o o dina y di e en ial equa ions he lis is longe , bu he e he e is a b ie o e iew:
•
Compu e amilies o s able and uns able pe iodic solu ions and compu e he Floque mul iplie s ha
de e mines s abili y along hese amilies. The s a ing da a o compu e his o bi s can be gene a ed
au oma ically a Hop bi u ca ion poin s.
•
Compu e olds, pe iod-doubling bi u ca ions, and bi u ca ions o o i, in wo pa ame e s, de ec ing 1:1,
1:2, 1:3 and 1:4 esonances.
•Follow cu es o homoclinic o bi s and de ec and con inue a ious codimension-2 bi u ca ions.
•Loca e ex ema along a amily o pe iodic solu ions and con inue hem in mo e pa ame e s.
4.2 Use -supplied elemen s
4.2.1 Files
AUTO needs wo ile ypes o a p oblem o be de ined, one is he equa ions ile, and he o he is he cons an s
ile.
23
24 Chap e 4. AUTO
The equa ions ile is ei he a
xxx. 90
,
xxx.
o
xxx.c
ile, depending on he p og amming language u ilised
(he e
xxx
s ands o he name o he p oblem). The e mina ion (
.
) co esponds o ixed- o m Fo an, (
. 90
)
is in ee- o m Fo an and (
.c
) is w i en in C. All o hem a e alid bu he o ma (
. 90
) is he one used in
his wo k.
The ile
xxx. 90
con ains he Fo an ou ines ha will be explained la e , so ha i any o hem is i -
ele an o he p oblem hen i doesn’ ha e o be comple ed. Many examples can be ound in he olde
au o/07p/demos
. Fo a new p oblem, he simples way o c ea e i s equa ion ile is o copy he one co e-
sponding o a simila p oblem and modi y he app op ia e lines o code.
The cons an s ile is a
c.xxx
ile, whe e
xxx
doesn’ necessa ily ha e o coincide wi h he name o he
equa ions ile, because he cons an s ile used can be speci ied when a p oblem is execu ed, hough i has o
be con ained in he same olde as he
xxx. 90
ile. The meaning o many o hese cons an s will be explained
in sec ion (4.2.3).
4.2.2 Rou ines
The pu pose o each o he use -supplied ou ines in he
xxx. 90
ile is desc ibed bellow, al hough a good way
o amilia ise you sel wi h hem is o check he code o he demo p oblem cusp, as i is ully documen ed.
•FUNC: De ines he unc ion (u,p)o he p oblem.
•ST PNT
: I de ines he s a ing solu ion
(u,p)
, which should no be a b anch poin . This ou ine is only
called when he cons an
IRS =0
, which is usually he case o he i s un o he p oblem, o he wise i
se s he label o he solu ion whe e he compu a ion is o be es a ed.
•BCND: De ines he bounda y condi ions, see demo exp o ka .
•ICND: De ines he in eg al condi ions, see demo in o lin.
•FOPT : De ines he objec i e unc ional i needed, see demo op o ops.
•PVLS
: This ou ine is o "solu ion measu es", meaning ha i is used when he use wan s o speci y
a gi en ou pu unc ion o he p oblem a iables, such as no m o he solu ion, minimum, alue a a
bounda y...
4.2.3 Cons an s and pa ame e s
As explained be o e, he cons an s ha de ine he p oblem a e expec ed o appea in he
c.xxx
ile, bu his ile
is no s ic ly necessa y when using he Py hon in e ace, because you can de ine all he cons an s inside he
sc ip s when calling o he p oblem o un.
In he case ha he ile is used, he o ma in i is ee, wi h
cons an = alue
sepa a ed by comas o spaces.
An example wi h de aul alues is in igu e (4.1), hough in eal cons an iles you only need o speci y he
alues ha di e om hese.
A b ie o e iew o some o he cons an s is nex .
P oblem cons an s
•NDIM: Dimension o he sys em o equa ions.
•NBC: Numbe o bounda y condi ions.
•NINT: Numbe o in eg al condi ions.
•NPAR: Maximum numbe o pa ame e s ha can be used in all use -supplied ou ines.
•JAC
: Used o indica e whe he de i a i es a e supplied by he use o o be ob ained by di e encing,
such ha when JAR =0no de i a i es a e p o ided by he use .
Disc e iza ion cons an s
•NTST
: The numbe o mesh in e als used o disc e iza ion, i emains ixed o any pa icula un
and i ’s ecommended o be as small as possible o main ain con e gence.
•NCOL
: The numbe o Gauss colloca ion poin s pe mesh in e al, (
2⩽NCOL ⩽7
).
NCOL =4
is a
ecommended alue in mos cases.
•IAD
: I con ols he mesh adap a ion,
IAD =0
o ixed mesh and
IAD >0
o adap mesh e e y
IAD
s eps along he amily.
4.2 Use -supplied elemen s 25
Figu e 4.1 De aul alues o he AUTO cons an s ile.
Tole ances
•EPSL: Rela i e con e gence c i e ion o equa ion pa ame e s in he New on/Cho d me hod.
•EPSU: Rela i e con e gence c i e ion o solu ion componen s in he New on/Cho d me hod.
•EPSS: Rela i e a c-leng h con e gence c i e ion o he de ec ion o special solu ions.
•IT MX: Maximum numbe o i e a ions allowed in he accu a e loca ion o special solu ions.
•NWTN
: A e
NWTN
New on i e a ions he Jacobian is ozen, because AUTO uses ull New on
o he i s
NWTN
i e a ions and he Cho d me hod o i e a ions
NWTN +1
o
IT NW
, which is he
maximum numbe o combined New on-Cho d i e a ions.
Con inua ion s ep size
•DS: I de ines he pseudo-a cleng h s ep size o be used o he i s a emp ed s ep along any amily.
•DSMIN: Minimum allowable absolu e alue o he pseudo-a cleng h s ep size.
•DSMAX: Maximum allowable absolu e alue o he pseudo-a cleng h s ep size.
•IADS: I con ols he equency o he s ep size adap a ion. Adap s he s ep size e e y IADS s eps.
