Homoclinic phenomena in conservative systems

Depa amen de Ma em`a ica Aplicada I
Uni e si a Poli `ecnica de Ca alunya
Homoclinic phenomena in conse a i e
sys ems
Ma ina Gonchenko
Ad iso s: Amadeu Delshams and Pe e Gu i´e ez
P og ama de Doc o a en Ma em`a ica Aplicada
Tesi p esen ada pe aspi a al ´ı ol de Doc o a
pe la Uni e si a Poli `ecnica de Ca alunya
Ba celona, Feb e de 2013
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iii
To my amily
i
Acknowledgemen s
Du ing he yea s I ha e been in Ba celona and wo ked in his PhD hesis I ha e been
lucky o mee many in e es ing people ha ha e helped me and gi en an un o ge able
expe ience. I would like o exp ess my since e g a i ude o all o hem.
Fi s wo ds o hanks a e, o cause, o my supe iso s Amadeu Delshams and Pe e
Gu i´e ez. I hank you bo h o you pa ience, guidance and all he ime we ha e spen
discussing he p oblems o he hesis and explaining e e y li le hing. Wi hou you
suppo his hesis would no ha e been w i en.
I am indeb ed o he depa men o Ma em`a ica Aplicada I and, especially, he
Dynamical Sys ems G oup o he Uni e si a Poli `ecnica de Ca alunya o he kind
hospi ali y. I exp ess my e y since e g a i ude o p o esso s Te e M. Sea a, Jo di
Villanue a, Tom´as L´aza o, Juan Ram´on Pacha, Me c`e Oll´e, Toni Sus´ın, Pau Ma ´ın,
Ra ael Ram´ı ez Ros, Cha a Pan azi, Pau Rold´an, Joaquim Puig, Yu i Fedo o , Ma
Gonz´alez, Ma a Casanellas, F ancesc Planas, Josep Masdemon , Be na Plans and
o he s o you help and he ui ul con e sa ions we ha e had. I am e y g a e ul
o my o icema es Imma Baldom´a, Jes´us Fe n´andez, Ma a Pe˜na, Abd´o Roig o you
suppo and he cons an willingness o help. I am hank ul o he o me and ac ual
PhD s uden s, including Alejand o Luque, Gemma Hugue , Ma cel Gua dia, Ma ´ı
Lahoz, Ab aham de la Rosa, Jos´e Vicen e Mand´e, Isabel Be na, O iol Cas ej´on, Anna
Tama i , Ad i`a Simon, V´ıc o Gonz´alez, o c ea ing a pleasan academic and social
en i onmen in he o ice and/o in he co ee oom o he depa men . I am also
g a e ul o Hilda Rod´on, Es e Pineda, Maika S´anchez, Gemma Bald ´ıs, Rosa Ma ia
Cue as as well as he sec e a ies o he Facul a de Ma em`a iques i Es ad´ıs ica Raquel
Capa ´os and Ca me Capde ila o all he adminis a i e and echnical suppo du ing
my s ay.
I am e y g a e ul o he p o esso s o he Dynamical Sys ems g oup om he
Uni e si a de Ba celona, especially Ca les Sim´o, Joan Ca les Ta je , A u o Viei o,
`
Angel Jo ba, and o he s o all he in e es ing sugges ions and ui ul discussions.
I would like o hank my i s eache s who in oduced me o he wo ld o dynamical
sys ems om he Resea ch Ins i u e o Applied Ma hema ics and Cybe ne ics (Nizhny
No go od, Russia): Le Le man, Se gey Gonchenko, Oleg S enkin, Leonid Belyako ,
Vyachesla G ines, Albe Mo ozo , Mikhail Malkin. A special hank is o Leonid
Pa lo ich Shilniko , I will ne e o gi e you eaching.
I app ecia e Vassili Gel eich, Ra ael de la Lla e, Jean-Pie e Ma co, Ana oly Neish-

i
ad , Ra ael O ega, Yu i Su is, Dmi y T esche , Dmi y Tu ae o you in e es in
my esea ch. I also hank pos g adua e and pos doc o al ellows Rena o Callejo, Ro-
d igo T e i˜no, Ma in Himmel, Denis Volk, Dmi y Vo o niko , I an O syanniko o
sha ing you expe ience and you iendship.
Many hanks o my iends: Oleg, G isha, Tanya, Ka ia, And ey, Ma co, Ania,
Juan, Toni, Vuc, o making me laugh always, o an in ini e numbe o ennis and
beach oley ma ches, cycling, skiing and simply o being he e. I hank Oleg and his
amily o all he suppo .
Wi h all o my hea I would like o hank my amily: my mum Albina o guiding
me and explaining e e y de ail abou he Ma hema ics in he P ima y and Seconda y
Schools, my a he Se gey o in oducing me in he wo ld o homoclinic bi u ca ions
and inspi ing on w i ing he hesis, my b o he s Vladimi and Alexande o you help
and suppo , my niece Anya o jus smiling when you call me ia skype. I lo e you
all despi e he dis ance sepa a es us.
Finally, I wish o hank he Spanish Minis y o Educa ion o gi ing me his oppo -
uni y o come o Ba celona and unding my PhD s udies h ough he FPU schola ship
AP2005-4492. Also I acknowledge he Spanish MINECO-FEDER G an s MTM2009-
06973, MTM2012-31714 and he Ca alan G an 2009SGR859. I also acknowledge he
use o he UPC Applied Ma h clus e sys em Eixam o esea ch compu ing.
Con en s
Acknowledgemen s
In oduc ion 1
I Bi u ca ions o homoclinic angencies in a ea-p ese ing
maps 29
1 Bi u ca ions o quad a ic homoclinic angencies o wo-dimensional
symplec ic maps 31
1.1 S a emen o he p oblem and main esul s . . . . . . . . . . . . . . . . 31
1.2 Th ee classes o symplec ic maps wi h homoclinic angencies. . . . . . . 41
1.2.1 Maps o he i s and second classes. . . . . . . . . . . . . . . . 43
1.2.2 Maps o he hi d class . . . . . . . . . . . . . . . . . . . . . . 44
1.3 Gene al un oldings and Rescaling Lemma . . . . . . . . . . . . . . . . . 47
1.4 P oo s o Theo ems 1.1, 1.2 and 1.3 . . . . . . . . . . . . . . . . . . . . 50
1.5 In a ian s o homoclinic angencies in symplec ic wo-dimensional maps 51
2 Dynamics and bi u ca ions o non-o ien able a ea-p ese ing maps
wi h quad a ic homoclinic angencies 57
2.1 S a emen o he p oblem and p elimina y cons uc ions . . . . . . . . 57
2.1.1 Fini e-smoo h no mal o ms o non-o ien able saddle a ea-p ese ing
maps................................. 58
2.1.2 S ips, ho seshoes and e u n maps . . . . . . . . . . . . . . . . 60
2.2 Main esul s: on cascades o ellip ic pe iodic o bi s . . . . . . . . . . . 61
2.3 The escaling lemmas in he non-o ien able case . . . . . . . . . . . . . 66
2.4 P oo o he main esul s . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.4.1 On bi u ca ions o ixed poin s in he conse a i e H´enon maps . 71
2.4.2 P oo o Theo em 2.1 . . . . . . . . . . . . . . . . . . . . . . . 73
2.4.3 P oo o Theo ems 2.2 and 2.3 . . . . . . . . . . . . . . . . . . 75
3 Bi u ca ions o cubic homoclinic angencies in a ea-p ese ing maps 79
3.1 P eambles.................................. 79
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3.2 On bi u ca ions o pe iodic o bi s . . . . . . . . . . . . . . . . . . . . . 83
3.2.1 The desc ip ion o bi u ca ions o ixed poin s in he cubic H´enon
maps................................. 85
3.2.2 Bi u ca ion Theo em . . . . . . . . . . . . . . . . . . . . . . . . 89
4 Fini ely smoo h no mal o ms o saddle a ea-p ese ing maps 93
4.1 P eambles.................................. 93
4.2 Fini ely smoo h no mal o ms o symplec ic saddle maps: he p oo o
Lemmas1.1and1.2............................. 95
4.2.1 P oo o Lemma 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2.2 P oo o Lemma 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.3 Fini ely smoo h no mal o ms o non-o ien able a ea p ese ing saddle
maps .................................... 99
Appendix A On s uc u e o 1:4 esonances in conse a i e H´enon-like
maps 103
A.1 The esonance 1 : 4 in a ea-p ese ing maps . . . . . . . . . . . . . . . 104
A.2 Conse a i e gene alized H´enon maps . . . . . . . . . . . . . . . . . . . 106
A.3 Conse a i e cubic H´enon maps . . . . . . . . . . . . . . . . . . . . . . 107
II Exponen ially small spli ing o sepa a ices o whiske ed
o i in Hamil onian sys ems 111
5 Se up 113
5.1 A singula Hamil onian wi h n+ 1 deg ees o eedom . . . . . . . . . . 113
5.2 The Poinca ´e-Melniko me hod . . . . . . . . . . . . . . . . . . . . . . 116
6 Exponen ially small spli ing o sepa a ices o whiske ed o i wi h
quad a ic equencies 119
6.1 Quad a ic equencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.1.1 Con inued ac ions o quad a ic numbe s . . . . . . . . . . . . 119
6.1.2 A i hme ic p ope ies . . . . . . . . . . . . . . . . . . . . . . . . 120
6.2 Asymp o ices ima es............................ 122
6.2.1 Asymp o ic es ima es o he spli ing dis ance . . . . . . . . . . 125
6.2.2 Asymp o ic es ima es o he ans e sali y o he spli ing . . . 125
6.2.3 Dominan ha monics o he Melniko po en ial . . . . . . . . . . 131
6.2.4 Dominan ha monics o he spli ing po en ial . . . . . . . . . . 137
6.2.5 Nondegene a e c i ical poin s o L................. 139
6.2.6 P oo o Theo ems 6.1 and 6.2 . . . . . . . . . . . . . . . . . . . 143
6.3 Con inua ion o ans e se homoclinic o bi s in he case Ω2....... 144
7 Exponen ially small spli ing o sepa a ices o whiske ed o i wi h
ix
cubic equencies 155
7.1 Cubic equencies.............................. 156
7.1.1 The cubic golden numbe . . . . . . . . . . . . . . . . . . . . . 161
7.2 Asymp o ices ima es............................ 164
7.3 Dominan ha monics o he Melniko po en ial . . . . . . . . . . . . . . 173
7.4 Dominan ha monics o he spli ing po en ial . . . . . . . . . . . . . . 178
7.5 C i ical poin s o he spli ing po en ial . . . . . . . . . . . . . . . . . . 180
7.5.1 The case ∆ = de (S1, S2, S3)6=0. ................. 180
7.5.2 The case ∆ = 0, bu ∆1= de (S1, S2, S4)6=0. .......... 185
7.6 P oo o Theo ems 7.1 and 7.2 . . . . . . . . . . . . . . . . . . . . . . . 189
Appendix B The ixed poin heo ems 191
6INTRODUCTION
h ough his bounda y (µ= 0), he complica ed dynamics appea s immedia ely, “by
explosion” ( o ha eason such bi u ca ions we e called (homoclinic) Ω-explosion)2:
be o e he angency (a µ < 0) he sys em has a simple dynamics: Nµ={O}; he e is
only one (non ans e sal) homoclinic o bi N0={O∪Γ0}a he momen o he an-
gency (µ= 0); and in ini ely many Smale ho seshoes appea jus a e he homoclinic
angency spli s (a µ > 0) in o wo ans e sal homoclinic o bi s: Nµis non i ial. In
mo e de ail, his phenomenon was la e s udied in pape s o S. Newhouse and J. Palis
[NP76], J. Palis and F. Takens [PT85], L. Shilniko and O. S enkin [SS98] e c.
Theo em on cascade o pe iodic sinks. This heo em is one o he undamen al e-
sul s in homoclinic dynamics and plays qui e impo an ˆole in he heo y o dissipa i e
chaos. I s a es ha , in he amily µ, he e exis (nonin e sec ing) in e als o alues
o µaccumula ing o µ= 0 such ha he co esponding di eomo phism o he amily
has an asymp o ically s able pe iodic o bi (pe iodic sink). This esul was ex ended
o he mul idimensional case by S. Newhouse [New74] and S. Gonchenko [Gon83], and
gene al c i e ia o he exis ence o s able pe iodic o bi s nea a homoclinic angency
we e poin ed ou by S. Gonchenko, L. Shilniko and D. Tu ae in [GST93a, GST96a].
Theo y o moduli o opological and Ω-conjugacy o di eomo phisms wi h homoclinic
angencies. The au ho s explained he impo ance o homoclinic angencies o di e en
ypes o he global dynamics o sys ems. As we said be o e, sys ems wi h homoclinic
angencies o he i s ype can belong o he bounda y o Mo se-Smale sys ems. In
[GS73] i was shown ha sys ems wi h homoclinic angencies o he second ype can
belong o he bounda y o hype bolic sys ems. Also in [GS73] i was es ablished ha
di eomo phisms wi h homoclinic angencies o he hi d ype possess Ω-moduli, i.e.
con inuous in a ian s o opological conjugacy on he se o non-wande ing o bi s o
µ. The main Ω-modulus θ=−ln |λ|/ln |γ|was in oduced in [GS73], whe e i was
shown ha a ying θleads o bi u ca ions o pe iodic o bi s o µ. Fu he in es iga-
ions o his opic (see e.g. [Gon89, GS90, GST91, GST93a, GST96a, GST99, Kal00,
DN05, GST08]) ha e laid o he c ea ion o a e y in e es ing and ich heo y o ho-
moclinic bi u ca ions which p o ides a heo e ical basis o he dynamical chaos.
Simul aneously, S. Newhouse ob ained a se ies o undamen al esul s [New70,
New74, New79] ela ed o he heo y o homoclinic bi u ca ions in wo-dimensional
nonconse a i e di eomo phisms. He wan ed o see wha happens in a one-pa ame e
un olding, when a homoclinic angency spli s, and disco e ed wild hype bolic se s, i.e.
non i ial, ansi i e and uni o mly hype bolic se s whose he s able and uns able in-
a ian mani olds ha e an i emo able nondegene a e angency (in he sense ha al-
2The i s example o he Ω-explosion was gi en by Shilniko in he wo k [Shi69] whe e bi u ca ions
o a h ee-dimensional low wi h se e al homoclinic loops o a saddle-saddle equilib ium we e s udied.
Recall ha he saddle-saddle is an equilib ium wi h eigen alues λ1= 0, λ2<0, λ3>0 ha ing
also nonze o he i s Lyapuno alue l1( he simples example is gi en by sys em ˙x1=l1x2
1,˙x2=
λ2x2,˙x3=λ3x3).

BACKGROUND AND STATE OF THE ART 7
hough he gi en homoclinic angency is emo ed by a small pe u ba ion o he sys em,
one canno a oid he appea ance o new homoclinic angencies). I is impo an o no e
ha he wild hype bolic se s exis o di eomo phisms close (in he C2- opology) o
any di eomo phism wi h a homoclinic angency, [New79], and, hence, he e exis open
egions, he so-called Newhouse egions, whe e di eomo phisms wi h homoclinic an-
gencies a e dense. La e , he exis ence o Newhouse egions nea any sys em wi h a
homoclinic angency was p o ed in [GST93b] o he gene al mul idimensional case.
The dynamics in Newhouse egions o a ious kinds o sys ems was s udied in a
se ies o pape s by S. Gonchenko, L. Shilniko and D. Tu ae [GST93c, GST97, GST99,
GST07], who es ablished he impossibili y o p o iding a comple e s udy o homoclinic
bi u ca ions wi hin he amewo k o ini e pa ame e amilies.
These esul s we e ob ained o gene al sys ems. Howe e , some gene ici y condi-
ions exclude om conside a ions such e y impo an classes o sys ems as conse -
a i e, e e sible, Hamil onian ones e c. The s udy o such sys ems wi h addi ional
s uc u es is o g ea in e es and equi es o en special ools and me hods. Some qui e
impo an esul s on homoclinic bi u ca ions o such sys ems we e also ob ained. We
men ion a se ies o pape s [GG00, GG04, Gon02, GKM05, GOT12] ela ed o bi u ca-
ions o di eomo phisms wi h quad a ic homoclinic angencies in he case σ= 1, whe e
e y in e es ing homoclinic phenomena passing be ween he cases σ < 1 and σ > 1 we e
s udied; in [GMO06] bi u ca ions o h ee-dimensional di eomo phisms wi h quad a ic
homoclinic angencies o a saddle- ocus ixed poin wi h Jacobian equal 1 was s udied
and he bi h o Lo enz-like s ange a ac o s was p o ed (see also [GST09, GO10]
whe e analogous esul s we e ob ained).
Ra he in e es ing esul s we e ob ained ecen ly o wo-dimensional e e sible
maps wi h homoclinic and he e oclinic angencies. Thus, J. Lamb and O. S enkin
[LS04] p o ed he exis ence o Newhouse egions (in he class o e e sible maps) in
which maps possessing simul aneously in ini ely many asymp o ically s able (a ac -
ing), saddle, comple ely uns able ( epelling) and ellip ic pe iodic o bi s a e dense,
ex ending he esul s o [GST97]. They conside ed he case o e e sible and a p io i
nonconse a i e maps (i.e. maps ha ing wo symme ic saddle ixed poin s wi h he
Jacobian di e en o 1). Symme y b eaking bi u ca ions leading o he appea ance
o a ac ing and epelling pe iodic o bi s in e e sible maps ha ing a non ans e sal
he e oclinic cycle con aining wo saddle ixed poin s on he symme y line we e s ud-
ied in [DGL06]. This pape ga e a me hod o de ec ing elemen s o nonconse a i e
dynamics in e e sible sys ems.
Conce ning he conse a i e case, we men ion, i s , he pape o Newhouse [New77],
whe e he appea ance o 1-ellip ic pe iodic poin s3unde bi u ca ion o homoclinic
angency was p o ed o symplec ic mul idimensional maps. A ea-p ese ing maps
(APMs) wi h homoclinic angencies we e s udied by L. Mo a, N. Rome o [MR97] who
p o ed he exis ence o a cascade o gene ic ellip ic poin s. Also in he pape s o
3Tha is, poin s ha ing exac ly one pai o mul iplie s e±iϕ. No e ha he bi h o 2-ellip ic poin s
in ou -dimensional symplec ic maps wi h homoclinic angencies was es ablished in [GST98, GST04].
8INTRODUCTION
S.Gonchenko, L. Shilniko [GS97, GS00], condi ions o he coexis ence o in ini ely
many gene ic ellip ic poin s we e ound o APMs wi h non ans e sal he e oclinic
cycles and in [GS01, GS03] a phenomenon o global esonance was disco e ed when
an APM wi h a homoclinic angency had in ini ely many gene ic ellip ic poin s o all
successi e (su icien ly la ge) pe iods.
In his hesis (Pa I) we con inue hese in es iga ions o APMs wi h homoclinic
angencies and gi e, in pa icula , in a sense a comple e desc ip ion o bi u ca ions
o single- ound pe iodic o bi s ( ead, ixed poin s o i s e u n maps de ined nea a
homoclinic angency) including cons uc ion o he co esponding bi u ca ion diag ams.
Mo eo e , non-o ien able APMs wi h homoclinic angencies a e also conside ed.
The me hods o he s udy o he o bi beha io nea homoclinic and he e oclinic
angencies a e based, i s o all, on he cons uc ion o e u n maps. In a se ies o
pape s [TY86, BS89, GST93a, GG00, GS01, GGT02, GSS02, GST02, GS03, GG04]
i was shown ha he co esponding escaled i s e u n maps a e o he o m o
H´enon-like maps (s anda d H´enon maps, gene alized H´enon maps, cubic H´enon maps,
h ee-dimensional H´enon maps, e c).
Exponen ially small spli ing o sepa a ices in Hamil onian
sys ems
In gene al, i a Hamil onian sys em wi h an objec ha ing he coinciden s able and
uns able in a ian mani olds (sepa a ix) is pe u bed, he in a ian mani olds in e sec
a poin s o homoclinic o bi s wi hou coincidence. This phenomenon has go he name
o spli ing o sepa a ices and he p oblem o measu ing he spli ing has become classic
since he wo k by H. Poinca ´e [Poi90] whe e his phenomenon was disco e ed. Many
esea che s de o ed o inding es ima es o he spli ing in di e en se ings bo h o
lows and maps. The spli ing o sepa a ices can be measu ed by se e al quan i ies such
as: he maximal dis ance be ween he wo in a ian mani olds, he angle be ween he
in a ian mani olds a a homoclinic poin , he a ea o he lobe be ween wo consecu i e
homoclinic poin s, he homoclinic (Lazu kin) in a ian as well as he wid h o he
chao ic zone.
The mos popula ool o measu e he spli ing is he Poinca ´e-Melniko pe u -
ba i e me hod, in oduced by Poinca ´e in [Poi90] and edisco e ed 70 yea s la e by
Melniko and A nold [Mel63, A n64] (also called sho ly as Melniko me hod). The dis-
ance be ween he in a ian mani olds is gi en by a unc ion called spli ing unc ion.
This me hod p o ides o i a i s o de app oxima ion wi h espec o a pe u ba-
ion pa ame e gi en by an in eg al known as Melniko unc ion, whose simple ze os
gi e ise o ans e sal in e sec ions be ween he s able and uns able pe u bed man-
i olds. Fo n-dimensional whiske ed o i, i was es ablished [Eli94, DG00] ha he
spli ing unc ion and he Melniko unc ion, which a e de ined on n-dimensional o i,
a e he g adien s o scala unc ions: he spli ing po en ial and he Melniko po en-
ial, espec i ely. This means ha he ans e se homoclinic o bi s co espond o he
BACKGROUND AND STATE OF THE ART 9
nondegene a e c i ical poin s o he spli ing po en ial.
In he case o exponen ially small spli ing, he e o o he me hod may o e come
he main e m and an addi ional s udy is equi ed o ensu e ha he Poinca ´e-Melniko
app oxima ion domina es he e o e m.
The i s exponen ially small uppe bound was ob ained by Neish ad [Nei84] in one
and a hal deg ees o eedom Hamil onian sys ems. La e , simila es ima es we e ound
in [HMS88, Fon93, Fon95] o he apidly pe u bed pendulum. Also Fon ich and Sim´o
[FS90] ob ained uppe bounds o he spli ing in he case o a ea-p ese ing maps close
o iden i y. In he case o whiske ed o i wi h 2 o mo e equencies, se e al au ho s
ga e also exponen ially small uppe bounds [Sim94, Gal94, BCG97, BCF97, DGJS97].
In [DGS04] accu a e uppe bounds o he case o Diophan ine n-dimensional whiske ed
o i we e ob ained by in oducing low-box coo dina es.
In gene al, es ablishing lowe bounds is usually mo e di icul , bu some esul s ha e
been ob ained also by se e al me hods.
Fi s , he case o one-dimensional whiske ed o i (pe iodic o bi s) was conside ed
[Laz84, DS92, Gel97, T e97, DS97, DR98]. He e, V.F.Lazu kin [Laz84] in oduced new
ools s udying spli ing o sepa a ices in he Chi iko s anda d map. The in a ian
mani olds a e pa ame e ized analy ically in a complex s ip whose size is de ined by he
singula i ies o he unpe u bed homoclinic o bi . Lazu kin used low box coo dina es
a ound one o he mani olds and ob ained in he complex s ip he spli ing unc ion,
an analy ic pe iodic unc ion. Using he analy ic p ope ies, one ob ains, in he eal
domain, exponen ially small bounds o he spli ing. The same echnique was used o
jus i y he Poinca ´e-Melniko me hod in a Hamil onian wi h one and a hal deg ees o
eedom [DS92, DS97] and an a ea-p ese ing map [DR98].
When he dimension o he whiske ed o us is g ea e han 1, i u ns ou ha he
a i hme ic p ope ies o i s equencies play impo an ˆole and in luence on he exp es-
sion o he quasipe iodic spli ing unc ion in which he small di iso s a e p esen ed.
This was i s de ec ed by Sim´o [Sim94] and hen igo ously p o ed by Delshams e al.
[DGJS97] in he quasi-pe iodically o ced pendulum.
La e , se e al au ho s s udied he spli ing o sepa a ices o wo-dimensional
whiske ed o i in 3 deg ees o eedom Hamil onian sys ems. Fo ins ance, Sim´o and
Valls [SV01] s udied he A nold example (in oduced by A nold in [A n64] o illus a e
he ansi ion chain mechanism which is c ucial in he s udy o he A nold di usion) and
also conside ed he homoclinic bi u ca ions ha can occu . Lochak, Ma co and Sauzin
[Sau01, LMS03], Rudne and Wiggins [RW00] used a di e en echnique, namely he
pa ame iza ion o he whiske s by wo di e en solu ions o Hamil on-Jacobi equa ion,
o s udy a gene alized A nold model and p o ed he exponen ial smallness o he spli -
ing o some in e als o he pe u ba ion pa ame e ε. P onin and T esche [PT00]
ga e exponen ially small bounds o a slow- as sys em using ano he me hod called
con inuous a e aging.
In [DG03, DG04] Delshams and Gu i´e ez s udied a gene aliza ion o he A nold’s
example, a Hamil onian sys em wi h 3 deg ees o eedom ha ing a wo-dimensional
10 INTRODUCTION
whiske ed o us whose equency a io is he golden mean (√5−1)/2 o o he ew
quad a ic numbe . They applied he heo y o con inued ac ions o selec p ima y
esonances ela ed o he small di iso s ha appea in he dominan ha monics o
he Melniko unc ion. I was shown ha he dominan ha monics o he spli ing
unc ion co espond o he dominan ha monics o he Melniko unc ion, p o iding
he asymp o ic es ima es (and, hence, lowe bounds) o he spli ing. Wi h hese
es ima es hey p o ed ha in he case o he quad a ic golden equencies, he e exis
exac ly ou ans e se homoclinic o bi s o he whiske ed o us o all he su icien ly
small alues o he pe u ba ion pa ame e .
The asymp o ic es ima es we e done o wo-dimensional whiske ed o i wi h ew
quad a ic equencies [DG03, DG04], and one o he objec i es in his hesis is o
gene alize hese esul s o o he quad a ic numbe s in he wo-dimensional case and
also o he h ee-dimensional case. I is wo h men ioning ha he e is no s anda d
heo y o con inued ac ions o he case o h ee o mo e equency ec o s. This is
he eason o conside a pa icula case o cubic equency ec o , jus o be able o
p o ide some esul s on exponen ially small spli ing o sepa a ices o 3 equencies
o he i s ime.
No ice ha when he Poinca ´e-Melniko app oach canno be alida ed, o he ech-
niques can be applied o ge exponen ially small es ima es. Fo example, he pa ame iza-
ion o he in a ian mani olds by solu ions o he so-called inne equa ion, in oduced
by Lazu kin [Laz84], wi h he subsequen applica ion o he complex ma ching ech-
nique [Bal06, OSS03, MSS11, MSS11b], and “beyond all o de s” asymp o ic me hods
[Lom00]. Also in [T e97] an asymp o ic o mula o he spli ing was gi en in he case
o a “pendulum wi h a suspension poin ” using con inuous a e aging.
S uc u e and main esul s
This hesis is o ganized in o wo pa s acco ding o he opic conside ed, and each pa
is subdi ided in o a numbe o chap e s which con ain he main esul s and appen-
dices wi h some complemen a y ac s. Usually e e y chap e is de o ed o a di e en
p oblem.
Bi u ca ions o homoclinic angencies in a ea-p ese ing maps
In he i s pa we s udy a ea-p ese ing maps (whose Jacobian is ±1) wi h a ho-
moclinic angency o a saddle ixed poin (see, o example, Figu e 3). In o de o
know how ajec o ies beha e in a neighbo hood o a non ans e sal homoclinic o bi ,
we s udy hei bi u ca ions, i.e. we conside pa ame e dependen amilies o maps
close o he ini ial one (which possesses he homoclinic angency) and obse e how he
beha io o nea by ajec o ies changes quali a i ely as he maps app oach o o mo e
away om he ini ial map ( a ying he pa ame e s). Usually he ini ial map co e-
STRUCTURE AND MAIN RESULTS 11
Figu e 3: An example o a ea-p ese ing map ha ing a quad a ic homoclinic angency along o a
homoclinic o bi Γ0.
sponds o a bi u ca ion alue o he pa ame e s and di ides he amily in o sub amilies
wi h quali a i ely di e en phase po ai s. In pa icula , we wan o see wha happens
wi h he so-called single- ound pe iodic o bi s, i.e. pe iodic o bi s which en i ely lie in
a neighbo hood o he non ans e sal homoclinic o bi and pass close o i only once.
To his end, we cons uc i s e u n maps, o which we use ini ely-smoo h no mal
o ms o he saddle maps, con aining only esonan monomials in nonlinea i ies up o
some o de n≥3, and in oduce c oss-coo dina es (see de ails in Chap e 4). The ixed
poin s o he i s e u n maps co espond o single- ound pe iodic o bi s o he maps
unde conside a ion. Applying escaling me hods (see he Rescaling Lemmas in e e y
chap e ) we de i e he i s e u n maps o he H´enon-like maps whose bi u ca ions a e
well known. Thus, ansla ing he esul s ob ained o he ixed poin s o he e u n
maps o he pe iodic o bi s, we p o e he main esul s. We also s udy he phenomenon
o he coexis ence o in ini ely many single- ound pe iodic o bi s o di e en la ge pe-
iods (called global esonance) and p o e a wo pa ame e e sion o he heo em on
cascades o ellip ic pe iodic poin s.
Mo e p ecisely, we conside he ollowing p oblems:
Chap e 1. We conside wo-dimensional symplec ic maps, i.e. a ea-p ese ing maps
which a e also o ien a ion-p ese ing ( he Jacobian is equal o 1). The ini ial map
0has a saddle ixed poin Owi h mul iplie s λand λ−1and possesses a quad a ic
homoclinic angency Γ0. Le Hsbe a (codimension one) bi u ca ion su ace composed
o symplec ic C -maps close o 0and such ha e e y map o Hshas a non ans e sal
homoclinic o bi close o Γ0. We conside one pa ame e gene al un oldings µo sym-
plec ic maps, whe e µis he pa ame e o spli ing o he homoclinic angency, and we
equi e ha amily µis ans e se o Hsa µ= 0.
No e ha he ini ial map 0possesses also a homoclinic in a ian τ(in oduced
in (1.15)) ha is esponsible o he p esence o he chao ic dynamics. The poin is
ha he alue τ= 0 can be “bi u ca ional”, e en wi hou spli ing he ini ial angency