•THL
: De ines he pa ame e s whose weigh s is o be modi ied in he pseudo-a cleng h con inua ion,
so ha no all he pa ame e s change in he same way (o canno change a all).
Limi he con inua ion
•STOP: This adds s opping condi ions, o example s opping a he hi d Hop bi u ca ion.
•NMX: The maximum numbe o s eps aken along any amily.
•RL0: The lowe bound on he p incipal con inua ion pa ame e .
•RL1: The uppe bound on he p incipal con inua ion pa ame e .
•A0: The lowe bound on he p incipal solu ion measu e.
•A1: The uppe bound on he p incipal solu ion measu es.
O he s s opping op ions will be shown la e on.
32 Chap e 4. AUTO
Figu e 4.9 S a iona y solu ion amily o A→B→Cp oblem in he phase space.
Figu e 4.10 Con en o he cons an s ile o he second o las un o he p oblem. Image om [16].
The
abc(’HB’)
is a lis o he Hop bi u ca ion solu ions in he py hon objec abc, so he o loop goes
h ough all hose solu ions. Inside he loop he line abc = abc + un(solu ion, c=’abc.2’) execu es a un in
which he solu ion ac s as ini ial poin ( he Hop bi u ca ions) om which he pe iodic solu ions a e con inued,
and hen conca ena es he esul wi h abc. The esul s o his calcula ions can be seen in he bi u ca ion
diag am o igu e (4.11) and he solu ions in (4.12). In his case he
L2-No m
o he pe iodic solu ion amilies
co esponds o he maximum no m o he s a es h ough he o bi s hemsel es. The o bi s ma ked as
L
in
(4.11) co espond o olds. In igu e (4.13) he pe iod o he solu ion amilies a e ep esen ed, no ice ha e en
a he s a o each o hem he pe iod isn’ ze o. Finally, as o cla i y how he pe iodic solu ions eme ge o m
he Hop bi u ca ions, a de ail o his is ep esen ed in igu e (4.14), whe e he amily o pe iodic solu ions
ha comes om HB1is pic u ed om he s a .
This whole p oblem is a good example o he me hod ha will ollow, and will help o cla i y some o he
de ails in ol ed. The CR3BP is s udied now wi h AUTO.

4.5 P oblem example 33
Figu e 4.11 L2-NORM o he pe iodic solu ion amilies in A→B→Cp oblem.
Figu e 4.12 Some o he pe iodic solu ions in A→B→Cp oblem.
34 Chap e 4. AUTO
Figu e 4.13 Pe iod o he pe iodic solu ion amilies in A→B→Cp oblem.
Figu e 4.14
De ail o he
HB1
pe iodic solu ion amily as i eme ges om i s Hop bi u ca ion. The pa ame e
p1changes om 0.204 (in he HB) o 0.207.
5 AUTO-based nume ical s udy o he
CR3BP
In his chap e he ool AUTO is used o s udy he ypes o pe iodic o bi s a ound he Ea h-Moon sys em as
well as he in a ian mani olds associa ed.
5.1 P oblem de ini ion
As s a ed in he in oduc ion, he CR3BP desc ibes he mo ion o a negligible mass in h ee dimensional
space as an e ec o he g a i a ional a ac ion o wo hea y bodies which o bi hei common cen e o
mass in a pe ec ci cula mo ion. The scheme o his si ua ion o a gi en posi ion o he mass unde s udy
is in igu e (5.1), whe e he pa ame e
µ
ep esen s he a io o he mass o he smalle p ima y o he o al
mass o he sys em (
µ=m2/(m1+m2)
). The p oblem will be s udied in he non-dimensional o m, so ha
he dis ance be ween he wo main bodies equals o
1
a all imes ( he ime is also adimensionalized wi h de
angula eloci y).
Figu e 5.1 CR3BP scheme (a) and nomencla u e o he i e lib a ion poin s (b). Image om [15].
Gi en he ame o e e ence cen ed in he ba ycen e o he wo massi e bodies ha o a es wi h he same
speed as hem (a cons an o he p oblem), he adimensionalized equa ions o mo ion a e well known [2]:
x00 =2y0+x−(1−µ)(x+µ) −3
1−µ(x−1+µ) −3
2,
y00 =−2x0+y−(1−µ)y −3
1−µy −3
2,
z00 =−(1−µ)z −3
1−µz −3
2,
(5.1)
35
36 Chap e 5. AUTO-based nume ical s udy o he CR3BP
whe e
1=q(x+µ)2+y2+z3,
2=q(x−1+µ)2+y2+z”.
(5.2)
The equa ion sys em (5.1) has one in eg al o mo ion, speci ically he ene gy o Jacobi cons an :
E=x02+y02+z02
2−U(x,y,z)−µ(1−µ)
2,(5.3)
whe e U(x,y,z)is he po en ial ene gy, combina ion o g a i y and he o a ion o he ame o e e ence:
U(x,y,z) = 1
2(x2+y2)+ 1−µ
1
+µ
2
.
As we a e mos in e es ed in he case o he Ea h-Moon sys em, h oughou much o his documen we
will conside µ=0.01215.
5.2 Se ing up he AUTO p oblem
The . 90 ile ha will be use in gene al is he co esponding o he AUTO demo
3b
, which is explained
now. The
FUNC
sub ou ine is in igu e (5.2), whe e he equa ions o mo ion a e coded in ee- o m Fo an.
No ice ha he e a e wo explici pa ame e s in he equa ions, one is
µ
, and he o he is mul iplying he
eloci ies in he accele a ion equa ions (
F(4)
,
F(5)
and
F(6)
in he code), which is wha we called un olding
pa ame e in sec ion (3.4.1). Tha las pa ame e will be he one used o con inue he pe iodic o bi s.
Nex , he
ST PTN
sub ou ine is in igu e (5.3). He e he s a e a iables and he pa ame e s a e se up o
he i s un ha will be execu ed, which will be done o compu e he s a iona y solu ions wi h
µ
. No ice ha
o he selec ed alue o
µ
he chosen s a e is ha in which he posi ion is on he uni ci cle in he
xy
plane
and he eloci ies a e ze o. Tha is he degene a e case in which he e is no seconda y body in he sys em,
whe e any posi ion in he uni ci cle is a s a iona y solu ion o (5.1).