12 INTRODUCTION
[GS01]: i τ > 0, 0has in ini ely many Smale ho seshoes, while i τ < 0, hen dynamics
o 0is i ial: he se N0o o bi s en i ely lying in a small neighbo hood o Γ0con ains
only he saddle poin Oand he homoclinic o bi Γ0, i.e. N0=OSΓ0.
By Rescaling Lemma 1.4, p. 48, we deduce he i s e u n maps o he no mal
o ms which ake he o m o a conse a i e H´enon-like map (wi h a small cubic e m)
and es ablish he one pa ame e heo em on cascade o ellip ic pe iodic poin s (see
mo e de ails in Theo em 1.1, p. 37):
Theo em 0.1. Le 0be a symplec ic map wi h a homoclinic angency o a saddle poin
and µbe a one pa ame e gene al un olding as desc ibed abo e. Then he ollowing
s a emen s ake place:
1. In any segmen [−µ0, µ0]o alues µ, he e a e in ini ely many open in e als δk,
k=¯
k, ¯
k+ 1, . . . (¯
kis some in ege ), such ha δk→0as k→+∞and he map
µhas a single- ound ellip ic pe iodic o bi a µ∈δk;
2. A he bo de poin s µ=µ+
kand µ=µ−
ko δk, µhas a single- ound pa abolic
pe iodic o bi wi h double mul iplie s +1 and −1, espec i ely;
3. The ellip ic o bi is gene ic (KAM-s able) o µ∈δk, excep o exac ly wo alues
co esponding o he s ong esonances 1:3 and 1:4, i.e. when he mul iplie s a e
e±i2π/3and e±iπ/2;
4. When τ6= 0, he in e als δiand δjdo no in e sec o su icien ly la ge in ege s
i6=j.
No e ha analogous esul s ela ed o i ems 1, 2 and 3 o Theo em 0.1 we e p o ed in
[Bi 87], [BS89] and [MR97], bu he coexis ence o single- ound ellip ic pe iodic o bi s
o di e en pe iods ( he global esonance) was no conside ed. I em 4 o Theo em 0.1
shows ha , in gene al (τ6= 0), such ellip ic o bi s o di e en and la ge pe iods canno
coexis .
The case τ= 0 is excep ional and equi es a u he s udy wi hin he amewo k o
wo-pa ame e un oldings: µ,τ . I u ns ou ha he phenomenon o he global eso-
nance depends s ongly on he geome y o he ini ial homoclinic angency o 0. We
dis inguish wo cases: he case I wi h homoclinic angencies simila o Figu e 2(a),(c)
and he case II wi h homoclinic angencies as in Figu e 2(b),(d). Thus, we p o e a
new wo pa ame e e sion o he heo em on cascade o ellip ic pe iodic poin s (The-
o em 1.2, p. 38):
Theo em 0.2. Le 0be a symplec ic map wi h a homoclinic angency o a saddle poin
and µ,τ be a wo pa ame e gene al un olding as desc ibed abo e. Then he ollowing
s a emen s ake place:
1. In any neighbo hood o he o igin in he (τ, µ)-plane, he e a e in ini ely many
open domains ∆k, o ks a ing wi h some in ege ¯
k, such ha he map µ,τ has
a single- ound pe iodic ellip ic o bi in ∆k;
STRUCTURE AND MAIN RESULTS 13
2. The domains ∆kaccumula e o he axis µ= 0 as k→ ∞;
3. The bounda ies o ∆ka e wo cu es L+
kand L−
kwhe e he map µ,τ has a
pa abolic single- ound pe iodic o bi wi h double mul iplie s ei he +1 and −1,
espec i ely;
4. The ellip ic o bi is gene ic (KAM-s able) o all alues o (τ, µ)∈∆k, excep o
hose which belong o cu es L2π/3
kand Lπ/2
kwhen esonances 1:3 and 1:4 occu ,
espec i ely;
5. In he case I, he domains ∆iand ∆jdo no in e sec o any su icien ly la ge
and di e en in ege s iand j;
6. In he case II, he domains ∆iand ∆ja e necessa ily c ossed and hey in e sec
he axis µ= 0; Mo eo e , all domains ∆kwi h su icien ly la ge kcon ain he
o igin (τ= 0, µ = 0), p o ided some condi ion (1.16) is sa is ied.
See Figu e 4 o an illus a ion o Theo em 0.2 whe e he plana domains ∆kin he
cases I and II a e ep esen ed.
Figu e 4: Domains ∆ko Theo em 0.2 in he cases I and II
In case II, i ollows om i em 6 o Theo em 0.2 ha a τ= 0 all domains ∆k, o
ks a ing wi h some in ege ¯
k, in e sec and, mo eo e , unde ce ain condi ions (see
Co olla y 1.1) all he domains con ain µ= 0 – his means ha he map 0has in ini ely
many coexis ing gene ic ellip ic pe iodic poin s o all successi e pe iods k=¯
k, ¯
k+1, . . .
( he global esonance).
In he nex heo em we desc ibe he cha ac e o bi u ca ions when µ a ies inside
he in e als δko Theo em 0.1 and ind he condi ions unde which he bi u ca ions
h ough he s ong esonances 1:3 and 1:4 a e non-degene a e (Theo em 1.3, p. 40).
14 INTRODUCTION
Theo em 0.3. The bi u ca ions o ixed poin s in he i s e u n map o µ ollow he
same scena io as he one obse ed in he conse a i e gene alized H´enon map
¯x=y, ¯y=M−x−y2+νky3,(1)
whe e M∼λ−2k(µ−αk)and αk∼λk,νk∼λka e small coe icien s. Fo his map
he esonance 1:3is non-degene a e o all alues o νk, while he esonance 1:4is
non-degene a e i νk6= 0
The eade is e e ed o equa ions (1.18) o see he exac o mulae o Mand νk
as well as o Figu e 1.7 o Chap e 1 o see he co esponding bi u ca ions o he map
(1).
In his chap e we also p o ide a classi ica ion o quad a ic homoclinic angencies
in he symplec ic case (see Sec ion 1.2).
Chap e 2. We conside 0an a ea-p ese ing map ha does no p ese e o ien a ion
( he Jacobian is −1). I has a saddle ixed poin Owi h mul iplie s 0 <|λ|<1<|γ|,
|λγ|= 1. The s able and uns able in a ian mani olds o Oha e a homoclinic angency
along a homoclinic o bi Γ0. We di ide such APMs 0in o 2 g oups:
• he globally non-o ien able maps wi h an o ien able saddle ( he saddle alue is
λγ = 1) on a non-o ien able mani old (M¨obius s ip, Klein bo le, e c), see an
example o such a map in Figu e5;
• he locally non-o ien able maps wi h non-o ien able saddles ( he saddle alue is
λγ =−1).
Figu e 5: An example o non-o ien able a ea-p ese ing map (on a M¨obius s ip) wi h a quad a ic
homoclinic angency along a homoclinic o bi Γ0.
We also conside one and wo pa ame e amilies µ, µ,α and µ,ˆα, whe e µis s ill he
pa ame e o spli ing o he homoclinic angency and αand ˆα(in oduced in (2.6))
STRUCTURE AND MAIN RESULTS 15
a e analogs o τin he symplec ic case, ha is homoclinic in a ian s esponsible o
he p esence o he chao ic dynamics in 0.
I u ns ou ha in he globally non-o ien able case he i s e u n maps do no
ha e ellip ic ixed poin s, bu a pe iod wo ellip ic poin appea s which co esponds o
a double- ound pe iodic o bi ; whe eas, in he locally non-o ien able maps, he e exis
in e als o he pa ame e µwhe e he i s e u n maps ha e ellip ic ixed poin s and
o he in e als whe e he i s e u n maps ha e pe iod 2 poin s. Thus, we es ablish
he exis ence o cascades o ellip ic poin s (see de ails in Theo em 2.1, p. 62):
Theo em 0.4. Le 0be a non-o ien able APM and µbe a one pa ame e amily o
close o 0APMs as desc ibed abo e. Fo any in e al (−µ0, µ0), he e exis s such a
posi i e in ege ¯
ksuch ha he ollowing holds:
1. (a). In he globally non-o ien able case, he maps µha e no single- ound ellip ic
pe iodic o bi s, while he e exis in ini ely many in e als e2
k,k=¯
k, ¯
k+ 1, . . . ,
whe e µhas a double- ound ellip ic o bi .
(b) In he locally non-o ien able case, he e exis in ini ely many al e na ing in-
e als e2mand e2
2m+1 such ha he map µhas a single- ound ellip ic pe iodic
o bi a µ∈e2mand has a double- ound ellip ic pe iodic o bi a µ∈e2
2m+1.
2. The in e als ekas well as e2
kaccumula e o µ= 0 as k→ ∞ and do no in e sec
o su icien ly la ge and di e en in ege ki α6= 0 and ˆα6= 0.
3. Any in e al ekhas bo de poin s µ=µ+
kand µ=µ−
kwhe e he map µhas a
single- ound pe iodic o bi wi h double mul iplie +1 and wi h double mul iplie
−1, espec i ely. Any in e al e2
khas bo de poin s µ=µ2+
kand µ=µ2−
kwhe e
he map µhas a single- ound pe iodic o bi wi h mul iplie s +1 and −1a µ=
µ2+
kand a double- ound pe iodic o bi wi h double mul iplie −1a µ=µ2−
k.
4. The ellip ic o bi is gene ic (KAM-s able) in ekand e2
k, excep o s ong eso-
nances 1:3 and 1:4.
We also conside he ques ion on he coexis ence o ellip ic pe iodic poin s by means
o wo pa ame e amilies µ,α and µ,ˆα.
Theo em 0.5. Fo wo pa ame e amilies µ,α and µ,ˆα he e exis in ini ely many
open domains, E2
kin he globally non-o ien able case and domains E2mand E2
2m+1 in
he locally non-o ien able case, such ha
1. The map µ,α and µ,ˆαha e a single- ound pe iodic ellip ic o bi in Ek, and ha e
a double- ound ellip ic pe iodic o bi in E2
k;
2. The domains Ekand E2
kaccumula e o he axis µ= 0 as k→ ∞;
22 INTRODUCTION
Theo em 0.8 (T ans e sali y o he spli ing).Unde he hypo heses o Theo em 0.7,
one has:
• he Melniko unc ion M(θ)has exac ly 4 ze os θ∗, all simple, o all εexcep o
some small neighbo hood o some geome ic sequences o ε(gi en in (6.13) and
(6.21)).
•The minimal eigen alue o ∂θM(θ∗)sa is ies
m∗∼µε1/4exp −C0h2(ε)
ε1/4
whe e h2(ε)is a posi i e pe iodic in ln ε unc ion.
In Figu e 10 he e is a schema ic illus a ion o he unc ions h1(ε) and h2(ε) p e-
sen ed as exponen s in he co esponding es ima es o Theo em 0.7 and 0.8. I is wo h
men ioning ha he exp ession o h1and h2depends on he speci ic quad a ic numbe
chosen om (5)
No ice ha he geome ic sequences men ioned in Theo em 0.8 a e hose whe e
he spli ing unc ion has mo e han 2 essen ial dominan ha monics because he
second essen ial dominan ha monic coincides wi h he hi d one, and his equi es
a special s udy. As an illus a ion, we ca y ou his s udy o he sil e numbe
Ω2= [2,2,2, . . .] = [2] = √2−1 and show (imposing some condi ions on he phases σk
o in (4)) he con inua ion (wi hou bi u ca ions) o he 4 homoclinic o bi s o all
alues o ε→0, see also Theo em 6.3, p. 153.
Theo em 0.9 (T ans e sali y o he spli ing o Ω2).Fo he Hamil onian sys em
(2-4) wi h n= 2 and Ω=Ω2in (3), assume ha ε > 0is small enough and µ=εp,
p>p∗wi h p∗= 2 i ν= 1 and p∗= 3 i ν= 0, hen i σk= 0 o all k∈Z2 {0}, one
has:
• he Melniko unc ion M(θ)has exac ly 4 ze os θ∗, all simple, o all ε;
•The minimal eigen alue o ∂θM(θ∗)sa is ies
m∗∼µε1/4exp −C0h2(ε)
ε1/4
whe e h2(ε)is a posi i e pe iodic unc ion in ln ε.
Chap e 7. We conside he case n= 3 o he Hamil onian sys em (2-4) and gen-
e alize he esul s ob ained o wo-dimensional o i wi h quad a ic equencies o a

STRUCTURE AND MAIN RESULTS 23
1
εn
εn+1 ε′n+1 ε′nεn−1
A2 = B0
A1
h1 (ε)
h2 (ε)
ε
Figu e 10: Plo s o he unc ions h1and h2 o Ω2.
h ee-dimensional whiske ed o us wi h cubic equencies. To ix ideas, we conside a
equency ec o o he o m
ω= (1,Ω,Ω2),
whe e Ω is a cubic i a ional numbe , i.e. a eal oo o a cubic polynomial wi h
in ege coe icien s. We conside he so-called complex case, i.e. he componen s o
he equency ec o lie in a cubic ield, gene a ed by a cubic i a ional numbe whose
wo conjuga es a e no eal, and show an oscilla o y beha io o hei p incipal small
di iso s, ha did no ake place o he quad a ic equencies.
Fi s , in Sec ion 7.1 we s udy he a i hme ic p ope ies o he cubic equencies and
gi e a classi ica ion o he associa ed esonances (k∈Z3 {0}such ha γk:= |hk, ωi||k|2
is small). The idea is o cons uc a unimodula ma ix Twi h he cubic equency
ec o ωas one o i s eigen ec o s, and, hus, classi y he esonances in o p ima y and
seconda y ones, see o mo e de ails Sec ion 7.1. Un o una ely, o cubic i a ional
numbe s he e is no s anda d heo y o con inued ac ions like he one ha was applied
o quad a ic numbe s (in he quad a ic case he pe iodici y o he con inued ac ions
24 INTRODUCTION
is used o cons uc he ma ix T). The e o e, only some conc e e cubic numbe s can
be conside ed o which he ma ix Tis known, see o example [Cha02]. In pa icula ,
we pay special a en ion o he cubic golden numbe , he eal oo o Ω3+ Ω = 1(Ω ≈
0.6823).
We p o e ha he Poinca ´e-Melniko me hod can be applied choosing an app o-
p ia e p>p∗and p o ide an asymp o ic es ima e o he maximal size o he spli ing
unc ion M(θ) (see also Theo em 7.1, p. 171):
Theo em 0.10 ((Maximal) spli ing dis ance).Fo he Hamil onian sys em (2-4) wi h
n= 3, assume ha εis small enough and µ=εp,p>p∗wi h p∗= 2 i ν= 1 and
p∗= 3 i ν= 0, hen he ollowing asymp o ic es ima e holds
max
θ∈T3|M(θ)| ∼ µ
3
√εexp −C0h1(ε)
ε1/6,
whe e C0is he cons an gi en in (7.11) and he unc ion h1(ε)sa is ies he ollowing
bounds:
•“Cons an bound”: 0< C−
1≤h1(ε)≤C+
2wi h cons an s C−
1and C+
2, de ined in
(7.22);
•“Pe iodic bound”: 0< h−
1(ε)≤h1(ε)≤h+
1(ε), whe e h−(ε),h+(ε)a e a 3 ln λ-
pe iodic unc ions in ln ε;min h−
1=C−
1,max h−
1=C−
2,min h+
1=C+
1,max h+
1=
C+
2, he cons an s C−
2, C+
1a e de ined in (7.22).
In con as o he quad a ic case, he unc ion h1(ε) is no pe iodic and has a mo e
complica ed o m (see Figu e 11 whe e one can suspec ha h1(ε) is a quasipe iodic
unc ion).
Also we es ablish he ollowing nume ical esul abou he exis ence and he ans e -
sali y o 8 homoclinic o bi s o he whiske ed o us. We p o e ha o εsmall enough
he 8 simple ze os o he spli ing unc ion Ma e de e mined by i s 3 dominan ha -
monics i he ec o s o indexes S1,S2,S3co esponding o hese e ms a e independen ,
and, o he wise, by 4 dominan ha monics i no (see also Theo em 7.2, p. 173):
Theo em 0.11 (T ans e sali y).Unde he hypo heses o Theo em 0.10, one has:
•I de (S1, S2, S3)6= 0, he Melniko unc ion M(θ)has exac ly 8 ze os θ∗, all
simple, o all εexcep o some small neighbo hood o a disc e e se o ε. The
minimal eigen alue o ∂θM(θ∗)sa is ies
m∗∼µε1/2exp −C0h3(ε)
ε1/6.
STRUCTURE AND MAIN RESULTS 25
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1
1
1.15
1.25
1.35
Cnum
C2
+
C1
+
C2
−
C1
−
Figu e 11: Plo s o he unc ions h1(ε) ( hick solid) and h±
1(ε) (solid) using a loga i hmic scale o
ε. No ice ha he uppe bound C+
2is no sha p, bu a nume ical sha p uppe bound Cnum can be
ob ained.
•I de (S1, S2, S3)=0, bu de (S1, S2, S4)6= 0, he Melniko unc ion M(θ)has
exac ly 8 ze os θ∗, all simple, o all εexcep o some small neighbo hood o a
disc e e se o ε. The minimal eigen alue o ∂θM(θ∗)sa is ies
m∗∼µε1/2exp −C0h4(ε)
ε1/6.
The p oo o his heo em is mo e di icul han he one o he quad a ic case, since
he unc ions p esen ed in he exponen s o he es ima es a e no pe iodic in ln εand,
ac ually, we canno poin ou exac ly (analy ically) he disc e e se whe e he heo em
ails.
I is wo h men ioning ha , as we know, hese a e he i s asymp o ic esul s o
he p oblem o spli ing o sepa a ices o whiske ed o i wi h 3 equencies, and ha
many open p oblems s ill emain.
26 INTRODUCTION
Appendix B. We p o e some auxilia y lemmas ela ed o he Fixed Poin heo em.
These lemmas enable us o ind he c i ical poin s o he spli ing po en ial L(θ) and
he ze os o he spli ing unc ion M(θ).
Conclusions and u u e wo k
In his sec ion we summa ize he main achie emen s o he hesis and sugges some
open p oblems o in es iga e in he nea es u u e.
Bi u ca ions o homoclinic angencies in a ea-p ese ing maps
•We ha e s udied bi u ca ions o a quad a ic homoclinic angency o wo-dimensional
symplec ic (Chap e 1) and a ea-p ese ing non-o ien able (Chap e 2) saddle
maps and p o ed he exis ence o cascades o ellip ic pe iodic o bi s nea he
homoclinic o bi wi hin he amewo k o one and wo pa ame e gene al un old-
ings.
•We ha e conside ed he ques ion o he coexis ence o ellip ic pe iodic o bi s
o di e en pe iods o he symplec ic and a ea-p ese ing non-o ien able saddle
maps and es ablished he phenomenon o he global esonance.
•We ha e s udied bi u ca ions o a cubic homoclinic angency o wo-dimensional
symplec ic maps and disco e ed he s uc u e o he bi u ca ional diag am in wo
pa ame e gene al un oldings (Chap e 3).
•We ha e cons uc ed ini ely-smoo h no mal o ms o wo-dimensional symplec-
ic and a ea-p ese ing non-o ien able saddle maps (Chap e 4).
•We ha e es ablished he s uc u e o 1 : 4 esonance o some conse a i e H´enon-
like maps (Appendix A).
Exponen ially small spli ing o sepa a ices o whiske ed o i wi h se e al
equencies in Hamil onian sys ems
•We ha e s udied exponen ially small spli ing o sepa a ices o wo-dimensional
whiske ed o i wi h quad a ic equencies. We ha e ound 23 new quad a ic
numbe s o which he Poinca ´e-Melniko me hod can be applied and es ablished
he exis ence o 4 ans e se homoclinic o bi s.
•We ha e s udied he con inua ion o he homoclinic o bi s o all ε→0 in he
case o he sil e numbe Ω2=√2−1.
CONCLUSIONS AND FUTURE WORK 27
•We ha e es ablished he exis ence o exponen ially small spli ing o sepa a i-
ces o h ee-dimensional whiske ed o i wi h cubic golden equency ec o and
de ec ed he ans e sali y o 8 homoclinic o bi s.
Fu u e wo k
In he closes u u e we plan o con inue in es iga ions in hese opics.
Rega ding he i s opic, we a e going o ansla e he ob ained esul s o he case
o e e sible maps. Namely, we would like
•To s udy bi u ca ions o cubic angencies in e e sible maps, pu ing a special
emphasis on symme y-b eaking bi u ca ions.
•To adap he esul s ob ained o symplec ic maps o he case o e e sible maps.
•To unde s and which mechanisms o asymme y a e caused by a ans e se ho-
moclinic ajec o y.
•To analyze global bi u ca ions o a ea-p ese ing maps wi h a ans e se homo-
clinic o bi o a pa abolic ixed poin .
Fo he second opic, we plan in he u u e
•In he wo-dimensional case, o s udy he con inua ion o he homoclinic o bi s
o all su icien ly small εin he case o he quad a ic numbe s (5) in oduced in
Chap e 6.
•To ind new quad a ic numbe s o which he echnique de eloped in Chap e 6
can be applied o de ec spli ing o sepa a ices.
•In he h ee-dimensional case, o conside o he conc e e cubic numbe s in he
complex case and apply he echnique o Chap e 7 o es ablish spli ing o sep-
a a ices.
•To conside cubic numbe s in he so-called eal case, (i.e. he componen s o he
equency ec o lie in a cubic ield, gene a ed by a cubic i a ional numbe whose
wo conjuga es a e eal). This s udy equi es a di e en app oach. In his case,
he beha io o he associa ed small di iso s seems o be di e en o he complex
case conside ed in Chap e 7, and will equi e in ensi e nume ical high-p ecision
simula ions in o de o es ablish he p ope ies o such ec o s, and hen y o
ob ain igo ous asymp o ic es ima es o he spli ing.
•To conside noble numbe s Ω = [a1, a2, . . . , am,1,1,1, . . .] ela ed o he golden
mean Ω1, o apply he esul s ob ained in his hesis o es ablish he phenomenon
o A nold di usion.