Figu e 5.2 FUNC sub ou ine in he CR3BP.
5.2 Se ing up he AUTO p oblem 37
Figu e 5.3 ST PNT sub ou ine in he CR3BP.
Figu e 5.4 PVLS sub ou ine in he CR3BP, pa 1.
Finally, he
PVLS
is in igu es (5.4) and (5.5). The i s pa calcula es he ene gy o he solu ion s a e
U
ha comes as an inpu using equa ion (5.3). No ice ha wo pa ame e s a e c ea ed, one s o es he ene gy o
he solu ion and he o he he alue o he posi ion in
y
. The
PAR(16)
will be a solu ion measu e. The second
pa o he ou ine has wo dis inguishable unc ions, one is o calcula e he maximum eal Floque mul iplie
and o s o e i in
PAR(4)
, his will be use ul in disce ning he s abili y o he pe iodic o bi s. I
PAR(4) = 0
hen i means ha wo Floque mul iplie s a e ou side o he uni ci cle, so no s abili y can be expec ed om
ha solu ion. To achie e his i is necessa y o use he unc ion GETP, a mo e de ailed desc ip ion o i s
usage is in igu e (5.6). The second unc ion e ie es he imagina y pa s o he eigen alues which only
ha e imagina y pa , and s o es he co esponding pe iod (
2π/w
, whe e
w
is he imagina y pa ) in
PAR(5)
,
PAR(6)
and
PAR(7)
. This will only be used in he s a iona y solu ion analysis a he beginning. Fo his
equa ions ile no o he sub ou ines will be used.

38 Chap e 5. AUTO-based nume ical s udy o he CR3BP
Figu e 5.5 PVLS sub ou ine in he CR3BP, pa 2.
Figu e 5.6 Desc ip ion o GETP unc ion ex ac ed om cusp demo p oblem.
5.3 S a iona y solu ion analysis 39
5.3 S a iona y solu ion analysis
Fi s , and as s a ed in he example o sec ion (4.5), we need he s a iona y solu ions om which he pe iodic
o bi s eme ge. As i is well known, he CR3BP has
5
s a iona y solu ions o lib a ion poin s ( igu e (5.1) (b)).
Th ee o he lib a ion poin s, deno ed L1, L2 and L3, a e collinea wi h he p ima y bodies. Each o he o he
wo lib a ion poin s, L4 and L5, o ms an equila e al iangle wi h he p ima ies.
The p ocedu e will be o s a a he s a iona y solu ion in he
ST PNT
sub ou ine and le AUTO sea ch
om he e un il i inds he desi ed poin s a
µ=0.01215
. To do ha , he cons an s ile in igu e (5.7) is he
one u ilized. The mos ele an cons an s alues o no ice om he e a e:
•NDIM =6, he dimension o he p oblem is now 6once exp essed as an ODE.
•IPS =1and IRS =0, as we a e looking o s a iona y solu ions s a ing om he poin in STPNT.
•ICP = [2,16,5,6,7]
, he main con inua ion pa ame e is
PAR(2)
(
µ
in he code o 3b. 90), bu
PAR(16)
,
PAR(5)
,
PAR(6)
and
PAR(7)
will be solu ion measu es. By knowing
PAR(16)
( he
y
coo dina e o he
poin ) we can know i i is a collinea o iangula solu ion, bu mo e impo an ly, i allows o con inue
he solu ion in he uni ci cle when
µ=0
. On he o he hand, knowing he pe iods associa ed will ell
us he numbe and ype o o bi s ha can emana e om hem.
•NPAR =16, his is necessa y in o de o use pa ame e s un il PAR(16).
•UZSTOP ={16 : 0.991,2 : [−0.1,1.1]}, he condi ions o he con inua ion o s op in each b anch.
Figu e 5.7 Cons an s ile used o he analysis o s a iona y solu ions in he CR3BP.
Maybe mo e impo an han he cons an s ha a e in he ile, he lack o any e e ence o he alue o
µ
a
which he ou pu is needed o he p oblem o con inue s ands ou . Tha will be sol ed when he p og am is
un.
Figu e 5.8
Sc ip ha compu es he lib a ion poin s o he
µ
in he Ea h-Moon sys em, as well as he numbe
o pe iodic o bi s ha emana e om hem ho ough he pe iods.
Then, o execu e he sea ch, he AUTO sc ip in igu e (5.8) has been used. Le ’s ocus i s in he lines
be o e he impo . A a iable wi h he alue o
µ
o he Ea h-Moon sys em is c ea ed, hen a message is
p in ed on he sc een o indica e ha he compu a ion o he lib a ion poin s begins. Then, by using 1 =
un(’ 3b’, UZR=
{
2:mu
}
), AUTO au oma ically se s he p oblem wi h he iles called 3b (equa ions and
cons an s iles), and adds he cons an UZR=
{
2:mu
}
o he p oblem, which is mean o o ce an ou pu
o he alue o
µ
p e iously gi en. This ew lines a e enough o make a comple e analysis o s a iona y
40 Chap e 5. AUTO-based nume ical s udy o he CR3BP
solu ions in he CR3BP wi h AUTO, and i is a e y comple e analysis as he igu e (5.9) shows, he esul
o he plo command. The sc een ou pu is in igu e (5.10), whe e we can see AUTO has un eil
4
b anches
in he solu ion amilies. The i s b anch is he one o he s a ing poin , he uni ci cle o
µ=0
, as he
alues o
PAR(2)
seems o indica e (ze o o nume ical p ecision). The 3 o he b anches a e accessed h ough
he b anching poin s wi h labels
2
,
3
and
4
. The second b anch co esponds o he iangula poin s wi h
posi i e
y
coo dina e, as he
PAR(16)
e eals, wi h he hi d b anch being he iangula poin s wi h nega i e
y
coo dina e. The las b anch is ha o he collinea poin s, blue in igu e (5.9). Nex hing o no ice is ha
b anch
4
has
3
use eques ed ou pu s wi h
PAR(2) = 0.01215
o nume ical p ecision. These a e he h ee
collinea poin s o he alue o
µ
eques ed, which we will call
”UZ7”
,
”UZ8”
and
”UZ6”
(
UZ
because hey
a e use eques ed and he numbe comes om hei o de o appea ance, hey a e
L1
,
L2
and
L3
espec i ely).
The iangula poin s a e also in hei co esponding b anches wi h ype
”UZ2”
and
”UZ4”
(
L4
and
L5
).
These labels a e needed o he nex s ep. Finally, as expec ed om his pa icula p oblem, he iangula
poin s seem o ha e he h ee alues o non-ze o pe iods o
µ=0.01215
, while he collinea poin s ha e
wo. All o i ells us he numbe o pe iodic o bi s ha eme ge om hem, ha is,
3
pe iodic o bi s om
L4
and L5 espec i ely, and 2 om L1,L2and L3.
We now con inue wi h he sc ip . Nex line impo s a py hon sc ip wi h a unc ion inside called
w i e
_
lag ange(), which can be seen in igu e (5.11). This unc ion p in s on he sc een in o ma ion ega ding
he pu ely imagina y eigen alues o he lib a ion poin s, i is in igu e (5.12). Al hough his las s ep is no
s ic ly necessa y o he nex one (calcula ion o pe iodic o bi s), i summa izes he in o ma ion ga he ed
abou he pe iodic o bi s ha will be compu ed on o wa d, as o example he long pe iod o bi s ha emana e
om L4and L5.
Figu e 5.9 Bi u ca ion diag am o he s a iona y solu ions in he CR3BP.
5.3 S a iona y solu ion analysis 41
Figu e 5.10 Sc een ou pu o he s a iona y solu ion analysis o he CR3BP.
Figu e 5.11 Py hon unc ion o inspec he s a iona y solu ions p ope ies.
Figu e 5.12 In o ma ion gi en by he w i e_lag ange() unc ion abou he lib a ion poin s on sc een.
48 Chap e 5. AUTO-based nume ical s udy o he CR3BP
Figu e 5.26 Simula ion o o bi LAB =10 o he L3 amily o se e al pe iods.
Figu e 5.27 S3 solu ion amily, gene al iew.
5.5 The Halo Families
Up un il now, all he o bi al amilies ha ha e been calcula ed a e he esul o con inuing he cycles ha
eme ge om he i e lib a ion poin s as has been explained. Bu in he p ocess AUTO has e ealed ha some
o he pa hs o con inua ion a e a ailable h ough he bi u ca ing o bi s. AUTO can de ec o he b anches by
moni o ing he ank o he sys em ma ix and compu es he bi u ca ion di ec ions, mo e in o ma ion on his
can be ound in [
18
]. Now we use hose b anching o bi s o accede o he Halo o bi al amilies, which a e he
subjec o s udy.
To achie e he b anch swi ching we need o eed he o bi o he un command as in he nex example o
he Halo o bi H1, ha which eme ges om he L1 amily:
H1 = un(L1(’BP1’), c=’c.H1’),
whe e, as in he case o he compu a ion o eme ging pe iodic o bi s, he equa ions ile isn’ necessa y (i is
implici in he solu ion o bi ha we a e eeding as inpu ), and he L1(’BP1’) e e s o he i s bi u ca ing o bi
o he
L
1 amily ha we ound, which was named a e
H
1, he Halo amily ha emana es om he Plana
Lyapuno amily o he L1 Lag ange poin . Fo he cons an s ile he e a e a couple o hings o commen .
Fi s is ha he cons an
ISW
has o be se o
−1
so ha AUTO au oma ically pe o ms he b anch swi ching,
and second ha he con inua ion s ep size
DS
is now sign sensi i e. This is because he Halo amilies ha e

5.5 The Halo Families 49
Figu e 5.28 Pe iod and Floque mul iplie s o he S3 amily.
Figu e 5.29 L4 solu ion amily, gene al iew.
Figu e 5.30 Pe iod and Floque mul iplie s o he L4 amily.
wo a ian s, he No he n and Sou he n case, a esul o he symme y in he zaxis in he equa ions o he
sys em. The es o he cons an s ha e he same ole as in he p e ious cases, so we now p oceed o show he
esul s o he Halo amilies.
50 Chap e 5. AUTO-based nume ical s udy o he CR3BP
Figu e 5.31 V1 (le ) and V2 ( igh ) solu ion amily, gene al iew.
Figu e 5.32 V3 (le ) and V4 ( igh ) solu ion amily, gene al iew.
Figu e 5.33 Pe iod and Floque mul iplie s o he V1 amily.
The esul s o he
H
1 amily a e in igu es (5.36) o (5.38). Figu e (5.36) shows how he
H
1 amily
eme ges om he L1, hen i de elops o a nea ec ilinea o bi a ound he moon o inally in e sec wi h he
C1 amily. Had he amily been con inued hen i would ha e esul ed in he sou he n a ian o his o bi al
5.6 Nea ec ilinea Halo o bi s 51
Figu e 5.34 Pe iod and Floque mul iplie s o he V3 amily.
Figu e 5.35 Pe iod and Floque mul iplie s o he V4 amily.
amily, whi he same p ope ies. The Floque mul iplie s in (5.37) only show ha his amily o o bi s is
eally uns able a he beginning, bu apidly goes o a nea s able beha iou . E en mo e, igu e (5.38) shows
how he e is an ac ual s e ch o he amily wi h comple e s abili y (in he CR3BP model). This o bi s will be
s udied in de ail in he nex subsec ion.
Fo he
H
2 amily esul s see igu es (5.39) o (5.41), whe e a simila beha iou o he Floque mul iplie s
can be seen a he end o he amily, mo e de ails in he nex subsec ion.