28 INTRODUCTION
Pa I
Bi u ca ions o homoclinic
angencies in a ea-p ese ing maps
29
Chap e 1
Bi u ca ions o quad a ic
homoclinic angencies o
wo-dimensional symplec ic maps
1.1 S a emen o he p oblem and main esul s
Conside a C -smoo h ( ≥3) symplec ic map 0sa is ying he ollowing condi ions:
A. 0has a saddle ixed poin Owi h mul iplie s λand λ−1, whe e |λ|<1.
B. 0has a homoclinic o bi Γ0a whose poin s he s able and uns able in a ian
mani olds o he saddle Oha e a quad a ic angency (see Figu e 1).
Le Hsbe a (codimension one) bi u ca ion su ace composed o symplec ic C -maps
close o 0such ha e e y map o Hshas a non ans e sal homoclinic o bi close o
Γ0. Le εbe a amily o symplec ic C -maps ha con ains he map 0a ε= 0. We
suppose ha he amily depends smoo hly on pa ame e s ε= (ε1, ..., εm) and sa is ies
he ollowing condi ion:
C. The amily εis ans e se o Hsa ε= 0.
Le Ube a small neighbo hood o O∪Γ0. I consis s o a small disk U0con aining
Oand a numbe o small disks su ounding hose poin s o Γ0 ha do no lie in U0
(see Figu e 1).
De ini ion 1.1. A pe iodic o homoclinic o bi en i ely lying in Uis called p- ound i
i has exac ly pin e sec ion poin s wi h any disk o he se U U0.
In his chap e we s udy bi u ca ions o single- ound (p= 1) pe iodic o bi s in
he amilies ε. No e ha e e y poin o such an o bi can be conside ed as a ixed
poin o he co esponding i s e u n map. Such a map is usually cons uc ed as a
31
38 1. HOMOCLINIC BIFURCATIONS IN 2D SYMPLECTIC MAPS
q is de ined in (1.1));
2. A he bo de poin s µ=µ+
kand µ=µ−
ko δk, µhas a single- ound pa abolic
pe iodic o bi wi h double mul iplie s +1 o −1, espec i ely;
3. The ellip ic o bi is gene ic o all alues o µ om δk, excep o exac ly wo
alues co esponding o he s ong esonances when he mul iplie s a e e±iπ/2and
e±i2π/3;
4. In he cases c < 0o c > 0and τ6= 0 (cis gi en in (1.14)), he in e als δiand
δjdo no in e sec when in ege s iand ja e di e en and su icien ly la ge.
No e ha some analogs ela ed o i ems 1, 2 and 3 o his heo em we e p o ed
in [Bi 87], [BS89] and [MR97]. Howe e , p oblems on he coexis ence o single- ound
ellip ic pe iodic o bi s we e no conside ed. The i em 4 o Theo em 1.1 shows ha , in
gene al, such o bi s o di e en and la ge pe iods can no coexis .1By “gene al” we
mean ha he case τ= 0 is excluded. Howe e , om he geome ical poin o iew,
his case looks o be qui e in e es ing. Indeed, as one can ex ac om Figu es 1.5 ha
i τ a ies nea ze o, he posi ion o in e als δkcan sha ply change and, mo eo e , he
in e als δiand δjwi h di e en iand jcan in e sec and hey can be e en “nes ed”.
The e o e, in o de o unde s and he co esponding phenomena we mus include τ
in o he se o pa ame e s.
Theo em 1.2 (On wo pa ame e cascades o ellip ic poin s).Le 0be a symplec ic
map sa is ying condi ions Aand Band µ,τ be a wo pa ame e amily which un olds
gene ally, unde condi ion C, he gi en homoclinic angency wi h τ= 0. Then he
ollowing s a emen s ake place:
1. In any neighbo hood o he o igin in he (τ, µ)-plane, he e a e in ini ely many
open domains ∆k, o k=¯
k, ¯
k+1, . . . (¯
kis some in ege ), such ha he map µ,τ
has a single- ound pe iodic (o pe iod k+q, whe e q is de ined in (1.1)) ellip ic
o bi a (τ, µ)∈∆k;
2. The domains ∆kaccumula e o he axis µ= 0 as k→ ∞;
3. The bounda ies o ∆ka e wo cu es L+
kand L−
ksuch ha he map µ,τ has a
pa abolic single- ound pe iodic o bi wi h double mul iplie s ei he +1 i (τ, µ)∈
L+
ko −1i (τ, µ)∈L−
k;
4. The ellip ic o bi is gene ic o all alues o (τ, µ)∈∆k, excep o hose which
belong o cu es Lπ/2
kand L2π/3
kwhen he mul iplie s o he o bi a e equal o
e±iπ/2and e±i2π/3, espec i ely;
1I is no he case o wo-dimensional symplec ic maps wi h non ans e sal he e oclinic cycles: as
i was shown in [GS97] and [GS00], maps ha ing simul aneously in ini ely many single- ound (gene ic)
ellip ic pe iodic o bi s a e dense in a bi u ca ion codimension-one su ace composed om maps wi h
non ans e sal he e oclinic cycles.

1.1. STATEMENT OF THE PROBLEM AND MAIN RESULTS 39
5. In he case c < 0(cis gi en in (1.14)), he domains ∆iand ∆jdo no in e sec
o any su icien ly la ge and di e en in ege s iand j;
6. In he cases c > 0, he domains ∆iand ∆ja e necessa ily c ossed and hey
in e sec he axis µ= 0; Mo eo e , i −3< s0<1/4, whe e
s0=dx+(ac + 20x+)− 11x+(1 −1
4 11x+),(1.16)
whe e all he coe icien s a e gi en in (1.11) hen all domains ∆kwi h su icien ly
la ge kcon ain he o igin (τ= 0, µ = 0).
See Figu e 1.6 o he illus a ion o he heo em. Fo example, you can see he
non-in e sec ing (in Figu e 1.6(a)–(c)) as well as he c ossed and in e sec ing µ= 0 (in
Figu e 1.6 (d)–( )) domains ∆iand ∆j,i6=j, in he case c < 0 and c > 0, espec i ely.
Figu e 1.6:
40 1. HOMOCLINIC BIFURCATIONS IN 2D SYMPLECTIC MAPS
Co olla y 1.1. Le he ollowing ela ions c > 0,τ= 0,−3< s0<1/4and
s06={0; −5/4} ake place o he map 0. Then he e exis s such k0>0 ha 0
has a coun able se o gene ic (KAM-s able) single- ound ellip ic pe iodic o bi s o all
successi e pe iods beginning wi h k0+q.
In he ollowing heo em we cla i y bo h a disposi ion o he in e als δk om
Theo em 1.1 and a cha ac e o he co esponding bi u ca ions when µ a ies inside
δk.2
Theo em 1.3. The in e als δk om Theo em 1.1 ha e o m δk= (µ+
k, µ−
k)i d < 0
(dis gi en in (1.14)) and δk= (µ−
k, µ+
k)i d > 0. Bi u ca ions o ixed poin s in he
i s e u n map Tk(µ) ollow, in gene al, he scena io obse ed in he conse a i e
gene alize H´enon map
¯x=y, ¯y=M−x−y2+νky3,(1.17)
whe e
νk= 03
d2λk,
M=−d(1 + ν1
k)λ−2kµ+λk(cx+−y−)(1 + kβ1λkx+y−)−s0+ν2
k
(1.18)
being 03 he coe icien gi en in (1.11), s0 he coe icien gi en by (1.16) and ν1
k=
O(λk),ν2
k=O(kλk)some asymp o ically small coe icien s. See Figu e 1.7 o d < 0
o gene al bi u ca ions o he map (1.17). The esonance 1 : 3 is non-degene a e o all
alues o νk, while he esonance 1 : 4 is non-degene a e o νk6= 0 (see Figu e 1.7(b)
o νk>0and Figu e 1.7(c) o νk<0).
No e ha i 03 6= 0, hen bo h scena ios o Figu e 1.7 (b) and (c) ake place
o λ < 0: we ha e he case (b) o e en kand he case (c) o odd k. Analogous
phenomenon was obse ed in [BS89] in bi u ca ions o appea ance (disappea ance) o
ho seshoes in h ee-dimensional conse a i e sys ems wi h homoclinic loops o saddle-
oci.
The con en o he es pa o his chap e is he ollowing. In sec ion 1.2 we s udy
he semi-local dynamics o symplec ic maps wi h quad a ic homoclinic angencies: we
selec h ee classes o such maps and desc ibe he s uc u e o o bi s en i ely lying in a
small neighbou hood U(O∪Γ0). In sec ion 1.3 we p o e ou main echnical esul , he
Rescaling Lemma 1.4, which shows ha one can educe he s udy o bi u ca ions o he
i s e u n maps o he qui e s anda d analysis o bi u ca ions o wo-dimensional con-
se a i e H´enon-like maps. Sec ion 1.4 con ains he p oo s o ou main Theo ems 1.1,
1.2 and 1.3.
2No e ha he in e als δkcan be also iewed as in e als ob ained on lines τ=cons when
in e sec ion wi h domains ∆k. The e o e, we can ex ac om Figu e 1.6 a ce ain in o ma ion on
disposi ion o hese in e als.
1.2. THREE CLASSES OF SYMPLECTIC MAPS 41
Figu e 1.7: Bi u ca ions o ixed poin s in he i s e u n map Tk(µ) o d < 0. (a) The main
scena io, he e µ+
k< µ−
k(i d > 0, hen µ+
k> µ−
k). (b)–(c) Bi u ca ions nea esonance 1 : 4 (β= 0
co esponds o µ=µπ/2
k) o he cases (b) νk>0 (he e he ixed poin is always ellip ic) and (c)
νk<0 ( o β= 0 he ixed poin is a saddle wi h eigh sepa a ices).
1.2 Th ee classes o symplec ic maps wi h homo-
clinic angencies.
Le 0be a symplec ic map sa is ying condi ions Aand B. E iden ly, any o bi o 0
en i ely lying in U, excep o O, mus isi bo h neighbo hoods Π−and Π+(o he wise,
i wouldn’ be close o Γ0). Mo eo e , such o bi s mus ha e poin s belonging o he
in e sec ions T1(σ1
j)∩σ0
i o all possible in ege iand j. Gi en a su icien ly la ge
in ege ¯
k > 0, we assume ha Π+and Π−con ain he s ips σ0
kand σ1
k, espec i ely,
only wi h numbe s k≥¯
k. In o he wo ds, we will conside only such en i ely lying in
Uo bi s o 0whose poin s om Π+ each Π− o a numbe o i e a ions ha is no
less han ¯
k. Deno e he se o such o bi s as N¯
k≡ N¯
k( 0).
In his sec ion we s udy he s uc u e o he se N¯
k( 0) and, hus, ex end he esul s
o [GS87], [GS01] and [GS03] o he symplec ic case. Recall he ollowing de ini ion
and esul .
De ini ion 1.2. [GS87], [GST96b] We say ha he ho seshoe T1(σ1
j) has a egula
in e sec ion wi h he s ip σ0
ii (see Figu e 1.2)
(i) he se T1(σ1
j)∩σ0
iconsis s o wo connec ed componen s ∆1
ij and ∆2
ij ;
(ii) he map T1Tj
0 es ic ed o he p eimage (T1Tj
0)−1∆α
ij ⊂σ0
jo he componen ∆α
ij,
whe e α= 1,2,is a saddle map (i.e., i is exponen ially con ac ing along one o
he coo dina es and expanding along ano he ).
42 1. HOMOCLINIC BIFURCATIONS IN 2D SYMPLECTIC MAPS
Lemma 1.3. [GS87] The e a e a cons an S1>0and a su icien ly la ge in ege ¯
k
such ha , o any i, j ≥¯
k, he ollowing asse ions a e alid.
(i) I
d(λiy−−cλjx+)> Sij(¯
k),(1.19)
whe e Sij =S1(|λ|i+|λ|j)|λ|¯
k/2, he ho seshoe T1(σ1
j)in e sec s egula ly he s ip σ0
i.
ii) I
d(λiy−−cλjx+)<−Sij(¯
k),(1.20)
hen T1(σ1
j)∩σ0
i=∅.
The inequali ies (1.19)–(1.20) ha e a a he simple geome ical sense. The s ip
σ0
iis a na ow ho izon al ec angle in Π+ha ing a cen al line y=λiy−, while, he
s ip σ1
jis a na ow e ical ec angle in Π−ha ing a cen al line x=λjx+. By
(1.10), he s ip σ1
jis mapped unde T1in o a ho seshoe which con ains a pa abola
y=cλjx++d(x−x+)2/b2. The inequali y d(λiy−−cλjx+)>0 means ha he s aigh
line y=λiy−and he pa abola a e c ossed in wo poin s, whe eas, he inequali y
d(λiy−−cλjx+)<0 implies ha hese cu es do no in e sec . By coe icien Sij(¯
k)
we ake in o accoun a non-ze o hickness o he s ips and ho seshoes.
I is con enien o e o mula e his lemma as ollows: i he ho seshoe T1(σ1
j) has
an i egula in e sec ion wi h he s ip σ0
i, hen he ollowing inequali ies mus hold
|d||λiy−−cλjx+| ≤ Sij(¯
k),(1.21)
and i T1(σ1
j)∩σ0
i6=∅, hen
d(λiy−−cλjx+)≥ −Sij(¯
k),(1.22)
I is e iden ha he cha ac e o in ege solu ions o inequali ies (1.19)–(1.20)
depends essen ially on he signs o he quan i ies λ, c and d. In u n, his means ha
1.2. THREE CLASSES OF SYMPLECTIC MAPS 43
he s uc u e o N¯
kdepends essen ially on he ype o homoclinic angency. By his
p inciple, he same as o he case o gene al di eomo phisms (see [GS73]), we can
subdi ide quad a ic homoclinic angencies in he symplec ic case in o h ee big classes
in he ollowing way:
•The i s class ela es o he angencies wi h λ > 0, c < 0 and d < 0 (see
Figu e 1.3a).
•The second class ela es o he angencies wi h λ > 0, c < 0 and d > 0 (see
Figu e 1.3b).
•The angencies o all o he ypes (wi h all o he combina ions o he signs o λ, c
and d) a e ela ed o he hi d class (see Figu es 1.3c and 1.4).
We will say also ha a gi en symplec ic map is o he i s , second o hi d class,
i i has he homoclinic angency unde conside a ion o be he i s , second o hi d
class, espec i ely.
1.2.1 Maps o he i s and second classes.
In he case o a map o he i s class, since λ > 0, c < 0 and d < 0, he inequali y
(1.20) holds o all i, j ≥¯
k. I ollows, by Lemma 1.3, ha T1(σ1
j)∩σ0
i=∅ o any
i, j ≥¯
kwhich implies
P oposi ion 1.1. [GS87] Le 0be a map o he i s class. Then he se N¯
khas he
i ial s uc u e: N¯
k={O, Γ0}.
Fo a map o he second class, since λ > 0, c < 0 and d > 0, we ha e now ha
inequali y (1.19) holds o all i, j ≥¯
k. I means, in u n, ha all he ho seshoes T1(σ1
j)
and s ips σ0
i( o any i, j ≥¯
k) ha e he egula in e sec ions. The e o e he se N¯
khas
a non-uni o mly hype bolic s uc u e and all o bi s om N¯
k, excep o Γ0, a e saddle
(see also [GS87]). Mo eo e , we can gi e he exac desc ip ion o he se N¯
kin his
case. Namely, le B3
¯
k+qbe a subsys em o he opological Be noulli scheme (shi ) on
h ee symbols (0,1,2) consis ing only o (bi-in ini e) sequences o o m
(..., 0, αs−1,
ks+q
z}| {
0, ..., 0, αs,
ks+1+q
z }| {
0, ..., 0, αs+1,0, ...),(1.23)
whe e αs∈ {1,2},ks≥¯
k o any s, and any sequence (1.23) does no con ain wo
successi e nonze o symbols. Le ˜
B3
¯
k+qbe he ac o -sys em ha is esul ed om B3
¯
k+qi
o iden i y wo homoclinic o bi s (..., 0, ..., 0,1,0, ..., 0, ...) and (..., 0, ..., 0,2,0, ..., 0, ...).
Deno e his iden i ied o bi as ˜ω.
P oposi ion 1.2. [GS87], [GS01] Le 0be a map o he second class. Then he sys em
0N¯
kis opologically conjuga e o ˜
B3
¯
k+qand all o bi s om N¯
k Γ0a e saddle.