Finally, he
H
3 amily, hough o less p ac ical in e es , has been ep esen ed in igu es (5.42) o (5.43). A
close look a he Floque mul iplie s shows ha i is quasi-s able a he beginning (low
PAR(4)
) bu becomes
comple ely uns able la e on.
Ou o hese h ee amilies he
H
1 and
H
2 ha e been p oposed as sui able candida es o an inhabi ed
acili y in cis-luna space [
19
]. A na ow s e ch whe e hey a e called Nea Rec ilinea Halo O bi s o
NRHOs p esen s some in e es ing p ope ies, hese a e s udied in some de ail nex .
5.6 Nea ec ilinea Halo o bi s
As hei name indica es, hese subg oup o o bi s a e cha ac e ized by an elonga ed pa h as seen om Ea h,
bu we a e now going o o mally delimi he s e ch o o bi s in bo h he
H
1 and
H
2 amilies we a e e e ing
o. In sec ion (5.4.1) i was al eady commen ed ha one o he p ope ies o he Floque mul iplie s is ha
he e is always a pai equal o 1( he i ial pai ), now ano he p ope y is equi ed o his classi ica ion. In
his con ex , he mul iplie s always appea in ecip ocal pai s (µi,1/µi), so i is use ul o de ine he s abili y
index
νi=1
2(µi+1/µi)
o he wo non- i ial pai s. Simila ly han wi h he Floque mul iplie s, his me ic
52 Chap e 5. AUTO-based nume ical s udy o he CR3BP
Figu e 5.36 H1 solu ion amily.
Figu e 5.37 Pe iod and Floque mul iplie s o he H1 amily.
indica es s abili y when bo h o hem a e less han one in modulus. Fo an uns able o bi a la ge alue o he
s abili y index co esponds o a as e depa u e om he nominal pa h.
In gene al, he NRHOs a e conside ed as he subse o Halo o bi s possessing s abili y indices wi hin some
small bound su ounding
±1
and simila in magni ude. Fo his wo k, he s abili y swi ches o his me ic
will be use o delimi he bounda ies o he NRHOs. Ac oss he
H
1 and
H
2 amilies and nea he Moon
egion a numbe o s abili y swi ches occu om linea ly s able o uns able when con inuing he amilies
om he ini ial bi u ca ions. Un o una ely, his wasn’ easily seen wi h he
PAR(4) s PAR(3)
plo s in las
sec ion, bu he me ic
νi
in ela ion wi h he closes dis ance o he Moon o each o bi will appea as a mo e
sui able way o disce n he e olu ion o he s abili y p ope ies. Tha is wha is ep esen ed in igu e (5.44)
o bo h he H1 and H2 amilies in he egion nea he Moon.
Then, o he
H
1 amily he o bi s a e conside ed o he NRHO subse be ween he i s and o h s abili y
swi ch ( om igh o le ), ma ked wi h a ows in (5.44). This egion s a s wi h he o bi s o a ound
17400Km
o closes dis ance and ends a bi a e he collision o bi (a e his he o bi s a e o no eal p ac ical in e es ).
Fo he
H
2 amily he subse is conside ed be ween he i s and hi d in e sec ion, hus being he egion
be ween
15500Km
and
90Km
o closes dis ance. This de ini ion applies o bo h he no he n and sou he n
membe s o he amilies. This plo also highligh s he s able egions in g een when bo h he indices a e below
1 in modulus.
5.6 Nea ec ilinea Halo o bi s 53
Figu e 5.38 De ail o he Floque mul iplie s in he H1 amily.
Figu e 5.39 H2 solu ion amily.
Fo his subse o o bi s he pe iod and ene gy alues a e ep esen ed in igu es (5.45) and (5.46), wi h he
pe iod now in days and he ene gy is non-dimensional (5.3).
The subse o o bi s unde s udy is now se , and he nex s ep is o s udy i s p ope ies ega ding he s abili y.
Figu e (5.44) shows us ha he s abili y inside he NRHO subse is signi ican ly be e han be o e i s a s. In
o de o es ima e he empo al scale o he dominan di e ging mo ion a new me ic is de i ed by conside ing
he ime cons an , ψ, as he numbe o e olu ions be o e depa u e:
ψ[ e ] = 1
|Re(ln(µmax))|
As i has been de ined he ime cons an is in ini e o a ma ginally s able o bi (
µmax =1
). The ime
cons an can be in e p e ed as an es ima ion o he numbe o e olu ions o an ini ial pe u ba ion o be
ampli y by a ac o o app oxima ely 3. Fo bo h he NRHOs amilies
ψ
is plo ed in igu e (5.47). This
igu e shows he p e iously indica ed possibili y o main aining a s a ion in NRHO mo ion o e long pe iods
o ime while consuming ew p opellan esou ces. In o de o be e show he beha iou o hese o bi s
simula ions o some hem a e in igu e (5.48). The
86 Km
o bi has been included as an example o s able
o bi e y close o he su ace, and he
290 Km
o bi , hough uns able, shows ha i is possible o hese
kind o o bi s o emain close o he e e ence o o e a e y long ime, as he simula ion shows e y li le
dispe sion o e he se en pe iods.

54 Chap e 5. AUTO-based nume ical s udy o he CR3BP
Figu e 5.40 Pe iod and Floque mul iplie s o he H2 amily.
Figu e 5.41 De ail o he Floque mul iplie s o he H2 amily.
Figu e 5.42 H3 solu ion amily.
5.6 Nea ec ilinea Halo o bi s 55
Figu e 5.43 Pe iod and Floque mul iplie s o he H3 amily.
Figu e 5.44 Region o s abili y swi ches in he icini y o he Moon o he H1 and H2 amilies.
Figu e 5.45 Pe iod in he NRHO subse .