44 1. HOMOCLINIC BIFURCATIONS IN 2D SYMPLECTIC MAPS
1.2.2 Maps o he hi d class
We ha e o mally 6 di e en combina ions o he signs o coe icien s λ, c and d ela ed
o he hi d class. Howe e , i λ < 0 we can always choose such pai s o he homoclinic
poin s M+and M− o which dis posi i e. Besides, since 0is symplec ic, he e is no
necessi y o dis inguish 0and −1
0. The e o e, he combina ions λ > 0, c > 0, d > 0
and λ > 0, c > 0, d < 0 can be ans o med one o ano he , i o conside −1
0ins ead
o 0.3Thus, we can educe he numbe o di e en ypes o homoclinic angencies o
he hi d class o he ollowing h ee ones.
1. maps wi h λ > 0, c > 0, d > 0 (see Figu e 1.4b);
2. maps wi h λ < 0, c > 0, d > 0 (see Figu e 1.4c);
3. maps wi h λ < 0, c < 0, d > 0 (see Figu e 1.3c).
Deno e by Hi
3, i = 1,2,3,codimension-one bi u ca ion su aces, in he space o C -
smoo h symplec ic maps, composed om maps wi h homoclinic angencies o poin ed
ou ypes.
I is ypical o maps o he hi d class ha he s uc u e o he se N¯
kdepends
essen ially on he alue o he in a ian τ(de ined in (1.15)). In pa icula , he ollowing
esul akes place o maps on H1
3(i was announced in [GS01], we gi e he e he p oo ).
P oposi ion 1.3. Le 0∈H1
3.
1) I τ < 0, hen he e exis s such ¯
k1=¯
k1(τ)→ ∞ as τ→0 ha he se N¯
k1has he
i ial s uc u e: N¯
k1={O, Γ0}.
2) I τ > 0, he se N¯
k, o any ¯
k, con ains non i ial hype bolic subse s.
3) I τ > 0and τ /∈Z+(whe e Z+is he se o posi i e in ege numbe s), hen he e
exis s ¯
k2=¯
k2(τ)→ ∞ as dis {τ, Z+} → 0such ha he se N¯
k2allows a comple e
desc ip ion in e ms o he symbolic dynamics and all o bi s o N¯
k2, excep o Γ0, a e
saddle.
P oo . 1) Taking loga i hm o he bo h hands o (1.20) we ob ain he inequali y
j−i+τ < −S|λ|¯
k/2,(1.24)
whe e Sis a posi i e cons an (independen o i, j and ¯
k). By lemma 1.3, i i≥¯
k
and j≥¯
ksa is y (1.24), hen he ho seshoe T1(σ1
j) has emp y in e sec ion wi h he
s ip σ0
i. No e ha in he case τ < 0, he inequali y (1.24) has solu ions only o o m
j > i. In pa icula , i means ha o all i≥¯
k he ho seshoes T1σ1
ilie abo e he own
s ips σ0
i(see Figu e 1.5a) and, he e o e, all o bi s, excep o Oand Γ0, lea e Uunde
posi i e i e a ions o 0.
3I is easy o check ha he ollowing ela ions ˜c=c−1,˜
d=−d(cb2)−1 ake place o he map
˜
T1=T−1
1.
1.2. THREE CLASSES OF SYMPLECTIC MAPS 45
2) The inequali y (1.19) can be w i en in he o m
j−i+τ > S|λ|¯
k/2.(1.25)
I τis posi i e and ¯
kis su icien ly la ge, inequali y (1.25) has always in ini ely many in-
ege solu ions o o m j≤iincluding solu ions j=i. The la e means, by lemma 1.3,
ha , o all su icien ly la ge i, he ho seshoes T1σ1
iha e he egula in e sec ions wi h
he own s ips σ0
i, see Figu e 1.5c. I implies ha i τ > 0, he co esponding map
0∈H1
3has in ini ely many Smale ho seshoes Ωi(i.e. o e e y su icien ly la ge i, he
i s e u n map Ti≡T1Ti
0:σ0
i→σ0
ihas he nonwande ing se which is conjuga e o
he Smale ho seshoe).
3) Conside he inequali y (1.21). A e aking loga i hm, i is ew i en as
|j−i+τ| ≤ S|λ|¯
k/2.(1.26)
I τ > 0 is no in ege , inequali y (1.26) has no in ege solu ions when ¯
k=¯
k(τ) is
su icien ly la ge. Thus, all he s ips and ho seshoes ha e only ei he egula o emp y
in e sec ions. I allows o gi e he comple e desc ip ion o N¯
k. Namely, le Bτ(¯
k) be
a subsys em o ˜
B3
¯
k+qsuch ha
(i) Bτ(¯
k) con ains he o bi s (. . . , 0,...,0, . . . ) and ˜ω;
(ii) in any sequence (1.23) he leng hs (ks+q) and (ks+1 +q) o any wo successi e
s ings composed om ze o symbols sa is y inequali y (1.25) wi h j=ks, i =ks+1.
Then 0N¯
kis conjuga e o Bτ(¯
k).
Now we conside he cases wi h λ < 0, i.e. 0∈H2
3and 0∈H3
3(see Figu e 1.8).
P oposi ion 1.4. Le 0∈H2
3∪H3
3. Then he se N¯
k( 0), o any ¯
k, con ains non-
i ial hype bolic subse s always, excep maybe o he “global esonance” case 0∈H2
3
wi h τ= 0.
P oo . Le 0∈H2
3. Since λ < 0, c > 0, d > 0, inequali y (1.19) o e en iand j, can
be w i en as
j−i+τ > S|λ|¯
k/2,whe e i, j ≥¯
kand i, j = 0(mod2),
I τ > 0, his inequali y has in ini ely many in ege solu ions o o m j≥iincluding
solu ions i=j. I implies ha , i τ > 0, he map 0∈H2
3has in ini ely many
ho seshoes Ωiwi h e en i.
On he o he hand, inequali y (1.19) o odd iand jis ew i en as
j−i+τ < −S|λ|¯
k/2,whe e i, j ≥¯
kand i, j = 1(mod2).
46 1. HOMOCLINIC BIFURCATIONS IN 2D SYMPLECTIC MAPS
Figu e 1.8:
I τ < 0, his inequali y has in ini ely many in ege solu ions o o m i≥j. Thus,
when τ < 0 he map 0∈H2
3has in ini ely many ho seshoes Ωiwi h odd i.
Le 0∈H3
3. Since λ < 0, c < 0, d > 0, inequali y (1.19) holds o any su icien ly
la ge e en iand j. I means ha all ho seshoes T1(σ1
j) in e sec egula ly wi h all
s ips σ0
iwhen he numbe s iand ja e e en (he e, a ce ain analogy wi h maps o he
second class is obse ed). In any case, map 0∈H3
3has in ini ely many ho seshoes Ωi
whe e i uns all su icien ly la ge e en in ege s.
Co olla y 1.2. 1) Le 0∈H2
3. Then, i τ > 0( esp., τ < 0), he map 0has in ini ely
many ho seshoes Ωiwi h e en i( esp., odd i) and has no ho seshoes wi h odd i( esp.,
wi h e en i).
2) Le 0∈H3
3. Then he map 0has in ini ely many ho seshoes Ωiwi h e en iand
has no ho seshoes wi h odd i.
P oposi ion 1.5. Le 0∈H2
3( esp., 0∈H3
3). I |τ|is no e en in ege ( esp., odd
in ege ), hen he e exis s such ¯
k3 ha he se N¯
k3allows he comple e desc ip ions in
e ms o he symbolic dynamics and all o bi s o N¯
k, excep o Γ0, a e saddle.
P oo . Conside he case 0∈H2
3. Suppose ha τ6= 0. I is easy o see ha , o
su icien ly la ge ¯
k=¯
k(τ) (¯
k→ ∞ as τ→0), he se N¯
kconsis s o o bi s in e sec ing
s ips σ0
jei he wi h only e en numbe s when τ > 0 o wi h only odd numbe s when
τ < 0. In he i s case (τ > 0), see Figu e 1.9a, o wa d i e a ions o any poin om
σ0
jwi h odd jcan no e u n on σ0
j. Indeed, he ho seshoe T1(σ1
i) can in e sec only
s ips wi h odd numbe s such ha j−i+τ < 0, i.e. i>j+τ; besides, any ho seshoe
T1(σ1
l) wi h e en ldoes no in e sec he s ips wi h odd numbe s. In he case τ < 0,
1.3. GENERAL UNFOLDINGS AND RESCALING LEMMA 47
see Figu e 1.9b, any he ho seshoe T1(σ1
j) wi h e en jcan in e sec only such s ip σ0
i
o which iis e en and j−i+τ > 0, i.e. i<jbecause τ < 0.
Figu e 1.9:
Now we can gi e a comple e desc ip ion o he se N¯
k o 0∈H2
3when |τ|is no
e en in ege . We no e only ha numbe s iand jsuch ha T1(σ1
j) has an i egula
in e sec ion wi h σ0
imus sa is y he inequali y (1.26). I |τ|is no e en in ege , he
inequali y (1.26) has no in ege solu ions iand jo he same pa i y. Conside a
subsys em B2+
τo ˜
B3
¯
k+qsuch ha in any sequence (1.23) he numbe s ksa e e en o
all sand sa is y inequali y ks−ks+1 +τ > 0. Analogously, le B2−
τbe such a subsys em
o ˜
B3
¯
k+q ha in any sequence (1.23) numbe s ksa e odd o all sand sa is y inequali y
ks−ks+1 +τ < 0. Suppose ha |τ|is no e en in ege . Then he sys em 0N¯
kis
conjuga e ei he o B2+
τin he case τ > 0 o o B2−
τin he case τ < 0.
Fo 0∈H3
3we ha e, due o he geome y, ha i egula in e sec ions o he
ho seshoes T1(σ1
j) and s ips σ0
ican exis only in hose cases whe e he numbe s iand
jha e opposi e pa i ies. Mo eo e , he inequali y (1.26) can ha e such solu ions only
o odd |τ|. Conside a subsys em B3
τo ˜
B3
¯
k+qsa is ying he ollowing condi ions:
(i) B3
τcon ains all sequences o o m (1.23) in which all numbe s ks≥¯
k(τ) a e e en;
(ii) B3
τdo no con ain he sequences wi h ksand ks+1 o be bo h odd;
(iii) B3
τcon ains all he sequences wi h e en ksand odd ks+1 such ha ks−ks+1 +τ < 0;
(i ) B3
τcon ains all he sequences wi h odd ksand e en ks+1 such ha ks−ks+1 +τ > 0.
Then 0N¯
kis conjuga e o B3
τwhen |τ|is no odd in ege .
1.3 Gene al un oldings and Rescaling Lemma
In his sec ion we calcula e he i s e u n maps Tk≡T1Tk
0:σ0
k7→ σ0
k o all su icien ly
la ge kand apply he esul s ob ained (Lemma 1.4) o s udying bi u ca ions o ixed
poin s. Mo eo e , we conside in his sec ion, and wha ollows, one and wo pa ame e
54 1. HOMOCLINIC BIFURCATIONS IN 2D SYMPLECTIC MAPS
Then we ha e
d0x+0=dx+, 0
11x+0= 11x+,
a0c0=ac + (x+)2β1c2+ 02x+−c2β1(x+)2=ac + 20x+.
Thus, by (1.16), s0=s0
0.
Take now a pai M+0=M+and M−0 =T−1
0(M−). Then he new global map T0
1is
de ined as T0
1=T1T0:T−1
0(Π−)→Π+and, hus, i can be w i en as
¯x=x++F(x0, y0−y−),¯y=G(x0, y0−y−),
whe e x0=λx(1+β1xy)+O(x3y2), y0=λ−1y(1−β1xy)+O(x2y3) and (x, y)∈T−1
0Π−.
Thus, we ha e ha x+0=x+, y−0 =λy−. Then we calcula e o he coe icien s as he
co esponding de i a i es a x= 0, y =λy−. We ha e
a0=∂F
∂x0
∂x0
∂x +∂F
∂y0
∂y0
∂x , c0=∂G
∂x0
∂x0
∂x +∂G
∂y0
∂y0
∂x ,
0
20 =1
2 ∂2G
(∂x0)2∂x0
∂x 2
+ 2 ∂2G
∂x0∂y0
∂y0
∂x
∂x0
∂x +∂2G
∂y02∂y0
∂x 2
+∂G
∂x0
∂2x0
∂x2+∂G
∂y0
∂2y0
∂x2!,
0
11 =∂2G
(∂x0)2
∂x0
∂x
∂x0
∂y +∂2G
∂x0∂y0∂y0
∂x
∂x0
∂y +∂x0
∂x
∂y0
∂y +∂2G
∂y02
∂y0
∂x
∂y0
∂y +∂G
∂x0
∂2x0
∂x∂y
+∂G
∂y0
∂2y0
∂x∂y,
d0=1
2 ∂2G
(∂x0)2∂x0
∂y 2
+ 2 ∂2G
∂x0∂y0
∂y0
∂y
∂x0
∂y +∂2G
∂y02∂y0
∂y 2
+∂G
∂x0
∂2x0
∂y2+∂G
∂y0
∂2y0
∂y2!.
Since o x= 0, y =λy−
∂G
∂y0= 0,∂
∂y x0,∂x0
∂x ,∂x0
∂y = 0,∂x0
∂x =λ, ∂y0
∂y =λ−1,∂y0
∂x =−λβ1(y−)2,
we ob ain ha
a0=λa −bβ1λ(y−)2, c0=λc, 0
11 = 11 −2dβ1(y−)2, d0=dλ−2,
0
20 = 20λ2− 11λ2β1(y−)2+dλ2β2
1(y−)4+cλ2β1y−,
Thus, we ha e
d0x+0=dx+λ−2,
a0c0=λ2ac −bcβ1λ2(y−)2=λ2ac +β1λ2(y−)2(since bc =−1,
0
11x+0= 11x+−2dβ1(y−)2x+,
d0=dx+λ−2,
0
20x+0= 20λ2x+− 11λ2β1(y−)2x++dλ2β2
1(y−)4x++cλ2β1y−x+.

1.5. INVARIANTS OF HOMOCLINIC TANGENCIES 55
We ob ain, by (1.16),
s0
0=dx+[ac + 20x+]−β1d(y−)2x+−d 11β1(y−)2(x+)2+d2β2
1(y−)4(x+)2
+dcβ1y−(x+)2+ 11x+−2dβ1(y−)2x+−1
4( 11x+)2+dβ1(y−)2(x+)2
−d2β2
1(y−)4(x+)2=s0+dβ1x+y−(cx+−y−)
Since cx+=y− o τ= 0, his comple es he p oo .
56 1. HOMOCLINIC BIFURCATIONS IN 2D SYMPLECTIC MAPS
Chap e 2
Dynamics and bi u ca ions o
non-o ien able a ea-p ese ing
maps wi h quad a ic homoclinic
angencies
In his chap e we s udy bi u ca ions o a ea-p ese ing and non-o ien able maps wi h
quad a ic homoclinic angencies. I seems ha i is a qui e new opic in homoclinic
bi u ca ions. Up o now, homoclinic angencies o non-o ien able maps we e s udied
only o gene al (dissipa i e) sys ems, see e.g. [GS73, GS86, PT87, GS07]. Howe e ,
his heme should be in e es ing o unde s anding dynamics o chao ic conse a i e
maps such as non-o ien able plana maps like he H´enon maps wi h he Jacobian −1
and symplec ic maps on wo-dimensional non-o ien able closed mani olds (like Klein
bo le).
2.1 S a emen o he p oblem and p elimina y con-
s uc ions
In his chap e we conside non-o ien able APMs close o he one wi h a quad a ic
homoclinic angency. Le 0be such a map. As in Chap e 1 we deno e by O he saddle
ixed poin o 0, by U0a small neighbou hood o Oand by Γ0 he non ans e sal o
Ohomoclinic o bi . We embed 0in o a pa ame e amily ε. We assume ha 0and
εsa is y he condi ions Band Co Chap e 1 (Sec ion 1.1).
Howe e , due o non-o ien abili y, ins ead o condi ion Awe suppose he ollowing
condi ion:
A0. 0has a saddle ixed poin Owi h mul iplie s λand γ, whe e 0 <|λ|<1<|γ|
and |λγ|= 1 . Mo eo e , we will conside wo di e en cases:
57
58 2. NON-ORIENTABLE APMS WITH HOMOCLINIC TANGENCIES
A0.1 he saddle is o ien able, i.e. λ=γ−1;
A0.2 he saddle is non-o ien able, i.e. λ=−γ−1.
No e ha in condi ion A0.1 we assume ha he global map T1de ined in Chap e 1,
Sec ion 1.1, is non-o ien able, i.e. he Jacobian o T1is equal o −1. This beha io o
o bi s o e u n maps is ypical o maps on wo-dimensional non-o ien able mani olds
(M¨obius s ip, Klein bo le, p ojec i e plane e c.), see an example o such a map in
Figu e 2.1.
Figu e 2.1: An example o non-o ien able a ea-p ese ing map (on he M¨obius band) ha ing a
quad a ic homoclinic angency a he poin s o a homoclinic o bi Γ0. Some o hese homoclinic
poin s a e shown as g ey ci cles. Also a small neighbo hood o he se O∪Γ0is shown o be he union
o a numbe o “squa es”.
The main goal is he same: o s udy bi u ca ions o single- ound pe iodic o bi s (see
De . ini ion 1.1) in he amilies ε. E e y poin o such an o bi is conside ed again
as a ixed poin o he co esponding i s e u n map Tk=T1Tk
0, whe e T0≡T0(ε)
is he local map and T1≡T1(ε) is he global map whose de ini ion is simila o he
symplec ic case (see Chap e 1). In pa icula , he coo dina e exp ession o T1is he
same as in Chap e 1, o mulas (1.10) and (1.11). Conce ning he local map T0, i s
no mal o m is symplec ic ( he same as in Lemma 1.1) in he case A0.1, while in he
case A0.1 we p o ide non-o ien able no mal o ms in he nex sec ion.
2.1.1 Fini e-smoo h no mal o ms o non-o ien able saddle
a ea-p ese ing maps
In he ollowing lemma we gi e he main no mal o m (o he i s o de ) o he local
map T0(ε) in he non-o ien able case.
2.1. STATEMENT OF THE PROBLEM 59
Lemma 2.1. [GST07]. Le T0(ε)be a C -smoo h, ≥3, saddle a ea-p ese ing map
ha has a ixed poin Owi h mul iplie s λand −λ−1, whe e |λ|<1. Then he e exis s
such C -smoo h local canonical coo dina e change, which is C −2wi h espec o he
pa ame e s, ha he map T0 akes he ollowing o m
¯x=λ(ε)x+o(x2y),¯y=−λ−1(ε)y+o(xy2),(2.1)
The ollowing lemma ela es o he n- h o de no mal o m.
Lemma 2.2. Fo any p= −2n+ 1, whe e n≥2is an in ege such ha n < /2
(i =∞, hen nis a bi a y), he e exis s such Cp-smoo h local canonical coo dina e
change, which is Cp−2wi h espec o he pa ame e s, ha he map T0 akes he ollowing
o m
¯x=λ(ε)x1 +
n
P
2
βi(ε)·(xy)i+o(xn+1yn),
¯y=−λ−1(ε)y1 +
n
P
2
˜
βi(ε)·(xy)i+o(xnyn+1),
(2.2)
whe e he coe icien s βiand ˜
βia e in a ian s o smoo h canonical changes o coo -
dina es p ese ing o m (2.2), and, mo eo e , βi(ε) = ˜
βi(ε)≡0 o all odd i≤n.
As in he symplec ic case, see Lemma 1.2, he no mal o ms o Lemmas 2.1 and 2.2
allow o ob ain a qui e simple coo dina e exp ession o i e a ions Tk
0 o all in ege k.
Namely, le (xi, yi)∈U0, i = 0, . . . , k −1,be such poin s ha (xi+1, yi+1) = T0(xi, yi),
hen he ollowing esul s hold.
Lemma 2.3. 1) I T0 akes he i s o de no mal o m (2.1), hen Tk
0can be w i en
as ollows
xk=λkx0+λ2kP1(x0, yk, ε), y0= (−λ)kyk+λ2kQ1(x0, yk, ε),(2.3)
whe e he unc ions P1and Q1a e uni o mly bounded along wi h all de i a i es up o
o de ( −2) and he ollowing es ima es akes place o he las de i a i es
k(xk, y0)kC −1=O(|λ|k),k(xk, y0)kC =o(1)k→∞.
2) I T0 akes he n- h o de no mal o m (2.2), hen Tk
0can be w i en as
xk=λkx0·R(k)
n(x0yk, ε) + λ(n+1)kP(k)
n(x0, yk, ε),
y0= (−λ)kyk·R(k)
n(x0yk, ε) + λ(n+1)kQ(k)
n(x0, yk, ε),(2.4)
whe e R(k)
n(x0yk, ε)is gi en by o mula (1.6) in which ˜
β1(k)=0,˜
β2(k) = β2k, . . . The
unc ions P(k)
n=o(xn+1
0yn
k), Q(k)
n=o(xn
0yn+1
k)a e uni o mly bounded in kalong wi h
all hei de i a i es wi h espec o x0and ykup o he o de ( −2n−1), besides,
k(xk, y0)kC −2n=O(|λ|k),k(xk, y0)kC −2n+1 =o(1)k→∞.

60 2. NON-ORIENTABLE APMS WITH HOMOCLINIC TANGENCIES
Rema k 2.1. In p inciple, we will use Lemma 2.3 only o he cases whe e T0 akes
he i s and second o de no mal o ms. In he la e case ( o he second o de no mal
o m) he i e a ions o Tk
0can be w i en as
xk=λkx0+O(kλ3k), y0= (−λ)kyk+O(kλ3k),(2.5)
We p o e Lemma 2.2 in Chap e 4. The p oo o Lemma 2.3 is he same as o
Lemma 1.2 om Chap e 1 and, he e o e, we omi i .
In wha ollows, we use in U0 he local no mal o m coo dina es (x, y) in oduced
in he Sec ions 1.1 and 2.1. In hese coo dina es bo h Ws
loc and Wu
loc a e s aigh ened
and, hence, we can pu M+= (x+,0), M−= (0, y−), whe e x+>0 and y−>0.
Wi hou loss o gene ali y, we assume ha x+>0 and y−>0. Then he global map
T1(ε)≡ q(ε) : Π−→Π+can be w i en in he o m (1.10), whe e ela ions (1.11)
hold.
In he a ea-p ese ing case, he Jacobian J(T1) o T1is equal o ±1 iden ically o
all alues o pa ame e s ε. We no e ha APMs wi h homoclinic angencies ha e a ious
nume ical in a ian s. In pa icula , we in oduce he ollowing impo an quan i ies
α=cx+
y−−1 and ˆα=cx+
y−+ 1 (2.6)
which a e some analogous o he in a ian τ o he symplec ic case (αis he same as
in he symplec ic case, see o mula (1.38)).
2.1.2 S ips, ho seshoes and e u n maps
We assume ha he neighbou hoods Π+and Π−a e su icien ly small and ixed, so
ha T0(ε)(Π+)∩Π+=∅and T−1
0(ε)(Π−)∩Π−=∅ o all small ε. Then he domain o
de ini ion o he successo map om Π+ o Π−unde i e a ions o T0(ε) consis s o in-
ini ely many nonin e sec ing s ips σ0
kbelonging o Π+and accumula ing a Ws
loc ∩Π+
as k→ ∞. Analogously, he ange o his map consis s o in ini ely many (nonin e -
sec ing) s ips σ1
k=Tk
0(σ0
k) belonging o Π−and accumula ing a Wu
loc ∩Π−as k→ ∞.
See Figu e 2.2 whe e a loca ion o he s ips is shown o a ious cases.
Acco ding o (1.10) and (1.11), he images T1(σ1
j) o he s ips σ1
jha e a ho se-shoe
shape o m and accumula e o he cu e lu=T1(Wu
loc) as j→ ∞. No e ha any o bi
s aying en i ely in Umus in e sec bo h he neighbo hoods Π−and Π+(o he wise, i
would no be close o Γ0). Thus, such o bi s mus ha e poin s belonging o in e sec ions
o he ho seshoes T1(σ1
j) and he s ips σ0
i o all possible in ege iand j.
When µ a ies he loca ion o he ho seshoes T1(σ1
j) is changed: hey mo e o-
ge he wi h T1(Wu
loc). I implies ha a cha ac e o mu ual in e sec ions o he s ips
and ho seshoes can ca dinally change. I ela es, in pa icula , o he s ips σ0
iand
ho seshoes T1(σ1
i) wi h he same numbe s i. Thus, a a ying µbi u ca ions o c e-
a ion/des uc ion Smale ho seshoes will occu in he i s e u n maps Tk(µ). We will
s udy he main accompanied bi u ca ions in Sec ions below.
2.2. MAIN RESULTS 61
Figu e 2.2: The s ips σ0
kand σ1
k o λand γo a ious signs.
2.2 Main esul s: on cascades o ellip ic pe iodic
o bi s
We di ide non-o ien able APMs unde conside a ion in o wo g oups:
(i) he globally non-o ien able APMs when T0is o ien able and T1is non-o ien able
(λγ = +1 and bc = +1), i.e. he condi ion A0.1 holds;
(ii) he locally non-o ien able APMs when T0is non-o ien able (λγ =−1), , i.e. he
condi ion A0.2 holds.
No e ha in he locally non-o ien able case he Jacobian o T1equals +1 o −1
depending on choice o a pai o he homoclinic poin s. Indeed, i he Jacobian o T1
is equal o −1 (i.e. bc = +1) o a gi en pai M+∈Ws
loc and M−∈Wu
loc o he poin s,
hen, o he pai ˜
M+=T0(M+) and M−o he homoclinic poin s, he Jacobian o
he new global map ˜
T1=T0T1will be equal +1, since J(T0) = −1. The e o e, in he
locally non-o ien able case, we will assume, o mo e de ini eness,
62 2. NON-ORIENTABLE APMS WITH HOMOCLINIC TANGENCIES
• ha he homoclinic poin s M+∈Ws
loc and M−∈Wu
loc a e such ha J(T1) = +1.
Rema k 2.2. I can show1 ha he quan i ies αand ˆα om (2.6) do no depend on
choice o pai s o homoclinic poin s M+∈Ws
loc and M−∈Wu
loc; condi ionally ha , in
he locally non-o ien able case, he poin s a e chosen in such a way ha he sign o
J(T1) is no changed.
Theo em 2.1 (One pa ame e cascades o ellip ic poin s in APMs).Le 0be APM
sa is ying condi ions A0and B and µbe a one pa ame e amily o close o 0APMs
ha un olds gene ally (unde condi ion C) a µ= 0 he quad a ic homoclinic angency.
Fo any in e al I= (−µ0, µ0) alues o µ, he e exis s such in ege and posi i e ¯
k ha
he ollowing holds:
1. (a) In he globally non-o ien able case, he maps µha e no single- ound ellip ic
pe iodic o bi s, while he e exis in e als e2
k⊂I,k=¯
k, ¯
k+ 1, . . . , whe e µhas a
double- ound ellip ic o bi , o pe iod 2(k+q), which co esponds o a pe iod wo poin
o he i s e u n map Tk.
(b) In he locally non-o ien able case, he e exis in e als e2mand e2
2m+1 in I o
any in ege msuch ha 2m≥¯
k, whe e he map µhas a single- ound ellip ic pe iodic
o (pe iod 2m+q) o bi a µ∈e2mand has a double- ound ellip ic pe iodic o bi a
µ∈e2
2m+1.
2. The in e als ekas well as e2
kaccumula e o µ= 0 as k→ ∞ and do no in e sec
o su icien ly la ge and di e en in ege ki α6= 0 in he globally non-o ien able case
as well as α6= 0 and ˆα6= 0 in he locally non-o ien able case.
3. Any in e al ekhas bo de poin s µ=µ+
kand µ=µ−
ksuch ha he map µ
has a single- ound pe iodic o bi (o pe iod k+q) wi h double mul iplie +1 a µ=µ+
k
and wi h double mul iplie −1a µ=µ−
k. Any in e al e2
khas bo de poin s µ=µ2+
k
and µ=µ2−
ksuch ha he map µhas a single- ound pe iodic o bi (o pe iod (k+q))
wi h mul iplie s +1 and −1a µ=µ2+
kand a double- ound pe iodic o bi (o pe iod
2(k+q)) wi h double mul iplie −1a µ=µ2−
k. See Figu e 2.3.
4. The angula a gumen ϕo he mul iplie s e±iϕ o he ellip ic poin s a µ∈ek
o µ∈e2
kdepends mono onically on µand he ellip ic poin is gene ic (KAM-s able)
o all such µ, excep o hose whe e ϕ(µ) = π
2,2π
3.
No e ha Theo em 2.1 does no gi e answe on he ques ion on a mu ual posi ion o
he in e als ekand e2
kin he c i ical cases α= 0 and ˆα= 0. Bu his momen is qui e
impo an , since ela es o he coexis ence o ellip ic o bi s o di e en pe iods. The
same as in he symplec ic case, we conside his ques ion by means o wo pa ame e
amilies.
We assume now ha 0is a map sa is ying condi ions A0and B wi h α= 0 in he
globally non-o ien able case and wi h α= 0 (when c > 0) o ˆα= 0 (when c < 0) in he
locally non-o ien able case. Deno e by H0,0and ˆ
H0,0codimension 2 bi u ca ion su aces
om he space o APMs consis ing o maps close o 0and ha ing a non ans e sal
1 he same as he in a iance o τin Lemma 1.5 om Chap e 1
2.2. MAIN RESULTS 63
Figu e 2.3: Bi u ca ion scena ios in he i s e u n maps Tkacco ding o i em 3 o Theo em 2.1.
We show he e ha he bi h o he ellip ic poin is happened when inc easing µ, while o some ypes
o homoclinic angencies i can occu a dec easing µ. (a) The map Tkis o ien able, hen he alue
µ=µ+
kco esponds o he appea ance o a ixed poin o Tk ha is a non-degene a e pa abolic ixed
poin wi h double mul iplie +1. This poin alls in o wo ixed poin s, saddle and ellip ic ones, when
µ∈ek. The momen µ=µ−
kco esponds o he pe iod doubling bi u ca ion wi h he ellip ic ixed
poin . (b) I Tkis he non-o ien able map, hen he alue µ=µ−
kco esponds o he appea ance o
a ixed poin wi h mul iplie s +1 and −1. This poin alls in o ou poin s, wo saddle ixed ones and
o he wo poin s compose an ellip ic cycle o pe iod 2, when µ∈e2
k. The momen µ=µ2−
kco esponds
o he pe iod doubling bi u ca ion o his pe iod 2 cycle.
homoclinic o bi close o Γ0and such ha he condi ion α= 0 and ˆα= 0 holds,
espec i ely. We will conside wo pa ame e amilies { µ,α}and { µ,ˆα}o APMs which
a e ans e se o H0,0and ˆ
H0,0a µ= 0, α = 0 and µ= 0,ˆα= 0, espec i ely.
Le Dand ˆ
Dbe su icien ly small neighbou hoods (o diame e  > 0) o he o igin
in he pa ame e planes (µ, α) and (µ, ˆα).
I ollows om Theo em 2.1 ha in ini ely many such open domains, E2
k o he
globally non-o ien able case and E2mand E2
2m+1 o he locally non-o ien able case,
exis in Dand ˆ
D ha he ollowing holds.
•I (µ, α)∈Ek, hen he map µ,α o µ,ˆαhas a single- ound ellip ic o bi o pe iod
(k+q) and i (µ, α)∈E2
k, hen he map µ,α o µ,ˆαhas a double- ound ellip ic
o bi o pe iod 2(k+q).
70 2. NON-ORIENTABLE APMS WITH HOMOCLINIC TANGENCIES
In he case 2) o he lemma, we p oceed in he same way as a he beginning o
Lemma 2.4 and since γ−k=−λk, ob ain he ollowing o mula, analogous o (2.12),
¯
ξ=aλkξ+bη +e02η2+Okλ2k|ξ|+|η|3+|λ|kO(|ξ||η|),
−λk¯η(1 + α2
k) + kλ2kO(|¯
ξ|+ ¯η2) + kλ3kO(|¯η|) =
=M1+cλkξ(1 + α3
k) + η2(d+λk 12x+) + λkη( 11x++α4
k) + λk 11ξη + 03η3+
+Oη4+k|λ|3k|ξ|+kλ2k(ξ2+η2) + λk|ξ|η2,
(2.25)
whe e
M1=µ+λk(cx++y−)(1 + kλkβ1x+y−) + λ2kx+(ac + 02x+) + O(kλ3k).(2.26)
( he di e ence is only ha he ac o −λks ands in he le side o he second equa ion
and (cx++y−) is in o mula o M1).
Now, we escale he a iables:
ξ=b(1 + α2
k)
d+λk 12x+λku , η =1 + α2
k
d+λk 12x+λk . (2.27)
Sys em (2.25) in coo dina es (u, ) is ew i en in he ollowing o m
¯u= +aλku+e02
bd λk 2+O(kλ2k),
¯ =M2+u(1 + α5
k)− 2+
− ( 11x++α6
k)− 11b
dλku − 03
d2λk 3+O(kλ2k),
(2.28)
whe e
M2=−d+λk 12x+
1 + α2
k
λ−2kM1.
The ollowing shi o coo dina es (we emo e he linea in e ms om he second
equa ion)
unew =u+1
2( 11x++α6
k), new = +1
2( 11x++α6
k),
b ings map (2.28) o he ollowing o m
¯u= +aλku+e02
bd λk 2+O(kλ2k),
¯ =M3+u− 2− 11b
dλku − 03
d2λk 3+O(kλ2k),
(2.29)
whe e
M3=M2+( 11x+)2
4.