Ano he impo an ac o when designing an o bi o a space s a ion ha is going o be inhabi ed a e
he he mal and powe es ic ions ha come wi h he eclipsing o he sun. In his case he a o emen ioned
s a ion can be eclipsed by bo h he Moon and he Ea h. To s udy he i s ype o eclipsing i is necessa y
o compa e he pe iod o he NRHO wi h he synodic pe iod o he Moon (
TM−S≈29.5306
days), ha is
ep esen ed in igu e (5.49) o he
y: 1
and
y: 2
esonances in he
H
2 NRHO amily. The
y: 1
esonance
means ha he o bi pe o ms
y
e olu ions o e one synodic pe iod, and
y: 2
means
y
e olu ions o e wo
56 Chap e 5. AUTO-based nume ical s udy o he CR3BP
Figu e 5.46 Ene gy in he NRHO subse .
Figu e 5.47 Time cons an in he NRHO subse .
synodic pe iod. As can be seen he e a e only wo
y: 1
esonances in he NRHO subse o he H2 amily
( he
4:1
and
3:1
esonances) and wo
y: 2
esonances ( he
9:2
and
7:2
), h ee o hese o bi s a e in igu e
(5.50). Fo his ep esen a ion a Moon cen ed Sun o ien ed ame o e e ence has been used, as i makes he
esonances ob ious. In his ame o e e ence he Sun is always in he same di ec ion, and hus he o bi s
wi h esonances o he
y: 1
ype allow o o ien a ions such ha he Moon’s shadow ne e cas s o e i , he
o bi o ien a ion is se by choosing he inse ion da e co ec ly. The Ea h’s shadow issue isn’ s udied he e,
bu a possible s a egy is o y se he spacec a apoapsis du ing each ull Moon phase, hus a oiding he
Ea h’s shadow. On he o he hand, he
9:2
esonance di icul ly a oids he luna eclipse because he shadow
doesn’ always cleanly pass h ough he gaps in he o bi , so epoch selec ion is mo e challenging in his case.
Finally, he 3:1 esonance has he added ad an age o being s able in he CR3BP con ex , making i a e y
in e es ing op ion o a space s a ion o bi (i no o he mission es ic ions a e conside ed).
I is also impo an o s udy he way in and ou om hese o bi s, and o ha he compu a ion o mani olds
wi h AUTO will be shown in he nex sec ion.
5.7 Compu a ion o mani olds wi h AUTO 57
Figu e 5.48 Simula ion o a ious H2 NRHOs o e 7 pe iods (sligh ly di e en in each o hem).
5.7 Compu a ion o mani olds wi h AUTO
Though he s a egy o compu e uns able mani olds has al eady been explained in sec ion (3.5), he de ails
o do i in AUTO a e now commen ed in some dep h. The gene al idea was o compu e he eigen unc ion
associa ed o he only uns able mul iplie . Once calcula ed, he mani old was app oxima ed as he pe u bed
s a e
(0) = u(0)+εw(0)
. This s a e was con inued using ime in eg a ion un il a ce ain condi ion was me
(some coo dina e in he inal s a e eaches a desi ed alue), and he esul ing o bi was used as he ini ial
i e a ion o con inue he whole mani old. The p ocess is now exempli ied wi h an o bi in he H1 amily.
The p ocess s a s wi h he al eady calcula ed amily o pe iodic o bi s
H
1. Fi s s ep is o p epa e he
ini ial solu ion o he eigen unc ion compu a ion, and in o de o do ha he py hon sc ip in igu e (5.51)
has been used. The code ecei es as inpu he solu ion ile o he
H
1 amily (p e iously sa ed), he label o
he desi ed o bi and he alue o
ε
in he i s app oxima ion o he mani old ha will be used (s ep a iable
in he code). I no Floque mul iplie is in oduced hen he code uses he alue o
PAR(4)
, which in he
p esen case is di e en om ze o and posi i e. Fo his example he chosen o bi has
PAR(4) = 10.59
and
he closes dis ance o he Moon is 23600 Km.
64 Chap e 5. AUTO-based nume ical s udy o he CR3BP
Figu e 5.60 S able and uns able mani old o he 23600 Km o bi , H1 amily.
Ano he ele an aspec is he ime equi ed o access he o bi using his me hod. In igu e (5.61) we can
see he o bi ha s a s close o he Ea h om hose in he s able mani old o igu e (5.60), which is abou
80000 Km
om he su ace. The do s a e placed one day o a el apa om each o he , meaning ha
10
days pass be o e he spacec a eaches he op o he o bi o he i s ime ( his pass is s ill a om he o bi
i sel ), a a he slow me hod o each he o bi . A possible sho cu o his will be shown la e on, hough i
is e en wo se o he o bi s in he H2 amily, as i is shown now.
The p ocess used un il now has also been applied o an o bi in he
H
2 amily wi h he closes dis ance
o he Moon o abou
20000 Km
, which has
PAR(4)≈10
, posi i e and ela i ely big, jus as be o e o he

5.7 Compu a ion o mani olds wi h AUTO 65
Figu e 5.61 O bi in he s able mani old o he 23600 Km o bi wi h he days ep esen ed as do s .
H
1 amily. The esul is in igu e (5.62), whe e i is ob ious ha now he dynamic o bo h he mani olds is
qui e di e en om he p e ious cases. Now, he mani olds g ow om behind he o bi (as seen om he
Ea h) and u n a ound un il hey ace he
x=0
plane e y a om he Ea h’s su ace ( he closes o bi ends
a hes han he Moon i sel ).
The nex case is he i s example o o bi wi h nega i e
PAR(4)
, i is he
6000 Km
o bi o he
H
2 amily
( his is inside he NRHO amily as de ined be o e). The only di e ence is he sign o he Floque mul iplie ,
bu his has a no iceable e ec on he shape o he mani old’s su ace. The esul o his o bi is in igu e
(5.63), whe e i is e iden ha now he su ace is wis ed a he han o ien able.
5.7.2 An al e na i e pa h o he NRHOs
The al e na i e is explained in [
20
], and i consis s in inding he ajec o y ha s a s a he pa king o bi
a ound he Ea h and ends in he inse ion poin in he s able mani old ha ul ima ely leads o he pe iodic
o bi . In o de o do ha one possible me hod is o ix he inse ion poin in he mani old and sea ch o he
impulse ha , by backwa ds ime in eg a ion, makes he ajec o y s a a he desi ed o bi (de ined by i s
al i ude, bu wi h he inclina ion ee). In pa icula , his wo k es ablish he inse ion poin in close p oximi y
o he Moon so ha he he spacec a akes ad an age o he g a i y pull.