2.4. PROOF OF THE MAIN RESULTS 71
We make he ollowing linea change o coo dina es
x=u+ ˜ν1
k , y = −˜ν2
ku , (2.30)
whe e
˜ν1
k=e02
bd λk,˜ν2
k=e02
bd λk+aλk.(2.31)
Then, sys em (2.29) is ew i en as
¯x=y−M3˜ν1
k+O(kλ2k),
¯y=M3+x−y2−aλky+ˆ
Rλkxy − 03
d2λky3+O(kλ2k),
(2.32)
whe e ˆ
R= (2a−2e02/bd −b 11/d)≡0.
Finally, make one mo e shi o coo dina es
X=x+1
2aλk+ ˜ν1
kM3, Y =y+1
2aλk,
in o de o nulli y he cons an e m in he i s equa ion and he linea in y e m in
he second equa ion o (2.32). A e his, we ob ain he inal o m (2.23) o map Tkin
he escaled coo dina es whe e o mula (2.24) akes place o he pa ame e M.
2.4 P oo o he main esul s
Theo ems 2.1, 2.2 and 2.3 a e p o ed ansla ing he esul s on bi u ca ions o ixed
poin s o he i s e u n maps Tk o single- ound pe iodic o bi s o µ, µ,α o µ,ˆα.
Bi u ca ions in i s e u n maps Tkcan be s udied wi h using hei no mal o ms
deduced by he escaling lemmas 2.4 and 2.5. Since hese no mal o ms coincide up
o asymp o ically small as k→ ∞ e ms wi h he non-o ien able conse a i e H´enon
map, we ecall in he nex sec ion some necessa y esul s on bi u ca ions o ixed poin s
in one pa ame e amilies o conse a i e H´enon map in non-o ien able case.
2.4.1 On bi u ca ions o ixed poin s in he conse a i e H´enon
maps
Thus, he Rescaling Lemma 2.4 and 2.5 show ha he uni ied limi o m o he i s
e u n maps Tkis he conse a i e H´enon map
¯x=y, ¯y=M+νx −y2.(2.33)
(o ien able, i ν=−1, and non-o ien able, i ν= +1). Bi u ca ions o ixed poin s in
he conse a i e H´enon amily a e well known.
72 2. NON-ORIENTABLE APMS WITH HOMOCLINIC TANGENCIES
In he case ν=−1, he H´enon map has a gene ic ellip ic ixed poin o e e y
M∈(−1; 3) excep o s0= 0 when ψ=π/2 and s0= 5/4 when ψ= 2π/3. These
cases co esponds o he s ong esonances 1 : 4 and 1 : 3, espec i ely. The conse a i e
esonance 1 : 3 is non-degene a e, while 1 : 4 is degene a e: he so-called case “A= 1”,
[A n96, AAIS86], is ealized he e. Howe e , i is impo an ha in he i s e u n
map (2.9), when he coe icien s 03 is non-ze o, he esonance 1 : 4 becomes non-
degene a e [Bi 87, Gon05], see also Appendix A. The conse a i e H´enon map has also
ixed pa abolic poin s, a M=−1 wi h mul iplie s ν1=ν2= +1 and a M= 3 wi h
mul iplie s ν1=ν2=−1. The co esponding bi u ca ions a e nondegene a e. See
Figu e 2.5 o an illus a ion.
Figu e 2.5: Bi u ca ions o ixed poin s in he H´enon map: (a) he main scena io, he e µ+
k< µ−
k
(i d > 0, hen µ+
k> µ−
k); (b)–(c) bi u ca ions nea esonance 1 : 4 in map (2.9) o he cases (b)
νk= 03d−2λk>0 (he e he ixed poin is always ellip ic) and (c) νk<0 ( o β= 0 he ixed poin
is a saddle wi h eigh sepa a ices); he e βis a pa ame e cha ac e izing a de ia ion o ψ om π/2.
In he case ν= +1, he H´enon map is non-o ien able and, hus i does no ha e
ellip ic ixed poin s. Howe e , ellip ic poin s o pe iod 2 exis o M∈(0,1). The
map has no ixed poin s o M < 0, i has one ixed poin ¯
O(0,0) wi h mul iplie s
ν1= +1, ν2=−1 a M= 0 and wo saddle ixed poin s ( ¯
O1(−√M, −√M) and
¯
O2(√M, √M)) a M > 0. Besides, an ellip ic o bi o pe iod 2 exis s o 0 < M < 1,
i consis s o wo poin s (p1(−√M, √M) and p2(√M, −√M)); he alue M= +1
co esponds o he pe iod doubling bi u ca ion o his o bi . See Figu e 2.6 o an
illus a ion. No e ha he ellip ic o bi o pe iod 2 is gene ic o all M∈(0,1) excep
o M=1
2,M=1
√2which co espond o he s ong esonances 1 : 4 and 1 : 3,
espec i ely, and M=5
8which co esponds o he ze o i s Bi kho coe icien a he
cycle {p1, p2}, see [DGGLO13].
I is also known (see, e.g., [DN78, AS82]) ha , i M > 5+2√5 ( his is only
2.4. PROOF OF THE MAIN RESULTS 73
Figu e 2.6: The main bi u ca ion scena io in he non-o ien able conse a i e H´enon map.
a su icien condi ion), hen he nonwande ing se o map (2.33) is Smale ho seshoe
which is o ien able o ν=−1 and non-o ien able o ν= +1.
2.4.2 P oo o Theo em 2.1
The p oo is deduced om he escaling lemmas 2.4 and 2.5. Indeed, since bi u ca ions
o ixed poin s o he H´enon map a e known, we can use his in o ma ion di ec ly o
eco e ing bi u ca ions o single- ound pe iodic o bi s in he amily µ. We need only
o know ela ions be ween he pa ame e s o he escaled map (2.9) and he ini ial
pa ame e s (i.e., in ac , be ween Mand µ).
In he globally non-o ien able case, he ela ions be ween Mand µa e gi en by
o mula (2.10) om which we ind µas ollows
µ=−λky−α(1 + kβ1λkx+y−)−1
d(M+sno
0+ ˆρ1
k)λ2k,(2.34)
whe e ˆρ1
k=O(kλk) is some small coe icien and α=cx+
y−−1 (see o mula (2.6)).
As i ollows om Lemma 2.4, he conse a i e non-o ien able H´enon map ¯x=
y, ¯y=M+x−y2,whe e Msa is ies (2.10), is no mal ( escaled) o m o he
i s e u n maps Tkwi h all su icien ly la ge k. This H´enon map has no ellip ic ixed
poin s, howe e , pe iod 2 ellip ic poin s exis s o 0 < M < 1. Thus, we ob ain, by
(2.10), ha he i s e u n map Tkhas a ixed poin wi h mul iplie s ν1= +1, ν2=−1
(i.e. when M= 0) i
µ=µ±
k=−λky−α(1 + kβ1λkx+y−)−1
d(sno
0+ ˆρk)λ2k,(2.35)
and a pe iod 2 poin wi h mul iplie s ν1=ν2=−1 (i.e. when M= 1) i
µ=µ2−
k=−λky−α(1 + kβ1λkx+y−)−1
d(sno
0+ 1 + ˆρk)λ2k,(2.36)
74 2. NON-ORIENTABLE APMS WITH HOMOCLINIC TANGENCIES
Thus, he i s e u n map Tkhas in his case a pe iod wo ellip ic pe iodic poin when
µ∈e2
k, whe e e2
kis he in e al o alues o µwi h he bo de poin s µ=µ±
kand
µ=µ2−
k. E iden ly, i α6= 0, he in e als e2
kwi h su icien ly la ge and di e en kdo
no in e sec .
In he locally non-o ien able case, as ou ag eemen , we ake such a pai o ho-
moclinic poin s M+and M− ha he global map T1is o ien able (i.e. J(T1) = 1).
Then, e iden ly, he i s e u n maps Tk≡T1Tk
0will be o ien able o e en kand
non-o ien able o odd k(i.e. J(Tk) = (−1)k).
Conside i s he case o e en k. By Lemma 2.5, he no mal escaled o m o Tk
is he H´enon map ¯
X=Y, ¯
Y=M−X−Y2, whe e Msa is ies (2.22). We ind om
he e ha µis gi en by he ela ion
µ=−λky−α−1
d(M+so
0+ ˆρ2
k)λ2k,(2.37)
o e en k, whe e ˆρ1
k=O(kλk), αis gi en by (2.6) and so
0by (2.7). In his case, since
he H´enon map has pa abolic ixed poin o M=−1 and M= 3, we ob ain ha he
in e al ekwi h e en khas bo de poin s µ=µ+
kand µ=µ−
k, whe e
µ+
k≡λky−α−1
d(so
0−1 + ˆρ2
k)λ2k,(2.38)
µ−
k≡λky−α−1
d(so
0+ 3 + ˆρ2
k)λ2k.(2.39)
He e, he map Tkhas a ixed poin which is pa abolic wi h mul iplie s ν1=ν2= +1
o µ=µ+
kand wi h mul iplie s ν1=ν2=−1 o µ=µ−
kand ellip ic o µ∈ek.
No e ha he e k uns all su icien ly la ge e en in ege s and, e iden ly, i α6= 0, he
in e als ekwi h di e en su icien ly la ge e en kdo no in e sec .
Conside now he case o odd k. Then he map Tkis non-o ien able and i s no mal
escaled o m is he non-o ien able conse a i e H´enon map ¯x=y, ¯y=M+x−y2,
whe e Msa is ies (2.24). Then we ind ha he in e al e2
khas bo de poin s µ=µ−
k
and µ=µ2−
k, whe e
µ±
k≡ −λkˆα−1
d(sno
0+ ˆρ3
k)λ2k,(2.40)
µ2−
k≡ −λky−ˆα−1
d(sno
0+ 1 + ˆρ3
k)λ2k,(2.41)
whe e ˆα=cx+/y−+ 1 and sno
0is gi en by (2.8). I ˆα6= 0, he in e als e2
kwi h
su icien ly la ge and di e en (odd) numbe s kdo no in e sec . I comple es he
p oo o Theo em 2.1.
2.4. PROOF OF THE MAIN RESULTS 75
2.4.3 P oo o Theo ems 2.2 and 2.3
P oo o Theo em 2.2. 1) In he globally non-o ien able case, by (2.35) and (2.36) he
equa ions o bi u ca ion cu es L2+
kand L2−
k, which a e bounda ies o he domain E2
k,
can be w i en as ollows
L2+
k:µ=−λky−cx+
y−−1(1 + kβ1λkx+y−)−sno
0+. . .
dλ2k,(2.42)
L2−
k:µ−λky−cx+
y−−1(1 + kβ1λkx+y−)−1 + sno
0+. . .
dλ2k.(2.43)
Since λ2kλk, i means ha he domains E2
kwi h su icien ly la ge ka e no mu ually
c ossed and do no in e sec he axis µ= 0, i cx+6=y−. Thus, he domains do no
in e sec always in he cases wi h c < 0 ((as in Figu e 2.7 (a)–(c)). Howe e , a he
global esonance α= (cx+/y−−1) = 0, as i ollows om (2.42) and (2.43), all domains
E2
kwi h su icien ly la ge ka e mu ually c ossed and all o hem in e sec he axis µ= 0
(as in Figu e 2.7 (d) and (e)).
2) In he locally non-o ien able case we ha e, by (2.38) and (2.39), ha he domains
E2mha e bounda ies
L+
2m:µ=−λ2my−cx+
y−−1+1−so
0+. . .
dλ4m,(2.44)
L−
2m:µ=−λ2my−cx+
y−−1−3 + so
0+. . .
dλ4m,(2.45)
co esponding o he exis ence o a pa abolic single- ound pe iodic o bi wi h double
mul iplie +1 a (µ, α)∈L+
2mo wi h double mul iplie −1 a (µ, α)∈L−
2m. A he
same ime, by (2.40) and (2.41), he domains E2
2m+1 ha e bounda ies
L2+
2m+1 :µ=−λ2m+1y−cx+
y−+ 1−sno
0+. . .
dλ4m+2,(2.46)
L2−
2m+1 :µ=−λ2m+1y−cx+
y−+ 1−1 + sno
0+. . .
dλ4m+2,(2.47)
co esponding o he exis ence o a single- ound pe iodic o bi wi h mul iplie s +1 and
−1 a (µ, ˆα)∈L2+
2m+1 o a double- ound pe iodic o bi wi h double mul iplie −1 a
(µ, α)∈L2−
2m+1.
Thus, o he di eomo phisms unde conside a ion wi h c > 0, he global esonance
occu s a α= 0. I co esponds o such si ua ion when all he domains E2m(wi h
su icien ly la ge m) mu ually in e sec and in e sec he axis µ= 0 nea he o igin
α= 0, µ = 0, whe eas, he domains E2
2m+1 do no in e sec o di e en and su icien ly
la ge m, accumula e o he axis µ= 0 as m→ ∞ bu do no in e sec i , see Figu e 2.8
(a). O he wise, o he case c < 0, he global esonance occu s a ˆα=cx+/y−+ 1 =

76 2. NON-ORIENTABLE APMS WITH HOMOCLINIC TANGENCIES
Figu e 2.7: Elemen s o bi u ca ion diag ams o amilies µ,α in he globally non-o ien able case.
0 and co espond o he si ua ion when he domains E2
2m+1 mu ually in e sec and
in e sec he axis µ= 0 (nea he o igin ˆα= 0, µ = 0), whe eas, he domains E2mdo
no in e sec o di e en and su icien ly la ge m, see Figu e 2.8 (b). I comple es he
p oo .
P oo o Theo em 2.3. Assume, o mo e de ini eness, ha d > 0 o all cases unde
conside a ion. The case d < 0 is ea ed in he same way.
1) Le 0be a globally non-o ien able map wi h α= 0. Then, o he one pa ame e
amily µwi h ixed α= 0, he in e als e2
kha e, by (2.35)–(2.36), a o m
e2
k= (−1−sno
0,−sno
0)λ2k
d.
E iden ly, i −1< sno
0<0, hese in e als will be nes ed and con aining µ= 0. I
implies ha he di eomo phism 0has in ini ely many double- ound ellip ic pe iodic
o bi s.
2.4. PROOF OF THE MAIN RESULTS 77
Figu e 2.8: Elemen s o bi u ca ion diag ams in he locally non-o ien able case: (a) o a amily
µ,α; (b) o a amily µ,ˆα.
2) In he locally non-o ien able case, le 0ha e cx+=y−and bc =−1, i.e. α= 0
and he global map T1is o ien able. In he amily µwi h ixed α= 0, by (2.44)–(2.45),
he in e als e2mha e a o m
e2m= (−3−so ,1−so )λ4m
d.
These in e als o alues o µwill be nes ed and all con ain µ= 0 o all su icien ly
la ge mi −3< so <1. I implies ha he di eomo phism 0has in ini ely many
single- ound ellip ic pe iodic o bi s.
In he locally non-o ien able case, le now 0be such a map ha cx+=−y−and
bc =−1, i.e. ˆα= 0 and he global map T1is o ien able again. Conside he one
pa ame e amily µwi h ixed ˆα= 0. Then, by (2.46)–(2.47), he in e als e2
2m+1+
ha e a o m
e2
2m+1 = (−1−so ,−so )λ4m+2
d.
I −1< sno <0, hese in e als a e nes ed and all con ain µ= 0. I implies ha he
di eomo phism 0has in ini ely many double- ound ellip ic pe iodic o bi s.
3) Fo he globally non-o ien able case wi h α= 0, as ollows om Lemma 2.4, all
he i s e u n maps Tk(wi h su icien ly la ge k) a e educed o he same escaled
no mal o m – he non-o ien able H´enon map ¯x=y, ¯y=−sno
0+x−y2. I is well
known ha , o −1< sno <0, he pe iod 2 ellip ic poin o his map is gene ic i
sno 6=−1
2;−1
√2;−5
8. The excep ional cases ela es, espec i ely, o esonances 1 : 4,
1 : 3 and such ellip ic poin whose i s Bi kho coe icien is ze o.
Fo he locally non-o ien able case wi h α= 0, as ollows om Lemma 2.4, all
maps T2ma e educed o he same escaled no mal o m – he o ien able H´enon map
¯x=y, ¯y=−so −x−y2. I is well known ha , o −3< sno <1, his map has a
78 2. NON-ORIENTABLE APMS WITH HOMOCLINIC TANGENCIES
ixed ellip ic poin which is gene ic (KAM-s able) i so 6={0; −5
√4}. The excep ional
cases ela es, espec i ely, o he s ong esonances ψ=π/2 and ψ= 2π/3.
Fo he locally non-o ien able case wi h ˆα= 0, all he i s e u n maps T2m+1 a e
educed, by Lemma 2.5, o he he non-o ien able H´enon map ¯x=y, ¯y=−sno
0+x−y2
and, hus, i sno 6=−1
2;−1
√2;−5
8, he co esponding double ound ellip ic pe iodic
o bi s a e gene ic. I comple es he p oo .
Chap e 3
Bi u ca ions o cubic homoclinic
angencies in a ea-p ese ing maps
In his chap e we s udy bi u ca ions o cubic homoclinic angencies in wo-dimensional
symplec ic maps. We dis inguish wo ypes o cubic homoclinic angencies, and each
ype gi es di e en i s e u n maps de i ed o di e se cubic conse a i e H´enon maps
wi h qui e di e en bi u ca ion diag ams. In his way, we es ablish he s uc u e o
bi u ca ions o single- ound pe iodic o bi s in wo pa ame e gene al un oldings. No e
ha he esul s o his chap e gene alize o he conse a i e case he esul s o [Gon85]
ob ained o he dissipa i e case.
3.1 P eambles
Bi u ca ions o cubic homoclinic angencies in gene al se ing we e s udied in [Gon85],
see also [GK88, Ta 91, GST96a]. Mo e p ecisely, in [Gon85] he e we e s udied bi u ca-
ions o a (m+ 2)-dimensional C -smoo h low X0, whe e m≥1 and ≥6, sa is ying
he ollowing condi ions:
•X0has a saddle pe iodic o bi L0wi h mul iplie s λ1, . . . , λmand γsuch ha
|λm| ≤ ··· ≤ |λ1|<1<|γ|and ei he
λ1is eal and |λ1|>|λ2|; o
λ1=¯
λ2=ρeiϕ,ϕ6= 0, π,|λ1|>|λ3|;
• he saddle alue σ=|λ1||γ|<1;
• he s able Ws
0and uns able Wu
0in a ian mani olds o L0ha e a angency o he
second o de (i.e. he cubic homoclinic angency) along a homoclinic cu e Γ0.
In his chap e we conside he case σ= 1. Mo eo e , we es ic ou sel o wo-
dimensional symplec ic di eomo phisms whose bi u ca ions o cubic homoclinic an-
gencies will be s udied.
79
86 3. BIFURCATIONS OF CUBIC HOMOCLINIC TANGENCIES IN APMS
Figu e 3.4: (a) The main elemen s o he bi u ca ion diag am o he map (3.13). The codimension
2 bi u ca ion poin s bia e as ollows: b1— a nonhype bolic saddle ixed poin wi h double mul iplie
+1 exis s; b2— wo pe iod 2 cycles wi h double mul iplie +1 exis ; b1,2
3– wo pa abolic pe iod 2
cycles wi h double mul iplie s −1 and +1, espec i ely, coexis ; b4– wo pe iod 2 cycles wi h double
mul iplie −1 coexis . Examples o (b) symme ic (M1= 0) and (c) asymme ic (M16= 0) bi u ca ions
a e shown.

3.2. ON BIFURCATIONS OF PERIODIC ORBITS 87
Figu e 3.5: (a) The main elemen s o he bi u ca ion diag am o he he map (3.13). The codi-
mension 2 bi u ca ion poin s bia e as ollows: b1and b0
1co espond o he exis ence o a ixed poin
wi h double mul iplie −1, ze o i s Lyapuno alue and nonze o second one; b2– a iple (s able)
ixed poin exis s ; b3and b0
3– wo ixed poin s wi h mul iplie s (−1,−1) and (+1,+1) coexis ; b4–
wo ixed poin s wi h mul iplie s (−1,−1) coexis ; b5– wo pe iod 2 poin s wi h mul iplie s (−1,−1)
coexis .
88 3. BIFURCATIONS OF CUBIC HOMOCLINIC TANGENCIES IN APMS
He e he ollowing bi u ca ion cu es a e indica ed: L+is he line o conse a i e old
bi u ca ion ( he bi h o a pa abolic ixed poin wi h double mul iplie +1); L−is he
line o conse a i e pe iod doubling (connec ed wi h he appea ance o a ixed poin
wi h double mul iplie −1); C+
1,2is he line o conse a i e old bi u ca ion o pe iod 2
poin s; C−
1,2is he line o conse a i e pe iod doubling bi u ca ion o pe iod 2 poin s
(second pe iod doubling).
Fo cubic H´enon maps o o m ¯x=y, ¯y=M1−bx +M2y±y3, main bi u ca ions
we e s udied in [GK88]. Howe e , he main a en ion in [GK88] was paid o he
dissipa i e case |b|<1. Ne e heless, one can show ha he bi u ca ion scena ios
in conse a i e case (b= 1), i.e. o maps (3.13) and (3.14), a e qui e simila , in
many aspec s, o he case 0 < b < 1. Bu we need o emembe ha he maps unde
conside a ion a e a ea-p ese ing and, hence, a big speci ic p esen s he e.1Then he
main bi u ca ions (bi u ca ions ela ed o ixed poin s) a e as ollows.
Bi u ca ion scena io in he map (3.13), see Figu e 3.4. Fo (M1, M2)∈1 he map
(3.13) has only one ixed poin p1which is a saddle-plus (wi h mul iplie s λand λ−1,
whe e 0 <λ<1). The ansi ion 1⇒2co esponds o he bi h o a pai (saddle and
ellip ic) o ixed poin s. When (M1, M2) = b1, he ixed poin p1is a non-hype bolic
saddle wi h double mul iplie +1, and his poin alls in 2on o 3 ixed poin s ( wo
saddle and one ellip ic) unde a conse a i e cusp-bi u ca ion. The ansi ion 2⇒3
co esponds o a nondegene a e pe iod-doubling bi u ca ion o he ellip ic ixed poin .
Thus, o egion 3, he map (3.13) has 3 saddle ixed poin s ( wo saddle-plus and one
saddle-minus) and one pe iod wo ellip ic o bi . Fu he p ima y bi u ca ions, when
c ossing he cu es L+
2and L−
2, ela e o poin s o pe iod 2 and mo e and, he e o e,
we do no obse e hem (see e.g. [GK88]).
Bi u ca ion scena io in he map (3.14), see Figu e 3.5. Fo (M1, M2)∈1 he map
(3.14) has only one ixed poin ˆp1which is a saddle-minus (wi h mul iplie s λand λ−1,
whe e −1< λ < 0). The ansi ion 1⇒2 h ough he segmen (b1, b0
1) o he cu e
L−co esponds o he pe iod-doubling bi u ca ion o he saddle poin ˆp1: he poin
becomes ellip ic ixed one and a saddle pe iod wo o bi is bo n in i s neighbou hood.
A ansi ion 1⇒20(as well as 1⇒2”) implies he bi h (unde conse a i e old
bi u ca ion) a pai o saddle and ellip ic pe iod wo poin s. A ansi ion 20⇒2
co esponds o a pe iod-doubling bi u ca ion unde which he pe iod 2 ellip ic o bi
me ged wi h a saddle ixed poin and becomes a ixed ellip ic poin . Thus, in he
egion 2 he map (3.14) has an ellip ic ixed poin and a saddle pe iod wo o bi .
We also illus a e in Figu e 3.6 bi u ca ions happened when a passage o (M1, M2)-
alues a ound he poin b1. T ansi ions c oss he cu e L+(such as 2⇒3,20⇒30
e c) co espond o he appea ance o wo new ixed poin s, saddle and ellip ic ones,
unde a conse a i e old bi u ca ion. The ellip ic ixed poin s unde go pe iod-doubling
bi u ca ion a ansi ions 3⇒30,3⇒3”, 30⇒4e c. We no e ha a (M1, M2) = b2
a iple (s able) ixed poin exis s which alls in 3on o 3 ixed poin s ( wo ellip ic and
1 o example, he p esence o homoclinic and he e oclinic s uc u es is qui e usual phenomenon in
conse a i e dynamics e en in he case o simple bi u ca ions...
3.2. ON BIFURCATIONS OF PERIODIC ORBITS 89
Figu e 3.6: Bi u ca ions a ound he poin b1o he bi u ca ion diag am o Figu e 3.5. Region HZ
(homo-he e oclinic zone) co esponds hose alues o (M1, M2) a which in a ian mani olds o all
saddles a e in e sec ed. In a ough app oxima ion, hese bi u ca ions a e simila bi u ca ions o wo
dimensional Hamil onian sys em whose po en ial unc ion is symme ic and changes as in he igu e.
one saddle).
3.2.2 Bi u ca ion Theo em
Due o he Rescaling Lemma 3.2, we can eco e bi u ca ions o single- ound pe iodic
o bi s in he ini ial amily µ1,µ2o symplec ic maps. As esul , we ob ain he ollowing
Theo em 3.2. 1) In any neighbou hood o he o igin in he (µ1, µ2)-plane, he e exis
in ini ely many bi u ca ion cu es L+
kand L−
kas well as Ck+
1,2and Ck−
1,2(see o mulas
(3.16)) which accumula e a he cu e B0 om Th. 3.1 as k→ ∞. The map Tk(µ)
has a pa abolic ixed poin wi h mul iplie s ν1=ν2= +1 a µ∈L+
kand a ixed poin
wi h mul iplie s ν1=ν2=−1a µ∈L−
k. I µ∈Ck+
1,2( esp., µ∈Ck−
1,2), hen he map
Tk(µ)has a pe iod wo poin wi h mul iplie s ν1=ν2= +1 ( esp., ν1=ν2=−1).
90 3. BIFURCATIONS OF CUBIC HOMOCLINIC TANGENCIES IN APMS
2) Fo any su icien ly la ge k, in he (µ1, µ2)-plane he e is a domain Ekbe ween
he cu es L+
kand L−
k(see Figu e 3.8) such ha he map Tk(µ)has a ixed ellip ic
poin a µ∈Ek. This poin is gene ic o all such µexcep hose ones which belong o
he cu es o s ong esonances when he mul iplie s a e ν1,2=e±iπ/2o ν1,2=e±i2π/3.
Figu e 3.7: Main elemen s o bi u ca ion diag am o he amilies µ1,µ2in di e en cases.
P oo . This heo em is a co olla y o he Rescaling Lemma 3.2 and he ansla ion o
he bi u ca ion scena ios o ixed poin s o i s e u n maps ( he conse a i e cubic
H´enon maps, see Sec ion3.2.1) o he co esponding single- ound pe iodic o bi s o
µ1,µ2.
We ind he equa ions o he cu es L+
k,L−
kand Ck+
1,2on he (µ1, µ2) -plane using
o mulas (3.15) and (3.10). We ob ain he ollowing o mulae:
L+
k:µ1=λk(y−−cx++. . . )±2
p|d|(2 − 11x+)λk−µ2
3˜ν3/2
(1 + . . . ),
L−
k:µ1=λk(y−−cx++. . . )±2
3p|d|−(2 + 11x+)λk−µ2
3˜ν1/2
×
×(4 − 11x+)λk−µ2(1 + . . . ),
Ck+
1,2:µ1=λk(y−−cx+. . . )±2
p|d|−(4 + 11x+
1)λk−µ2
33/2
(1 + . . . )
in he case dλk>0,
Ck+
1,2:µ1=λk(y−−cx+. . . )±2
p|d|4 + 11x+λk+µ2
33/2
(1 + . . . ),
µ2λ−k(1 + . . . )≥ −4
3in he case dλk<0,
(3.16)
3.2. ON BIFURCATIONS OF PERIODIC ORBITS 91
Figu e 3.8: Bi u ca ion cu es o he map Tk(µ): a) he case d > 0, λk>0, b) he case
d < 0, λk>0. The shaded egion Ekwi h bounda ies L+
kand L−
kco esponds o hose
alues o µa which he map Tk(µ) has an ellip ic ixed poin . The sizes o a speci ic
“sp ing-a ea” zone in Ekhas an o de λ3k/2×λkin µ1×µ2.
whe e ˆν=sign(dλk). We do no w i e a o mula o Ck−
1,2because o i s la geness.
Some elemen s o he bi u ca ion diag am o he amily µ1,µ2a e shown in Fig-
u e 3.7 o di e en cases. The domains Eko s abili y o single- ound ellip ic pe iodic
o bi s, wi h bounda ies L+
kand L−
k, a e illus a ed in Figu e 3.8. No e ha ypical sizes
o s abili y “sp ing-a ea zones” (nea he bi u ca ion poin µ∗
k) ha e o de λ3k/2×λk
and can be obse ed in nume ical expe imen s, [GSV13], (whe eas, such zones nea
quad a ic homoclinic angencies a e e y na ow, wi h wid h ∼λ2k).
No e also ha local bi u ca ions a s ong esonances ( o ixed poin s o Tkwi h
mul iplie s ν1,2=e±iπ/2o ν1,2=e±i2π/3) a e no degene a e. Such bi u ca ions we e
s udied in [Gon05] o cubic H´enon maps and we can apply he co esponding esul s
o ou case. See also Appendix A.

92 3. BIFURCATIONS OF CUBIC HOMOCLINIC TANGENCIES IN APMS
Chap e 4
Fini ely smoo h no mal o ms o
saddle a ea-p ese ing maps
4.1 P eambles
In his chap e we p o e he main echnical esul s, Lemmas 1.1, 1.2 and 2.2. No e ha
Lemma 2.3 is p o ed in he same way as Lemma 1.2 and we omi i s p oo . The p oo s
o Lemmas 1.1 and 2.2 di e only in some de ails. Howe e , due o he impo ance o
hese esul s, we p o e hese lemmas independen ly.
Be o e p o ing he lemmas, we ecall some necessa y ac s.
1) Conside a change o coo dina es (x, y)7→ (ξ, η) o he ollowing o m
ξ=∂V (x, η, ε)
∂η , y =∂V (x, η, ε)
∂x
whe e he unc ion V(x, η, ε) is some su icien ly smoo h unc ion o a iables x,ηand
pa ame e s εsa is ying condi ions
V(0,0,0) = 0,∂2V(0,0,0)
∂x∂η 6= 0.
I is well-known ha such a change is an a ea-p ese ing map (when x,ηand εa e
small and Vis su icien ly smoo h, C2a leas ) . I is called he canonical change o
coo dina es and he unc ion Vis called he gene a ing unc ion.
In wha ollows, we will make only canonical changes o coo dina es wi h canonical
unc ions o he o m V=xη(1 + O(|x|+|y|). Thus, in ac , we conside close o
iden ical and symplec ic changes, independen ly whe he he o ien a ion is p ese ed
o no .
2) Le Fεbe a pa ame e amily o wo-dimensional a ea-p ese ing maps which is
C -smoo h in bo h a iables and coo dina es. Le e e y map Fεha e a saddle ixed
poin Oεwi h eigen alues λ(ε) and γ(ε) such ha whe e |λ|<1 and |λγ|= 1. We
93
94 4. FINITELY SMOOTH NORMAL FORMS FOR SADDLE APMS
can always assume ha , o all su icien ly small ε, he ixed poin Oεis in he o igin
and ha he coo dina es, xand y, a e such ha he axes xand yco espond o he
p ope subspaces o eigen alues λand γ, espec i ely. Then he local map Tε≡Fε|U,
whe e Uis a small ixed neighbou hood o he o igin, can be w i en in he o m
¯x=λ(ε)x+φ(x, y, ε),¯y=γ(ε)y+ψ(x, y, ε)(4.1)
whe e unc ions φand ψand hei i s de i a i es in coo dina es anish a x=y= 0
o all small ε. In his case he equa ions o he local s able and local uns able mani olds
can be w i en as y=hs(x, ε) and x=hu(y, ε) , espec i ely, whe e hsand hua e C
and such ha
hs(0, ε) = ∂hs(0, ε)
∂x = 0, hu(0, ε) = ∂hu(0, ε)
∂y = 0.
I o make wo consecu i e changes o a iables o he o m ξ=x−hu(y, ε), η =y
and ξ=x , η =y−hs(x, ε), hen he map Tεis b ough o he ollowing o m
¯x=λ(ε)x+ (x, y, ε)x , ¯y=γ(ε)−1y+g(x, y, ε)y(4.2)
whe e (0,0, ε) = 0, gs(0,0, ε) = 0. Fo m (4.2) co esponds o he case whe e bo h he
local s able and local uns able in a ian mani olds o he poin Oεa e s aigh ened:
he equa ion o Ws
loc(Os) and Wu
loc(Os) a e y= 0 and x= 0 , espec i ely ( o all
su icien ly small ε). No e ha bo h he changes a e C -smoo h and canonical wi h
gene a ing unc ions V=xη −Rhu(η, ε) and V=xη +Rhs(x, ε), espec i ely.
3) Fo m (4.2) o he map Tεis mo e con enien han (4.1) bu i s use gi es some
echnical di icul ies. This is connec ed, in pa icula , wi h he ac ha ” oo much”
esonance e ms a e in he igh side o (4.2). Thus, he e is e y impo an he ques ion
on a educ ion o he map (4.2) o a mo e simple o m by means su icien ly smoo h
and a ea-p ese ing changes o coo dina es. I is clea ha he simples o m is he
linea o m o Tε. Bu only C1-linea iza ion is possible he e.
On he o he hand, o he analy ical case, J.Mose [Mos56] has been es ablished
ha he map T0may be educed o he ollowing no mal o m
¯x=λB(xy)x , ¯y=λ−1B−1(xy)y , (4.3)
whe e
B(xy)≡1 + β1·xy +β2·(xy)2+... +βn·(xy)n+...
and
B−1(xy)≡1 + ˜
β1·xy +˜
β2·(xy)2+... +˜
βn·(xy)n+...
a e se ies con e ging in some neighbo hood o he o igin. The Jacobian o (4.3) is equal
o one. Thus, coe icien s βiand ˜
βia e connec ed by some ela ions. Fo example,
β1=−˜
β1,˜
β2=β2
1−β2e c. Mo eo e , le
Bn(xy)≡1 + β1·xy +... +βn·(xy)n,
B−1
n(xy)≡1 + ˜
β1·xy +... +˜
βn·(xy)n(4.4)
4.2. PROOF OF LEMMAS 1.1 AND 1.2 95
be he segmen s o he se ies Band B−1, espec i ely. Then
∂Bn
∂x ·∂B−1
n
∂y −∂Bn
∂y ·∂B−1
n
∂x = 1 + O((xy)n+1)(4.5)
We will use hese p ope ies o he Bi kho -Mose no mal o m.
4) Lemma 2.1 o n= 1 has been p o ed in [GST07] whe e he exis ence o
he co esponding C -smoo h canonical changes we e de i ed, a ian s wi h C −1-
changes we e es ablished in [GS90, MR97, GS00]. Fini e smoo h no mal o ms, as
he ones om Lemma 2.2, nea saddle equilib ia o wo-dimensional lows we e ound
by E.A.Leon o ich [Leo51, Leo88]. He e we use, in ac , he Leon o ich me hod adap -
ing o he disc e e case. Howe e , we combine he Leo o ich me hod wi h he so-called
“A aimo ich me hod”, [A 84], when he exis ence o he app op ia e gene a ing unc-
ions is p o ed wi h using he no mal hype bolici y heo y [HPS77], i.e. we ind his
unc ion as an equa ion o some s ong s able (uns able) in a ian mani old. In [Leo88]
hese unc ions a e ound as solu ions o some homological equa ions.
4.2 Fini ely smoo h no mal o ms o symplec ic
saddle maps: he p oo o Lemmas 1.1 and 1.2
Resul s o his sec ion can be ea ed as an ex ension o he ini ely smoo h case o
he classical Mose ’s heo ems [Mos56] on he exis ence o analy ical no mal o ms o
a ea-p ese ing saddle maps.
4.2.1 P oo o Lemma 1.1
No e i s ha he main no mal o m (n= 1) o Lemma 1.1 was ound ea lie : he
exis ence o a C −1-smoo h no malized canonical change was p o ed in [GS00] and such
ype C -change was ound in [GST07]. The e o e we need o p o e he exis ence o
no mal o ms wi h n≥2. Howe e , we s a wi h he map T0in he ini ial o m (1.2).
This map is C and can be ep esen ed in he ollowing “n- h o de ex ended o m”
¯x=λ(ε)x{1 + α(1)
0(x, y, ε)+[β(1)
1+α(1)
1(x, y, ε)] ·xy +. . .
+[β(1)
n+α(1)
n(x, y, ε)] ·(xy)n}+O(xn+2yn+1),
¯y=λ−1(ε)y{1 + α(2)
0(x, y, ε)+[β(2)
1+α(2)
1(x, y, ε)] ·xy +. . .
+[β(2)
n+α(2)
n(x, y, ε)] ·(xy)n}+O(xn+1yn+2),
(4.6)
whe e β(ν)
1, . . . , β(ν)
na e some coe icien s, α(ν)
i≡[ϕ(ν)
i(x, ε)+ψ(ν)
i(y, ε)], i= 0, . . . , n, ν =
1,2, a e unc ions such ha ϕ(ν)
i(0, ε)≡0, ψ(ν)
i(0, ε)≡0. Since T0∈C , i ollows
om (4.6) ha α(ν)
i∈C −2i−1.
The lemma s a es ha he e exis canonical changes which annihila e unc ions
α(1,2)
i o i= 0,1, . . . , n. We will make hese changes consequen ly, s ep by s ep. Then
102 4. FINITELY SMOOTH NORMAL FORMS FOR SADDLE APMS
Thus, a e he canonical changes wi h he gene a ing unc ions V(i)
1and V(i)
2 om
(4.22), he map Tε akes he ollowing o m
¯x=λ(ε)x{1 + β1(ε)·xy +... +βi(ε)·(xy)i}+
+˜
ψ(i)
1(y, ε)·xi+1yi+O(xi+2yi+1),
¯y=γ(ε)y{1 + ˜
β1(ε)·xy +... +˜
βi(ε)·(xy)i}+
+ ˜ϕ(i)
2(x, ε)·xiyi+1 +O(xi+1yi+2)
(4.28)
Le us show ha he equali y J(Tε)≡1 implies ˜
ψ(i)
1≡0 and ˜ϕ(i)
2≡0 . Really, we
may ep esen he map (4.28) as
¯x=λ(ε)xBi(xy) + ˜
ψ(i)
1(y, ε)·xi+1yi+O(xi+2yi+1),
¯y=γ(ε)yB−1
i(xy) ˜ϕ(i)
2(x, ε)·xiyi+1 +O(xi+1yi+2)(4.29)
whe e Biand B−1
ia e he segmen s o he Bikho -Mose no mal o m. Taking in o
accoun he p ope y (4.5), one can w i e Jacobian o (4.29) in he o m
J=±1+(i+ 1)(λ˜ϕ(i)
2(x, ε) + γ˜
ψ(i)
1(y, ε)) ·xiyi+O((xy)i+1)
I ollows om he e ha ˜ϕ(i)
2≡0 and ˜
ψ(i)
1≡0.
In he nono ien able case λγ =−1, he monomials o (2.2) wi h βi,˜
βi≡0 o odd
ia e no esonan . The e o e, hey can be killed (inside o he e e y co esponding
s ep o he p oo ) by he canonical polynomial coo dina e ans o ma ions wi h he
gene a ing unc ions ˜
Vi=xη +νi(xη)i+1. One can check ha i in (4.4) all e ms βi
and ˜
βi anish o odd i, excep o he las ones βnand ˜
βn o odd n, hen βn=−˜
βn.
Then he change wi h he gene a ing unc ions ˜
Vnkills bo h hese e ms simul aneously.
This comple es he p oo o he lemma.