66 Chap e 5. AUTO-based nume ical s udy o he CR3BP
Figu e 5.62 S able and uns able mani old o he 20000 Km o bi , H2 amily.
5.7 Compu a ion o mani olds wi h AUTO 67
Figu e 5.63 Mani olds in he 6000 Km o bi o he H2 amily.
6 Conclusions and u u e wo k
The o iginal goal o showing how AUTO wo ks and applying i in he con ex o s udying he pe iodic
mo ion in he CR3BP has been accomplished. The main impo an concep s abou non-linea dynamics and
bi u ca ions ha e been p esen ed. This has been complemen ed whi a b ie o e iew o he way con inua ion
is nume ically pe o med inside he AUTO en i onmen o he p oblem ypes a hand, which has se he
g oundwo k o he nex chap e , he guide on he AUTO usage o he con ex o his wo k. AUTO has hen
been used as a ool o s udying he Ci cula es ic ed h ee body p oblem. This way, i has been ela i ely
easy o s udy he pe iodic mo ions ha eme ge om he Lag ange poin s which has ul ima ely lead o he
s udy o he Halo o bi s and i s p ope ies. Inside o his g oup o o bi s he Nea ec ilinea halo o bi s ha e
been s udied o he al eady p oposed DSG p ojec in some o he ele an aspec s o his mission (s abili y,
depa u e imes, synodic esonances, eclipsing...). In ela ion o he las , he mani olds associa ed o hese
o bi s ha e been calcula ed using AUTO as well, wi h some commen a ies abou i s po en ial o being he
dynamical s uc u e used as access o he cycles.
The AUTO so wa e has p esen ed i sel as an ex emely powe ul and e sa ile so wa e ha can pe o m
o he wise complex calcula ions in a e y simple way om he use side, up o a poin ha he compu a ion o
pe iodic o bi s can be done wi h a couple o Py hon code lines once he p oblem has been se up p ope ly.
This has allowed o a e y comple e s udy o he dynamical p ope ies o he cycles ha emana e om he
equilib ia in he CR3BP, as well as a mo e in dep h s udy o he Halo o bi s ha a e suppose o be he u u e
place o he Deep Space Ga eway, all o i wi h de ailed desc ip ions in how he compu a ions ha e been
pe o med. On he o he hand, his s udy has he d awback o being a simpli ica ion o he eal dynamical
en i onmen ha he DSG will ace, which includes he nex added di icul ies: The mo ion o he Moon
a ound he Ea h has di e en om ze o, albei small, eccen ici y, which changes he p oblem ype om
au onomous (easily s udied wi h AUTO) o non-au onomous. Al hough i is echnically possible o s udy
ime dependen p oblems wi h AUTO i adds di icul y in he sense ha i would be necessa y o include an
ex a equa ion o he ime (now a s a e a iable), and he s udy o pe iodic mo ion has o be explici ly se as
a BVP o he ype
IPS =4
. I could hen be aced as a con inua ion p oblem wi h he o bi o
e=0
as ini ial
i e a ion. The second di icul y o be add essed is ha o he o he pe u ba ions ha a e po en ially c i ical in
long e m o bi s, such as he Sun’s o e en Jupi e ’s. This could be s udied sepa a ely whi highe ideli y
models on he o bi s ha ha e al eady been compu ed o see he consequences o his simpli ica ion.
Aside om he s udy ha has been conduc ed he e i could be in e es ing o go deepe in he unde s anding
on how o access he NRHOs and speci ically how o pe o m endez ous wi h a spacec a al eady in i .
69

Lis o Figu es
1.1 Gene al equila e al iangle solu ion. F om [2] 2
1.2 Gene al in a ian collinea solu ion. F om [2] 2
1.3 Equila e al iangle solu ion wi h ci cula o bi s. F om [2] 2
1.4 Collinea solu ions wi h ci cula o bi s. F om [2] 3
2.1 This image om [9] shows some o he mos impo an ea u es o a phase po ai : ixed poin s (A,
B, C), cycles (D) and he di e en ypes o beha iou nea ixed poin s 8
2.2 This image om [8] shows an in a ian wo-dimensional o us T2o a con inuous- ime dynamical
sys em in R38
2.3 (a) Lyapuno s abili y e sus (b) asymp o ic s abili y. F om [8] 8
2.4 The Poinca é map associa ed wi h a cycle. F om [8] 10
2.5 Hop bi u ca ion ep esen a ion 12
3.1 A solu ion b anch wi h wo olds. F om [12] 15
3.2 In e p e a ion o pa ame e con inua ion. F om [12] 15
3.3 G aphical in e p e a ion o pseudo-a cleng h me hod. F om [12] 16
3.4 He e a e shown he colloca ion poin s and he ’ex ended-mesh poin s’ o he case m=3. Also, wo
o he local Lag ange basis polynomials a e shown. F om [12] 17
3.5 S uc u e o he sys em o he case o n=2di e en ial equa ions wi h numbe o mesh in e als
N=3, numbe o colloca ion poin s pe mesh in e al m=3, he numbe o bounda y condi ions
nb=2, and he numbe o in eg al cons ain s nq=1, being he las ow he pseudo-a chleng h
equa ion. F om [12] 18
3.6 G aphical in e p e a ion o he Poinca é phase condi ion. F om [12] 19
3.7 Bi u ca ion diag am in αwi h he e ical b anch o pe iodic solu ions. F om [12] 20
4.1 De aul alues o he AUTO cons an s ile 25
4.2 Resul o uning demo p og am wi h cons ans=’ab.1’. Image om [16] 28