Appendix A
On s uc u e o 1:4 esonances in
conse a i e H´enon-like maps
We s udy bi u ca ions o ixed poin s wi h mul iplie s e±iπ/2( he main 1:4 esonances)
in some conse a i e H´enon-like maps. We analyze he 1:4 esonance in he cases
o conse a i e gene alized H´enon maps (GHMs) and cubic H´enon maps. We ind
condi ions o nondegene acy o he co esponding esonances and gi e a desc ip ion o
accompanying bi u ca ions.
In oduc ion
The H´enon map [Hen76]
¯x=y, ¯y= 1 −bx +ay2,(A.1)
is one o he mos popula a i icial maps demons a ing a complica ed chao ic dynam-
ics. In he coo dina es xnew =−ax, ynew =−ay map (A.1) is w i en in he so-called
s anda d o m
¯x=y, ¯y=M1−M2x−y2,(A.2)
whe e M1=−aand M2=ba e new pa ame e s. Bo h maps (A.1) and (A.2) ha e
he cons an Jacobian, J=b, and, he e o e, hey a e degene a e wi h espec o
(And ono -Hop ) bi u ca ions o bi h o in a ian ci cles. Mo eo e , i we es ic ou -
sel es o he conse a i e case (J≡1), hen maps (A.1) and (A.2) a e degene a e again
wi h espec o bi u ca ions o ixed poin s wi h mul iplie s e±iπ/2. Howe e , i is well
known ha he s anda d H´enon map has also ”homoclinic o igina ion”. I appea s
as a model map o escaled i s e u n maps nea quad a ic homoclinic angencies
[GS73, GG04]. Thus, he degene acy shows ha map (A.2) is only ” i s app oxi-
ma ion” o he e u n map. The co esponding ”second app oxima ions”, so-called
gene alized H´enon maps o o m
¯x=y, ¯y=M1−M2x−y2+Rxy +Sy3,(A.3)
103
104 A. 1:4 RESONANCE IN CONSERVATIVE H´
ENON-LIKE MAPS
whe e Rand Sa e some coe icien s, we e de i ed in [GG00, GG04, GGT02, GSS02,
GST02] o a ious si ua ions wi h quad a ic homoclinic and he e oclinic angencies.
Usually, he coe icien s Rand S(some in a ian s o he homoclinic o he e oclinic
s uc u e) a e small and depend on ” ime o he e u n” k: i kis he numbe o
i e a ions o he di eomo phism such ha he image o an ini ial poin is in i s some
small neighbou hood, hen R=Rkand S=Sk end o 0 as k→ ∞.
When he ini ial homoclinic angency is cubic, he cubic H´enon maps o o m
¯x=y, ¯y=M1+M2y−Bx ±y3(A.4)
na u ally appea [GST96a] as no mal o ms o escaled i s e u n maps. The signs
”+” and ”−” co espond o di e en maps which appea , in u n, nea cubic angencies
o di e en ypes (see Figu e 3.1). Besides, he maps wi h ”+” and ”−” ha e a a he
di e en s uc u e o bi u ca ions [Gon85].
Main bi u ca ions o GHMs we e s udied in [GG00, GG04, Gon02, GKM05] and
bi u ca ions o he cubic H´enon maps we e conside ed in [Gon85, GK88]. In his
appendix we pay a en ion only o bi u ca ions o ixed poin s wi h mul iplie s e±iπ/2.
We explain a cha ac e o he conse a i e bi u ca ion in cases o he GHMs wi h
M2= 1, R = 0 and he cubic H´enon maps wi h B= 1.
A.1 The esonance 1:4in a ea-p ese ing maps
Le a plana a ea-p ese ing map ha e a ixed poin wi h mul iplie s e±iπ/2. Then, i
is well known [A n96], ha he co esponding complex local no mal o m is w i en as
¯
ζ=i(1 + β)ζ+D0
21ζ2ζ∗+D0
03ζ∗3+O(|ζ|4),(A.5)
whe e βis a pa ame e which cha ac e izes a de ia ion o he angle a gumen ϕo
mul iplie s o he ixed poin s om π/2 (ϕ > π/2 a β > 0 and ϕ < π/2 a β < 0 ),
he coe icien s D0
21 and D0
03 (depending on β) a e eal and, in gene al,
|D0
21|+|D0
03| 6= 0.(A.6)
This condi ion implies ha O(|ζ|4) e ms in (A.5) do no in luence on a cha ac e o
he local bi u ca ions. In his case, main de ails o econs uc ions o phase po ai s
can be desc ibed by means o he analysis o bi u ca ions in he ollowing low no mal
o m ˙
ζ= 4iβζ +Aζ|ζ|2+ζ∗3,(A.7)
whe e
A=−iD0
21
|D0
03|.(A.8)
Fo m (A.7) is a esul o embedding ou h deg ee o map (A.5) in o low up o e ms
o o de O(|ζ|4). The nondegene acy condi ion o he conse a i e 1 : 4 esonance
A.1. THE RESONANCE 1:4IN APMS 105
is |A| 6= 1. In his case, bi u ca ions o he i ial equilib ium o he Hamil onian
low (A.7) unde changing βlook as in Figu es 2 and 3 in he cases |A|>1 and
|A|<1, espec i ely.1When |A|= 1 in he c i ical momen , possible bi u ca ions can
be desc ibed wi hin wo-pa ame e amilies
˙
ζ= 4iβζ +i(A+µ)ζ|ζ|2+ζ∗3(A.9)
(whe e |A| ≡ 1) wi h pa ame e s βand µ. The co esponding bi u ca ional diag am
o low (A.9) is shown in Figu e 4.
Thus, e u ning o he case o map (A.5), we can desc ibe main econs uc ions o
phase po ai in he ollowing way:
1) In case |A|>1, he ixed poin Ois always ellip ic, bu when β > 0 wo pe iod
4 cycles appea in i s neighbo hood: one cycle is saddle and he o he is ellip ic (see
Figu e 2).
Figu e A.1: Bi u ca ion o he i ial equilib ium o |A|>1.
2) In case |A|<1, he poin Ois a saddle wi h 8 sepa a ices when β= 0. Main
bi u ca ions a e connec ed he e wi h a econs uc ion o pe iod 4 saddle cycles (see
Figu e 3).
M1<0 M1>0M1=0
Figu e A.2: Bi u ca ions o he i ial equilib ium o |A|<1.
3) In case |A|= 1 he e exis h ee bi u ca ion cu es L1, L2and L3which di ide
a neighbou hood o he o igin o he pa ame e plane (β, µ) on o h ee egions wi h
di e en local phase po ai s (see Fi u e 4). No e ha cu e L3co esponds o he
exis ence o pe iod 4 pa abolic poin .
1No e ha case D0
03 = 0 is no special. In his case (i also D0
21 6= 0), one can conside , ins ead o
(A.7), he low o o m ˙
ζ= 4iβζ +iζ|ζ|2+ε(β)ζ∗3(whe e ε(0) = 0) whose bi u ca ions o he ze o
equilib ium a e he same as in Figu e 2 (case |A|>1).
106 A. 1:4 RESONANCE IN CONSERVATIVE H´
ENON-LIKE MAPS
M1
S
L2
L3
L1
III
III
Figu e A.3: Bi u ca ion diag am in case |A|= 1.
We can apply hese heo e ical esul s o ou conc e e conse a i e maps: he gen-
e alized H´enon map and cubic H´enon maps.
A.2 Conse a i e gene alized H´enon maps
In he case o he conse a i e gene alized H´enon map
¯x=y;,¯y=M1−x−y2+Sy3.(A.10)
we ind ha i has a ixed poin wi h mul iplie s e±iπ/2a M1= 0. This poin is in he
o igin and he co esponding complex local no mal o m (A.5) has he coe icien s (see
[GKM05]): 8D0
21 = 1 + 3S, 8D0
03 =−1 + Sand, hus,
A=−i1+3S
1−Sand |A|= 1 + 4S
1−S.
The e o e, |A|>1 i S > 0 and |A|<1 i S < 0 ( ecall ha we conside case o small
S). In his case, local bi u ca ions can be desc ibed by means o low no mal o ms
(A.7) o (A.9) whe e β=M1/2 + O(M2
1) and µ= 4S+O(S2).
A.3. CONSERVATIVE CUBIC H´
ENON MAPS 107
A.3 Conse a i e cubic H´enon maps
Conside now he conse a i e cubic H´enon map o o m
¯x=y, ¯y=M1+M2y−x+y3(A.11)
I has one ixed poin M∗wi h mul iplie s e±iπ/2a alues o pa ame e s M1and M2
belonging o he cu e L+
π/2whose equa ion is
M1=±2 −M2
31−1
3M2.(A.12)
The coe icien s o he local complex no mal o m (A.5) a e
8D0
21 = 3 −3M2,8D0
03 = 1 + 3M2
and, hus,
A=−i3−3M2
|1+3M2|.
Since M2≤0, i implies ha |A|is always g ea e han 1 he e. The main local
bi u ca ions which occu he e a ansi ion o he pa ame e s c oss he cu e L+
π/2a e
shown in Figu e 5.
Conside now he ollowing cubic H´enon map
¯x=y, ¯y=M1+M2y−x−y3.(A.13)
I has a ixed poin M∗∗ wi h mul iplie s e±iπ/2when he pa ame e s M1, M2belong
o he ollowing cu e L−
π/2
M1=±2 M2
31−M2
3.(A.14)
No e ha cu e L−
π/2has a sel -in e sec ion poin (M1= 0, M2= 3), and only in his
momen he map has simul aneously wo ixed poin s wi h mul iplie s e±iπ/2. The
coe icien s o he local complex no mal o m (A.5) a e
8D0
21 =−3+3M2,8D0
03 =−1−3M2
and, hus
A=−i3−3M2
1+3M2
.(A.15)
I implies ha he e a e wo poin s P+and P−on L−
π/2(wi h M2=1
3and M1= 16/27
and M1=−16/27, espec i ely,) whe e |A|= 1. Mo eo e , |A|<1 i M2>1
3and
|A|>1 i 0 ≤M2<1
3.

108 A. 1:4 RESONANCE IN CONSERVATIVE H´
ENON-LIKE MAPS
Figu e A.4: Local bi u ca ions a ansi ion o he pa ame e s c oss he cu e L+
π/2in he case o
map (A.11).
Then, in acco dance o he obse a ion abo e o conse a i e bi u ca ions a 1:4
esonance, we can desc ibe a cha ac e o hese bi u ca ions in he case unde consid-
e a ion. We explain his wi h he help o Figu e 8 whe e h ee bi u ca ional cu es
L−
π/2,L3and L+1 a e shown. The cu e L3co esponds o he exis ence o a pe iod 4
pa abolic poin nea he ixed poin . The cu e L+1 co esponds o he appea ance o
a pa abolic ixed poin , i s equa ion is
M1=±2
3M2−2
33/2
.
No e ha we es ic ou sel by he conside a ion o a small neighbou hood o he cu e
L−
π/2, This neighbou hood is di ided by he cu es L−
π/2,L3and L1in o 16 domains
o alues o pa ame e s M1and M2. We show in Figu e 8 he co esponding phase
po ai s.
A.3. CONSERVATIVE CUBIC H´
ENON MAPS 109
Figu e A.5: Local bi u ca ions a ansi ion o he pa ame e s c oss he cu e L+
π/2in he case o
map (A.11).
110 A. 1:4 RESONANCE IN CONSERVATIVE H´
ENON-LIKE MAPS
Pa II
Exponen ially small spli ing o
sepa a ices o whiske ed o i in
Hamil onian sys ems
111
118 EXPONENTIALLY SMALL SPLITTING OF SEPARATRICES
The exponen s q1,q2,q3,q4a e gi en by:
in he case n= 2
q1= 8, q2= 4, q3= 14, q4= 12,i ν= 0,
q1= 6, q2= 2, q3= 10, q4= 8,i ν= 1,
and in he case n= 3
i ν= 0 : q1= 12, q2= 6, q3= 20, q4= 17;
i ν= 1 : q1= 8, q2= 3, q3= 14, q4= 11;
No e ha in [DG04, DG03] a simila Hamil onian wi h 3 deg ees o eedom was con-
side ed. The mos accu a e esul s we e ob ained o ω= (1,Ω) wi h Ω = (√5−1)/2,
he so-called quad a ic golden numbe . and he exis ence o 4 ans e se homoclinic
poin s o he whiske ed o us was p o ed o all alues ε→0.
Due o he quasipe iodici y (5.11) o M(s, θ) and he o he unc ions in ol ed we
can es ic ou sel es in o he sec ion s= 0 and ede ine he unc ions as:
M(θ) := M(0, θ),L(θ) := L(0, θ), M(θ) := M(0, θ), L(θ) := L(0, θ),
and hen ex end he esul s ob ained o s= 0 o any s∈R.

Chap e 6
Exponen ially small spli ing o
sepa a ices o whiske ed o i wi h
quad a ic equencies
In his chap e we s udy he spli ing o in a ian mani olds o whiske ed o i in a
nea ly-in eg able Hamil onian sys em wi h 3 deg ees o eedom. We conside wo-
dimensional o i whose equency a ios a e quad a ic i a ional numbe s. We deal
wi h numbe s whose con inued ac ions sa is y ce ain a i hme ic p ope ies which
gi e us 24 cases o conside a ion. We show ha he Poinca ´e-Melniko me hod can
be applied o es ablish he exis ence o 4 homoclinic o bi s o he whiske ed o i and
p o e ha hese homoclinic o bi s a e ans e se. We also p o e he con inua ion o
hese homoclinic o bi s o he sil e numbe √2−1.
We conside he Hamil onian sys em (5.1-5.3) o n= 2. He e he equency ec o
ωis gi en by a quad a ic ec o
ω= (1,Ω),(6.1)
whe e he equency a io Ω is a quad a ic i a ional numbe , i.e. an i a ional eal
oo o a quad a ic polynomial wi h in ege coe icien s. I is well known ha quad a ic
equency ec o s sa is y he Diophan ine condi ion (5.4) wi h he exponen τ= 1.
6.1 Quad a ic equencies
6.1.1 Con inued ac ions o quad a ic numbe s
I is well known ha all he quad a ic i a ional numbe s Ω ∈(0,1), i.e. he eal oo s
o quad a ic polynomials wi h a ional coe icien s, ha e he con inued ac ions
Ω = 1
a1+1
a2+1
a3+. . .
= [a1, a2, a3, . . .] (6.2)
119
120 EXPONENTIALLY SMALL SPLITTING OF SEPARATRICES
ha a e pe iodic s a ing wi h some elemen ai. We conside only he numbe s wi h
he pu ely pe iodic con inued ac ions [a1, a2, a3, . . . , am] and deno e hem acco ding
o hei pe iodic pa by Ωa1,a2,a3,...,am, whe e a1, a2, a3, . . . , amis he co esponding
pe iodic pa o he con inued ac ion, and we say ha his con inued ac ion is m-
pe iodic. Fo example, he amous golden numbe is Ω1= [1] = (√5−1)/2, he sil e
numbe Ω2= [2] = √2−1. Ano he in e es ing case is ha o 2-pe iodic con inued
ac ions, as o example: Ω1,2= [1,2] = √3−1.
Rema k 6.1. The same esul s apply o pe iodic bu no pu ely pe iodic con inued
ac ions, say b
Ω = [b1, . . . , bn, a1, . . . , am], since o small enough ε, we only need o
conside he pe iodic pa o he con inued ac ion o b
Ω. We will call hese con inued
ac ions wi h he same pe iodic pa as equi alen con inued ac ions. These numbe s
ha e a ela ion o ype
b
Ω = a+bΩa1,...,am
c+dΩa1,...,am
wi h de a b
c d =±1.
6.1.2 A i hme ic p ope ies
F om he Diophan ine condi ion (5.4) we de ine he quan i y γk=|hk, ωi||k|. We aim
o ind wo-dimensional non-ze o in ege ec o s kwhich gi e he smalles alues γk,
we call hese ec o s kas p ima y esonances, and o s udy hei sepa a ion om he
o he ec o s, seconda y esonances.
We say ha he in ege ec o kis admissible i |hk, ωi| <1/2 and deno e by A he
se o admissible ec o s. We es ic ou sel es o he se A, since o any k /∈ A we
ha e |hk, ωi| >1/2 and γk≥ |k|/2.
I is a well known ac ha o equency ec o s he e exis s a unimodula ma ix T
(wi h in ege en ies and de e minan ±1) ha ing he equency ec o as an eigen ec o
wi h he associa ed eigen alue λo modulus g ea e han 1. In he wo-dimensional
case, such ma ix Tcan be de i ed om he con inued ac ion o he numbe Ω. Fo
ins ance, o he sil e numbe Ω2 he ma ix Tis
T=2 1
1 0 
(see Sec ion 6.3).
We de ine he ma ix U= (T−1)> ha sa is ies he ollowing equali y
hUk, ωi=hk, U>ωi=1
λhk, ωi.(6.3)
QUADRATIC FREQUENCIES 121
Thus, i k∈ A, hen also Uk ∈ A. We say ha kis p imi i e i k∈ A bu U−1k /∈ A.
F om (6.3) we deduce ha kis p imi i e i and only i
1
2|λ|<|hk, ωi| <1
2.
Admissible ec o s can be p esen ed in o m
k0(j)=(− in (jΩ), j),
whe e j=Z {0}and in (a) is he closes in ege o a. We say ha an in ege jis
p imi i e i k0(j) is p imi i e. Le Pbe he se o p imi i e in ege s j.
Fo each p imi i e j, we de ine he ollowing esonan sequences o in ege ec o s:
s(j, n) = Un−1k0(j), n ≥1.(6.4)
I u ns ou ha such esonan sequences co e he whole se o admissible ec o s.
P oposi ion 6.1 (DG03).Fo any p imi i e j, he e exis s he limi
γ∗
j= lim
n→∞γs(j,n)=|hk0(j), ωi|K(j), K(j) := |k0(j)−hk0(j), ωi
hu, ωiu|,
and one has
(a) γs(j,n)=γj∗+O(λ−2n), n ≥1;
(b) |s(j, n)|=K(j)|λ|n−1+O(|λ|−n), n ≥1;
(c) (1 + Ω)|j|−a
2|λ|< γ∗
j<(1 + Ω)|j|+a
2|λ|, a =1
21 + |u|
|hu, ωi|
P o ided he bounds (c) o γ∗
j o each p imi i e j, we can selec he minimal o
hem, say γ∗
j0. We ge
γ∗= lim in
|k|→∞ γk= min
j∈P γ∗
j=γ∗
j0.(6.5)
The co esponding sequence s(j0, n) gi es us he p ima y esonances. Deno e s0(n) =
s(j0, n). We call seconda y esonances in ege ec o s belonging o any o he emaining
sequences s(j, n), j6=j0. We no malize he limi s γ∗
jdi iding by γ∗
j0
˜γ∗
j0= 1,˜γ∗
j=γ∗
j
γ∗
j0
.(6.6)
and de ine a pa ame e ˜γ∗∗ measu ing he sepa a ion be ween he p ima y and sec-
onda y esonances
˜γ∗∗ = min
j∈P {j0}˜γ∗
j
Rema k 6.2. We implici ly assume he hypo hesis ha he p imi i e j0is unique, and
hence ˜γ∗∗ >1. In ac , his happens o all he cases we ha e explo ed, p o ided we
choose he ma ix T sui ably.
122 EXPONENTIALLY SMALL SPLITTING OF SEPARATRICES
6.2 Asymp o ic es ima es
We need o show ha he Poinca ´e-Melniko me hod is applied in ou singula case
µ=εp(p > 0). To do his we p o ide asymp o ic es ima es (o lowe bounds) o he
dominan ha monics o he Melniko po en ial L(θ) and p o e ha he co esponding
dominan ha monics o he spli ing po en ial L(θ) o e come he emaining e ms as
well as he e o e m o he Poica ´e-Melniko app oach. I is con enien o us o
wo k wi h he scala unc ions Land L, bu we s a e ou main esul s in e ms o he
spli ing unc ion M( ecall M(θ) = ∂θL(θ)) whose alues coincide wi h he dis ance
be ween he in a ian mani olds o he whiske ed o us.
We pu ou unc ions and hde ined in (5.6) and (5.7) in o he in eg al (5.12) and
ge he Fou ie expansion o he Melniko po en ial
L(θ) = X
k∈Z {0}
Lkcos(hk, θi−σk)
wi h he Fou ie coe icien s
Lk=2π|hk, ˆωεi|e−ρ|k|
bsinh |π
2hk, ˆωεi|.(6.7)
Rema k 6.3. No e ha due o he p esence o sinh |·|in (6.7) coe icien s Lk u n
ou o be exponen ially small in ε.
Recall ha
|hk, ˆωεi| =|hk, b0ω
b√εi| =b0γk
b|k|√ε.
We p esen he coe icien s Lkin he exponen ial o m
Lk=αke−βk, k ∈Z2 {0},(6.8)
whe e
αk=4πb0γk
b2|k|√ε(1 −e−{πb0γk
b|k|√ε})
, βk=ρ|k|+πb0γk
2b|k|√ε.(6.9)
Thus, he la ges coe icien s Lka e gi en by he smalles exponen s βk. We can p esen
βkin mo e con enien o m
βk=Cµ
ε1/4gk(ε),(6.10)
whe e we w i e he unc ions gkin he o m
gk(ε) = √˜γk
2"ε
εk1/4
+εk
ε1/4#, ε1/4
k=Cµ
2ρ
√˜γk
|k|, Cµ= 2πb0
bργ∗
j0,˜γk=γk
γ∗
j0
.
(6.11)
ASYMPTOTIC ESTIMATES 123
He e we deno e Cµ o depend implici ly on µ, since band b0a e µ-close o 1. Indeed,
Cµis µ-close o he cons an
C0=q2πργ∗
j0(6.12)
wi h γ∗
j0gi en in (6.5). No ice ha he unc ions gkcon ain he main in o ma ion on
he size o βk.
Fo any ε ixed we ha e o ind he dominan e ms Lkand he co esponding
ec o s k. Since Lka e exponen ially small in ε, i is mo e con enien o wo k wi h
he unc ions gkwhose smalles alues co espond o o he la ges Lk. To his aim we
ep esen he unc ions gk,k∈Z2 {0}, in a igu e (see, o example, he Figu e 6.1 o
Ω2) and o e e y ε ixed we ind he ec o s Si=Si(ε), i= 1,2, . . . minimizing he
unc ions such ha
gS1(ε)≤gS2(ε)≤gS3(ε)≤. . .
Hence he dominan ha monics o L(θ) will be LS1,LS2,LS3, e c.
1
εn
εn+1 ε′n+1 ε′nεn−1
A2 = B0
A1
Figu e 6.1: G aphs o he unc ions gk(ε), k∈Z2 {0}, o Ω2using a loga i hmic scale o ε. The
ones wi h solid lines a e p ima y unc ions bgn(ε)
No e ha he unc ions gkha e hei minimum a ε=εkand he co esponding