4.3 Poin ypes in a bi u ca ion diag am s uc u e. Image om [16] 28
4.4 FUNC sub ou ine in he abc demo p oblem. Image om [16] 30
4.5 STPNT sub ou ine in he abc demo p oblem. Image om [16] 30
4.6 Con en o he cons an s ile o he i s un o he p oblem. Image om [16] 31
4.7 Resul o unning s a iona y solu ion amily sea ch in he A→B→Cp oblem 31
4.8 L2-NORM o he s a iona y solu ion amily in A→B→Cp oblem 31
4.9 S a iona y solu ion amily o A→B→Cp oblem in he phase space 32
4.10 Con en o he cons an s ile o he second o las un o he p oblem. Image om [16] 32
4.11 L2-NORM o he pe iodic solu ion amilies in A→B→Cp oblem 33
4.12 Some o he pe iodic solu ions in A→B→Cp oblem 33
4.13 Pe iod o he pe iodic solu ion amilies in A→B→Cp oblem 34
4.14 De ail o he HB1pe iodic solu ion amily as i eme ges om i s Hop bi u ca ion. The pa ame e
p1changes om 0.204 (in he HB) o 0.207 34
5.1 CR3BP scheme (a) and nomencla u e o he i e lib a ion poin s (b). Image om [15] 35
71
72 Lis o Figu es
5.2 FUNC sub ou ine in he CR3BP 36
5.3 ST PNT sub ou ine in he CR3BP 37
5.4 PVLS sub ou ine in he CR3BP, pa 1 37
5.5 PVLS sub ou ine in he CR3BP, pa 2 38
5.6 Desc ip ion o GETP unc ion ex ac ed om cusp demo p oblem 38
5.7 Cons an s ile used o he analysis o s a iona y solu ions in he CR3BP 39
5.8 Sc ip ha compu es he lib a ion poin s o he µin he Ea h-Moon sys em, as well as he numbe
o pe iodic o bi s ha emana e om hem ho ough he pe iods 39
5.9 Bi u ca ion diag am o he s a iona y solu ions in he CR3BP 40
5.10 Sc een ou pu o he s a iona y solu ion analysis o he CR3BP 41
5.11 Py hon unc ion o inspec he s a iona y solu ions p ope ies 41
5.12 In o ma ion gi en by he w i e_lag ange() unc ion abou he lib a ion poin s on sc een 41
5.13 Cons an s ile o he compu a ion o he L1 amily 42
5.14 Sc een ou pu o he L1 amily 43
5.15 L1 solu ion amily, gene al iew 44
5.16 L1 solu ion amily, de ailed iews 44
5.17 Pe iod and bigges Floque mul iplie o he L1 amily 44
5.18 Floque mul iplie s 3,4,5and 6o he L1 amily 45
5.19 De ail o Floque mul iplie s 5and 6o he L1 amily 45
5.20 Simula ion o o bi s LAB =1(le ) and LAB =6( igh ) o he L1 amily 45
5.21 Simula ion o o bi wi h lowes M6o he L1 amily 46
5.22 L2 solu ion amily, gene al iew 46
5.23 Pe iod and Floque mul iplie s o he L2 amily 47
5.24 L3 solu ion amily, gene al iew 47
5.25 Pe iod and Floque mul iplie s o he L3 amily 47
5.26 Simula ion o o bi LAB =10 o he L3 amily o se e al pe iods 48
5.27 S3 solu ion amily, gene al iew 48
5.28 Pe iod and Floque mul iplie s o he S3 amily 49
5.29 L4 solu ion amily, gene al iew 49
5.30 Pe iod and Floque mul iplie s o he L4 amily 49
5.31 V1 (le ) and V2 ( igh ) solu ion amily, gene al iew 50
5.32 V3 (le ) and V4 ( igh ) solu ion amily, gene al iew 50
5.33 Pe iod and Floque mul iplie s o he V1 amily 50
5.34 Pe iod and Floque mul iplie s o he V3 amily 51
5.35 Pe iod and Floque mul iplie s o he V4 amily 51
5.36 H1 solu ion amily 52
5.37 Pe iod and Floque mul iplie s o he H1 amily 52
5.38 De ail o he Floque mul iplie s in he H1 amily 53
5.39 H2 solu ion amily 53
5.40 Pe iod and Floque mul iplie s o he H2 amily 54
5.41 De ail o he Floque mul iplie s o he H2 amily 54
5.42 H3 solu ion amily 54
5.43 Pe iod and Floque mul iplie s o he H3 amily 55
5.44 Region o s abili y swi ches in he icini y o he Moon o he H1 and H2 amilies 55
5.45 Pe iod in he NRHO subse 55
5.46 Ene gy in he NRHO subse 56
5.47 Time cons an in he NRHO subse 56
5.48 Simula ion o a ious H2 NRHOs o e 7 pe iods (sligh ly di e en in each o hem) 57
5.49 y: 1 and y: 2 esonances along he NRHOs in he H2 amily 58
5.50 Some esonan H2 NRHOs in a Moon-Sun o a ing ame o e e ence 58
5.51 Py hon code o ex ac he in o ma ion o he pe iodic o bi and p epa e i o he eigen unc ion calcula ion 58
5.52 Added lines o he BVP in he eigen unc ion compu a ion 59
5.53 Bounda y and in eg al condi ions o he BVP in he eigen unc ion calcula ion 60
5.54 Py hon code o compu e he linea app oxima ion o he uns able mani old 60
5.55 Bounda y condi ions in he ime in eg a ion con inua ion p oblem and mani old compu a ion p oblem 61
5.56 Uns able mani old compu ed o an o bi in he H1 o bi al amily 61
5.57 Ini ial o bi in he mani old o he 23600 Km H1 o bi 62
Lis o Figu es 73
5.58 Uns able mani old compu ed o a quasi-s able o bi in he H1 o bi al amily 62
5.59 Uns able mani old o he 23600 Km o bi , H1 amily, wi h he ela ion be ween he en ance poin
o he mani old and he des ina ion in he x=0plane 63
5.60 S able and uns able mani old o he 23600 Km o bi , H1 amily 64
5.61 O bi in he s able mani old o he 23600 Km o bi wi h he days ep esen ed as do s 65
5.62 S able and uns able mani old o he 20000 Km o bi , H2 amily 66
5.63 Mani olds in he 6000 Km o bi o he H2 amily 67