124 EXPONENTIALLY SMALL SPLITTING OF SEPARATRICES
minimal alues a e gk(εk) = √˜γk. Recall (see he Sec ion 6.1) ha all he admissible
kcan be subdi ided in o he sequences o ec o s s(j, n) de ined in (6.4) acco ding
o hei limi alues ˜γ∗
j. The p ima y in ege ec o s kbelonging o he sequence
s0(n) = s(j0, n) play an impo an ˆole he e, since ˜γ∗
j0= 1, and, hus, hey gi e he
smalles gk(a leas nea he minimum poin s εs0(n)). Deno e he p ima y unc ions by
bgn(ε) := gs0(n)(ε)
and also he minimum poin s o bgnby εnand he in e sec ion poin s be ween bgnand
bgn−1by ε0
n. Hence, we ge he geome ic sequences o he poin s
εn:= εs0(n)=C0
2ρK(j0)|λ|n−14
=ε1
λ4n, ε0
n:= √εnεn−1=ε1
λ4n−2.(6.13)
No ice ha he ollowing scaling p ope y is ul illed:
bgn(ε) = bgn−1(λ4ε) = bg0(λ4nε).
This implies ha as a unc ion o ln ε, he g aph o gnis simply he g aph o g0 ans-
la ed a dis ance 4nln |λ|, hus, he ep esen a ion in Figu e 6.1 ( ha uses a loga i hmic
scale o ε) is 4 ln |λ|-pe iodic. Thus, i is su icien o d aw igu es o a wid h 4 ln |λ|
as in Figu es 6.2-6.2.2.
We de ine he cons an s ( he so-called le els)
Ai=1
2(|λ|i/2+|λ|−i/2).
No e ha A0=bgn(εn) = 1 is he minimal alue o bgnand, o i≥1, Aiis he alue o bgn
a he in e sec ions poin s o bgnand bgn+i, o example A1=bgn(ε0
n). Deno e by B0 he
minimal alue o he seconda y unc ions gs(j,n),j6=j0. I is clea ha B0= ˜γ∗∗ >1.
We a e in e es ed in he equencies Ω sa is ying he condi ion
B0≥A1,(6.14)
ha ensu es ha he mos dominan ha monic o all εis ound among he p ima y
esonances. Nume ical explo a ions indica e ha he condi ion (6.14) is sa is ied only
by a ini e numbe o 1 o 2-pe iodical con inued ac ions, namely, o he 24 quad a ic
i a ional numbe s
Ω1,...,Ω13,Ω1,2,...,Ω1,12.(6.15)
Rema k 6.4. I is clea ha , o example, Ω2,1also sa is ies he condi ion (6.14), bu
we can p esen i by means Ω1,2as
Ω2,1= [2,1] = [2,1,2] = Ω1,2
1 + 2Ω1,2
.
Thus, hey a e equi alen (see Rema k 6.1) and, o εsmall enough, he g aphs o gk(ε)
in he case o Ω2,1will be simila o he ones o Ω1,2.
ASYMPTOTIC ESTIMATES 125
We in oduce he pa ame e ind sa is ying he equa ion
1
2(|λ|ind/2+|λ|−ind/2) = B0.(6.16)
Then he condi ion (6.14) may be exp essed as ind ≥1.
6.2.1 Asymp o ic es ima es o he spli ing dis ance
Unde he condi ion (6.14), we can ensu e ha he unc ion gi ing he alues o he
minimum
h1(ε) = min
kgk(ε) = gS1(ε)
is gi en by he p ima y ec o s S1=s(j0,·) (see, o example, Figu e 6.2 o Ω2) and
we ind ha j0= 1 o he 24 numbe s o (6.15). We can ew i e he unc ion as (no e
ha ˜γ∗
j0= 1)
h1(ε) = bgn(ε) = 1
2 ε
εn1/4
+εn
ε1/4!, ε ∈[ε0
n+1, ε0
n], n ≥1,(6.17)
ex ended as a 4 ln |λ|-pe iodic unc ion o ln ε. This unc ion is con inuous o all
0<ε<ε0
1and min h1(ε) = h1(εn) = 1 and max h1(ε) = h1(ε0
n) = A1>1. No e ha
he ec o S1changes a he poin s ε0
n om s(1, n + 1) o s(1, n).
In he ollowing heo em we p o ide an es ima e o he maximal dis ance be ween
he s able and uns able in a ian mani olds in e ms o he maximum alue o he
spli ing unc ion M(θ). The maximum alue o Mis gi en by he mos dominan
ha monic ha ing he unc ion h1(ε) as exponen . The es ima e ob ained shows ha
he spli ing o sepa a ices exis s. No e ha we ha e in oduced he no a ion o ’∼’
jus be o e Theo em 5.1 in Chap e 5.
Theo em 6.1 ((Maximal) spli ing dis ance).Fo he Hamil onian sys em (5.1-5.7)
wi h n= 2, assume ha ε1and µ=εp,p > p∗wi h p∗= 2 i ν= 1 and p∗= 3 i
ν= 0, hen in he 24 quad a ic numbe s (6.15), he ollowing es ima e holds
max
θ∈T2|M(θ)| ∼ µ
√εexp −C0h1(ε)
ε1/4
whe e he cons an C0is de ined in (6.12) and he unc ion h1(ε)is he pe iodic unc ion
in ln εde ined in (6.17) which sa is ies min h1(ε)=1and max h1(ε) = A1>1.
6.2.2 Asymp o ic es ima es o he ans e sali y o he spli -
ing
In o de o show ha L(θ) has nondegene a e c i ical poin s, we need o conside a
leas 2 dominan ha monics o i s Fou ie expansion. Howe e , o some alues o εi
126 EXPONENTIALLY SMALL SPLITTING OF SEPARATRICES
1
εn
εn+1 ε′n+1 ε′nεn−1
A2 = B0
A1
h1 (ε)
Figu e 6.2: G aphs o he unc ions gk(ε) and h1(ε) de ined in (6.17) o Ω2(ind = 2 whe e pa ame e
ind is in oduced in (6.16)).
is necessa y o conside 3 and e en mo e dominan ha monics i he second and some
consecu i e ha monics a e o he same magni ude. Also i can happen ha he co e-
sponding ec o s S1and S2o he 2 dominan ha monics a e linea ly dependen , hus,
o p o e he nondegene aci y o he c i ical poin s we ha e o conside enough consec-
u i e dominan e ms LS1, LS2, . . . , LSm, LS0 o ha e 2 linea ly independen ec o s S1
and S0, while S2, . . . , Sma e dependen wi h S1( he numbe m≥1 depends on ε).
De ini ion 6.1. We will call LS1and LS0essen ial dominan ha monics i hey sa is y:
(i) LS1is he mos dominan ha monic,
(ii) S1and S0a e independen ,
(iii) i he e a e ha monics LS2, . . . , LSmsuch ha LS1≤LS2≤. . . ≤LSm≤LS0, hen
Si=ciS1, i = 2, . . . , m (6.18)
wi h cons an s ci>1.
We will de ine he numbe m(index o non-essen iali y) as
m=1,i he e a e no non-essen ial ha monics be ween LS1and LS0
l+ 1,i he e a e lnon-essen ial ha monics be ween LS1and LS0(6.19)
ASYMPTOTIC ESTIMATES 127
In he wo s cases o 24 numbe s (6.15), Ω13, Ω1,11, Ω1,12,m= 6 and ci=i o
i= 2,...,6. As we show la e , he e ms LS2, . . . , LSma e no ele an o he ans e -
sali y.
1
εn
εn+1
ε′n+1 ε′nεn−1
A2
B0
A1
h1 (ε)
h2 (ε)
Figu e 6.3: Func ions h1and h2 o Ω4(ind = 1.8245) in he loga i hmic scale o ε
The e o e, we de ine he unc ion
h2(ε) = gS0(ε),
whe e S0is he i s ec o linea ly independen wi h S1minimizing he unc ions gk.
Thus, h2(ε) is de ined by 2 ec o s S1and S0. No e ha S0=S0(ε) changes i ε a ies,
and la e we will discuss hese c i ical alues o εa which S0changes.
We s ess ha S1is always a p ima y ec o , whe eas S0can be ei he a p ima y
ec o o a seconda y one. Such si ua ions can be checked om he g aphics co e-
sponding o he 24 numbe s (6.15). We ha e ound wo di e en si ua ions:
(a) Ω1,Ω2,...,Ω13 – o all ε he ec o S0is also a p ima y ec o . Mo e p ecisely,
S0=s(1, n ±1), and he c i ical alues o εwhen S0changes a e ε0
n+1, εn, ε0
n(see,
134 EXPONENTIALLY SMALL SPLITTING OF SEPARATRICES
he ones p e ious o s(j, N) (n≥N) and he ones a e s(j, N) (n<N).
X
k∈{s(j,n)}|k|lLk=X
l1≥0|s(j, N +l1)|lLs(j,N+l1)+
N−1
X
l2=1 |s(j, N −l2)|lLs(j,N−l2).
The i s sum is
Σ1=P
l1≥0|s(j, N +l1)|lLs(j,N+l1)=P
l1≥1|s(j, N +l1)|lαs(j,N+l1)exp n−C0gs(j,N+l1)
ε1/4o
 |s(j, N)|lexp −C0
ε1/4gs(j,N)P
l1≥1|λ|l1lαs(j,N+l1)exp −C0Cλ
ε1/4p˜γ∗
jl1
whe e we used he exponen ial o m o Ls(j,n)(6.8), he inequali y (6.24) and he ac
ha |s(j, N +l1)|∼|λ|l1|s(j, n)|. Mo eo e , due o (6.9) we ha e
αs(j,N+l1)∼˜γ∗
j
|λ|l1|s(j, N)|√ε(1 −exp{− C˜γ∗
j
|λ|l1|s(j,N)|√ε}).
I we conside he sequence al1=|λ|l1lαs(j,N+l1), we ob ain ha
al1+1
al1
=|λ|(l1+1)lαs(j,N+(l1+1))
|λ|l1lαs(j,N+l1)∼|λ|(l1+1)l
|λ|l1l|λ|l1(1 −exp{− C˜γ∗
j
|λ|l1|s(j,N)|√ε})
|λ|l1+1(1 −exp{− C˜γ∗
j
|λ|l1+1|s(j,N)|√ε})≤ |λ|l,
since 1−exp{−x}
1−exp{−x/|λ|} ≤ |λ| o x > 0. The e o e, al1+1 ≤ |λ|lal1(o al1≤ |λ|l1la0,a0=
αs(j,N)) and he sum is bounded abo e by a geome ic se ies ha can be es ima ed by
he i s e m
Σ1 |s(j, N)|lαs(j,N)exp n−C0gs(j,N)
ε1/4oP
l1≥0|λ|l1lexp −C0Cλ√˜γ∗
j
2ε1/4l1
=|s(j, N)|lLs(j,n)1−|λ|lexp −C0Cλ√˜γ∗
j
2ε1/4−1
≤2|s(j, N)|lLs(j,N).
The inequali y is ue o εsmall enough (ε1/4< C0Cλp˜γ∗
jln(2|λ|l)/2).
Fo he second sum we p oceed analogously, using (6.25) and |s(j, N −l2)| ∼
|λ|−l2|s(j, N)|
Σ2=
N−1
P
l2=1 |s(j, N −l2)|lLs(j,N−l2)
 |s(j, N)|lexp −C0
ε1/4gs(j,N)N−1
P
l2=0 |λ|−l2lαs(j,N−l2)exp −C0Cλ
ε1/4p˜γ∗
jl2,
whe e
αs(j,N−l2)∼˜γ∗
j|λ|l2
|s(j, N)|√ε1−exp n−C˜γ∗
j
|λ|−l2|s(j,N)|√εo.

ASYMPTOTIC ESTIMATES 135
Conside ing al2=|λ|−l2lαs(j,N−l2)and ha ing
al2+1
al2
=|λ|−lαs(j,N−(l2+1))
αs(j,N−l2)∼ |λ|−l|λ|l2+1 1−exp n−C˜γ∗
j|λ|l2
|s(j,N)|√εo
|λ|l21−exp n−C˜γ∗
j|λ|l2+1
|s(j,N)|√εo ≤ |λ|−l|λ|
=|λ|1−l≤ |λ|,
since 1−exp{−x}
1−exp{−|λ|x}≤1 o |λ|>1, we deduce ha al2+1 ≤ |λ|l2a0=|λ|l2αs(j,N).
As be o e, we ob ain
Σ2 |s(j, N)|lLs(j,N)
N−1
P
l2=0 |λ|l2exp −C0Cλ
ε1/4p˜γ∗
jl2
=|s(j, N)|lLs(j,n)1
1−|λ|expn−C0Cλ√˜γ∗
j/(2ε1/4)o≤2|s(j, N)|lLs(j,N).
The bo h sums comple e he p oo o (b).
To p o e (c), we ecall ha o he 24 numbe s om (6.15) he wo essen ial domi-
nan e ms a e LS1and LS0whe e S1is p ima y while S0can be p ima y o seconda y.
Also he e can be non-essen ial e ms LSibe ween LS1and LS0. We conside 4 cases:
(1) S0is p ima y and no non-essen ial e ms (numbe s Ω1,Ω2,Ω3); (2) S0is p ima y
and he e a e non-essen ial e ms (numbe s Ω4,...,Ω13); (3) S0is seconda y and no
non-essen ial e ms (numbe Ω1,2); (4) S0is seconda y and he e a e non-essen ial e ms
(numbe s Ω1,3,...,Ω1,12
Then in e e y case we conside he sums o Ls(j,n) o he sequences o which S1,S0
and (in he case 2 and 4) Sibelong o, i.e. s(j0, n), s(j0, n), s(ji, n), n≥1. Acco ding
o (b) hese sums a e bounded by he dominan e m Ls(j,N)whe e s(j, N) is S1i
j=j0,S0i j=j0(cases 3 and 4), Sii j=ji(cases 2 and 4), espec i ely. Then
excluding hese dominan e ms om he sums, he co esponding uppe bounds a e
es ima ed by he nex dominan ha monic which is one o Ls(j,N−1) o Ls(j,N+1)
X
n≥1,n6=N|s(j0, n)|lLs(j0,n) |s(j0, N ±1)|lLs(j0,N±1)
X
n≥1,n6=N|s(j0, n)|lLs(j0,n) |s(j0, N ±1)|lLs(j0,N±1)
X
n≥1,n6=N|s(ji, n)|lLs(ji,n) |s(ji, N ±1)|lLs(ji,N±1)
One o hese ha monics coincides wi h LSm+2 , while he o he ones a e smalle , and,
hence, X
k∈{s(j,n)}
j=j0,j0,ji
k6=S1,S0,Si,i=1,...,m
|s(j, n)|lLs(j0,n)∼1
εl/4LSm+2 (6.26)
Fu he , we calcula e he sum o he emaining sequences s(j, n), j6=j0, j0, ji;n6= 1.
Due o (b), o each j he sum o he coe icien s Ls(j,n)o he same sequence is es ima ed
136 EXPONENTIALLY SMALL SPLITTING OF SEPARATRICES
essen ially by Ls(j,N)(no e ha N=N(j, ε), and o di e en j he numbe Nis
di e en ).
Σ3=X
j∈P
j6=j0,j0,ji
n≥1
|s(j, n)|lLs(j,n)X
j∈P
j6=j0,j0,ji
|s(j, N)|lLs(j,N)
=X
j∈P
j6=j0,j0,ji
|s(j, N)|lαs(j,N)exp −C0gs(j,N)(ε)
ε1/4.
Recall ha he se Pis he se o p imi i e numbe s jde ined in Sec ion 6.1. We
use he es ima es |s(j, N)| ∼ p˜γ∗
jε−1/4,gs(j,N)∼p˜γ∗
j o ε≈εs(j,N), and, o εsmall
enough (ε1/4≤Cp˜γ∗
j/ln 2), αs(j,N)≤√˜γ∗
j
2ε1/4. Also o p imi i e jwe apply he bound
(6.1) o γj, we will ha e o ˜γ∗
j:
(A|j|−B)/|λ| ≤ ˜γ∗
j≤A|j|+B, whe e A=1+Ω
2γ∗
j0
, B =a
2γ∗
j0
.
Hence, we ge
Σ3ε(l−1)/4X
j∈P
j6=j0,j0,jiq˜γ∗
jl+1 exp (−C0p˜γ∗
j
ε1/4)
ε(l−1)/4X
|j|≥¯
j
(pA|j|+B)l+1 exp (−C0pA|j|−B
p|λ|ε1/4).
We deno e ¯= min
j∈P
j6=j0,j0,ji
|j|. I u ns ou ha i sa is y ¯≤2 (and ¯ > 2 o Ω 6= Ω3,Ω1,2)
We can bound A|j|+B≤χ(A|j|−B) wi h a cons an χsa is ying 1 < χ ≤A¯−A+B
A¯−A−B
o |j| ≥ ¯−1, and, hence, he sum is es ima ed by he co esponding in eg al
Σ3ε(l−1)/4
∞
Z
¯−1
(√Ax −B)l+1 exp (−C0√Ax −B
p|λ|ε1/4)dx
=1
Aε(l−1)/4
∞
Z
A¯−A−B
(√u)l+1 exp (−C0√u
p|λ|ε1/4)du
=2(p|λ|)l+3
ACl+3
0
ε(l+1)/2
∞
Z
C0√A¯−A−B
√|λ|ε−1/4
(z)l+2 exp{−z}du
ASYMPTOTIC ESTIMATES 137
Thus, we ge he ollowing bounds
l= 0 Σ3(2ε1/2+ 2ε1/4+ 1) exp (C0√A¯−A−B
p|λ|ε−1/4)
l= 1 Σ3(6ε+ 5ε3/4+ 3ε1/2+ε1/4) exp (C0√A¯−A−B
p|λ|ε−1/4)
l= 2 Σ3(24ε3/2+ 24ε5/4+ 12ε+ 4ε3/4+ε1/2) exp (C0√A¯−A−B
p|λ|ε−1/4)
. . .
These dec easing bounds a e smalle han (6.26).
Finally, we conside non-admissible ec o s k o which γk≥ |k|/2. Also we ha e
γk=|hk, ωi||k|≤|k|2|ω|. F om (6.9), we ob ain
αk≤|k||ω|
√ε(1 −exp{−Cε−1/2}), βk≥ρ|k|+C
4ε1/2
The e o e,
X
k/∈A |k|lLk≤exp{−Cε−1/2/4}
√ε(1 −exp{−Cε−1/2})X
k/∈A |k|l+1 exp{−ρ|k|}.
This sum can be bounded om abo e by he sum o a geome ic se ies which is smalle
han (6.26). This comple es he p oo o Lemma.
6.2.4 Dominan ha monics o he spli ing po en ial
To s udy he nondegene a e c i ical poin s o he whole spli ing po en ial L(θ) ( he
simple ze os o he spli ing unc ion M(θ)) we can use as a i s app oxima ion he
es ima es ob ained in Lemma 6.1. We p o e ha assuming µ=εp, o a sui able p > 0,
he dominan ha monics don’ change essen ially in Li we add he e o e m o o de
O(µ2).
Recall ha M=∂θLand ake in o accoun ha Lis ˆωε-quasipe iodic, we can
conside he ollowing Fou ie expansion:
L(s, θ) = X
k∈Z2L∗
keihk,θ−ˆωεsi=X
k∈Z Lkcos(hk, θ −ˆωεsi−τk),
whe e Lk, τka e eal, Lk≥0 and Zis de ined in (5.8). Fo e e y k∈ Z, he exponen ial
and he igonome ic o ms a e ela ed by L∗
k=1
2Lke−iτk. Then he co esponding
Fou ie coe icien s o he spli ing unc ion M(s, θ) and he Melniko unc ion M(s, θ)
a e (in he exponen ial o m) M∗
k=ikL∗
kand M∗
k=ikL∗
k, espec i ely.
138 EXPONENTIALLY SMALL SPLITTING OF SEPARATRICES
Lemma 6.2. Assuming ε1,µ=εpwi h p>p∗, wi h p∗= 2 i ν= 1 and p∗= 3 i
ν= 0, one has:
(a) LS1∼µ
ε1/4exp{−C0h1(ε)
ε1/4},
LS0∼µ
ε1/4exp{−C0h2(ε)
ε1/4},
LSi∼ciµ
ε1/4exp{−ciC0h1(ε)
ε1/4}, i = 2, . . . , m;
(b) |τk−σk−s(0)hk, ˆωεi|  µ
εp∗, k =S1, S2, . . . , Sm, S0;
(c) P
k6=S1,S0,Si|k|lLk1
εl/4LSm+2 , i = 1, . . . , m, l ≥0.
P oo . The p oo is simila o he one o Lemma 5 in [DG04], he e we jus adap i o
he quad a ic numbe s om (6.15) The spli ing unc ion M(s, θ) can be de ined on
a complex domain: |Im s|<π
2−δ,|Im θ|< ρ, whe e δis a small educ ion ( o be
chosen). On his domain, he uppe bound (5.14), poin ed ou in Theo em 5.1 ( o he
wo-dimensional case), can be applied o he e o e m (5.13). Also we can deduce om
(5.13) ha he Fou ie coe icien s o he e o e m a e R∗
k=ik(L∗
k−µL∗
ke−is(0)hk,ˆωεi),
k6= 0. Taking modulus and a gumen , we ge
|Lk−µLk|  |R∗
k|
|k|,|τk−σk−s(0)hk, ˆωεi|  |R∗
k|
|k|µLk
.
Since Ris ˆωε-quasipe iodic, a s anda d esul (Lemma 11 o [DGS04]) can be applied
o i o ge bounds o i s Fou ie coe icien s:
|R∗
k|  µ2
δq3+µ2
δq4√εe−˜
βk(ε),˜
βk(ε) = (ρ−δ)|k|+(π/2−δ)b0γk
b|k|√ε.
We p esen he unc ion ˜
βkas
˜
βk(ε) = Cµ,δ√˜γk
2ε1/4"ε
εk1/4
+εk
ε1/4#,
whe e
ε1/4
k=Cµ,δ√˜γk
2(ρ−δ)|k|, Cµ,δ = 4(π/2−δ)b0
b(ρ−δ)γ∗
j0=C0+O(µδ−q2, δ).
The di e ence wi h (6.10) is ha o ˜
βkwe w i e π/2−δand ρ−δins ead o π/2 and
ρin βk(ε). In ac , we conside ˜
βkas a pe u ba ion o βk. Indeed, p oceeding as in
he p oo o Lemma 6.1, we ge o he mos dominan e m:
˜
βS1=C0h1(ε) + O(√ε, µδ−q2, δ)
ε1/4.
ASYMPTOTIC ESTIMATES 139
We can neglec he pe u ba ion e m i µδ−q2ε1/4,δε1/4. So we choose δ=ε1/4.
The smallness condi ions on µbecome µεq1/4( he condi ion con aining he exponen
q2can be igno ed, since q1≥q2+ 3). Then using (6.23), we conclude
|LS1−µLS1|  µ2
ε(q3−1)/4exp −C0h1(ε)
ε1/4.
The e o e, he e m |µLS1| ∼ µ
ε1/4exp{−C0h1(ε)
ε1/4}(es ima ed in Lemma 6.1) domina es
i µ
ε1/4µ2
ε(q3−1)/4.
I one akes µ=εp, he las condi ion is ul illed a p > (q3−2)/4. We ge p∗=
max{(q3−2)/4, q1/4}= (q3−2)/4, since q3−2≥q1. In ac , o ν= 0 we ha e
q3= 14 and hence p∗= 3, and o ν= 1 we ha e q3= 10 and p∗= 2. The emaining
s a emen s o (a) o he o he dominan e ms Si,i= 2, . . . , m and S0as well as he
pa (b) a e p o ed in a simila way.
To p o e (c), we bound he sum o |k|lLk,k6=Si, S0,i= 1, . . . , m, by he sum o a
geome ic se ies analogously as i was done in he p oo o Lemma 6.1.
6.2.5 Nondegene a e c i ical poin s o L
To p o e he nondegene ici y o he c i ical poin s o L(θ) we need o conside a
leas 2 essen ial dominan ha monics LS1and LS0(see De ini ion 6.1 o essen ial dom-
inan ha monics), also we ha e o ake in o accoun all he non-essen ial ha monics
LS2,...,LSm, i necessa y, ha LS1≥ LS2≥. . . ≥ LSm≥ LS0and he co esponding
ec o s S2, . . . , Sma e linea ly dependen wi h S1( ecall ha S1and S0a e indepen-
den ). Since hese ec o s a e linea ly dependen wi h S1, hey sa is y he ela ions
(6.18).
Fi s we conside he app oxima ion o Lby hese m+ 1 ha monics
L(m+1)(θ) =
m
X
i=1 LSicos(hSi, θi−σSi) + LS0cos(hS0, θi−σS0).
A e he linea change
ψ1=hS1, θi−σS1, ψ2=hS0, θi−σS0(6.27)
which can be w i en as
ψ=Anθ−b, whe e An=S>
1
(S0)>, b =σS1
σS0,
he unc ion L(m+1) becomes
K(m+1)(ψ) = LS1cos ψ1+
m
X
i=2 LSicos(ciψ1+4τi) + LS0cos ψ2,
whe e 4τi=σSi−ciσS1,i= 2, . . . , m.

140 EXPONENTIALLY SMALL SPLITTING OF SEPARATRICES
Lemma 6.3. 1. K(m+1)(ψ)has 4 nondegene a e c i ical poin s ψ(1) = (O(η),0), ψ(2) =
(O(η), π), ψ(3) = (π+O(η),0), ψ(4) = (π+O(η), π), whe e
η=0,i m= 1,
LS2/LS1,i m > 1(6.28)
and mis de ined in (6.19).
2. de D2K(m+1)(ψ(j)) = LS1LS0(1 + O(η)).
P oo . The c i ical poin s o K(m+1) a e he solu ions o he sys em:
LS1sin ψ1+
m
X
i=2
ciLSisin(ciψ1+4τi) = 0,LS0sin ψ2= 0.
F om he second equa ion we ge ψ2= 0 and ψ2=π, while he i s equa ion can be
w i en in he o m
sin ψ1=−
m
X
i=2
ciLSi
LS1
sin(ciψ1+4τi) = η (ψ1),
whe e (ψ1) is bounded wi h i s de i a i e
| (ψ1)| ≤ M=
m
X
i=2
ciLSi/LS21,| 0(ψ1)| ≤ N=
m
X
i=2
c2
iLSi/LS21.(6.29)
Hence, o η < 1
√M2+N2small enough he condi ion η2( 0)2+η2 2<1 is sa is ied
and, he e o e, by Lemma B.1 he e exis wo simple solu ions o he equa ion nea o
ψ1= 0 and ψ1=π.
These solu ions gi e ise o he 4 c i ical poin s ψ(1), ψ(2), ψ(3), ψ(4) o K(m+1).
The de e minan is easily compu ed. We ha e
de D2K(m+1)(ψ) = LS1LS0(cos ψ1−η 0(ψ1)) cos ψ2
o any ψ∈T2. Hence, a he c i ical poin s we ge
|de D2K(m+1)(ψ(i))|=LS1LS0(1 + O(η)).
Rema k 6.6. No e ha i he e a e no non-essen ial ha monics be ween LS1and
LS0, i.e. m= 1 and S0=S2, hen we pu η= 0 and he c i ical poin s a e ψ(1) =
(0,0), ψ(2) = (0, π), ψ(3) = (π, 0), ψ(4) = (π, π).
ASYMPTOTIC ESTIMATES 141
A e ha ing s udied he c i ical poin s o he app oxima ion K(m+1) we ha e o
s udy hei pe sis ence i we add he emainde . Applying he linea change (6.27) o
he whole spli ing po en ial L(θ), we ge
K(ψ) = K(m+1) +LSm+2 G(ψ1, ψ2),
whe e he e m LSm+2 G(ψ1, ψ2) co esponds o he sum o all he non-dominan e ms
o L(i is bounded acco ding o pa (b) o Lemma 6.2) and LSm+2 ( he nex a e
LS0) is he maximum o non-dominan ha monics. No e ha unc ion Gis ob ained
ia he linea change (6.27) applied o he emaining e ms o L. Taking in o accoun
he bound (c) o Lemma 6.2 o he non-dominan e ms, one ge s he ollowing bounds
o G:|G|  1,|∂ψ1G|  ε−1/2,|∂ψ2G|  ε−1/2,
|∂2
ψ1ψ1G|  ε−1,|∂2
ψ1ψ2G|  ε−1,|∂2
ψ2ψ2G|  ε−1.(6.30)
Lemma 6.4. I η= max η, LSm+2 /LS0ε2,ηis gi en in (6.28), hen he unc ion
K(ψ)has 4 c i ical poin s, all nondegene a e: ψ(j)
∗=ψ(j),0+O(η),j= 1,2,3,4, whe e
ψ(1),0= (0,0),ψ(2),0= (π, 0),ψ(3),0= (0, π),ψ(4),0= (π, π). A he c i ical poin s,
|de D2K(ψ(j)
∗)|=LS1LS01 + Oη
ε.
P oo . The c i ical poin s o Ka e he solu ions o
sin ψ1=η (ψ1) + η0g1(ψ1, ψ2),sin ψ2=η00g2(ψ1, ψ2),(6.31)
whe e η co esponds o non-essen ial dominan e ms ( he same as in he p oo o
Lemma 6.3), η0=LSm+2
LS1,η00 =LSm+2
LS0a e exponen ially small (no e ha η0< η00 1)
and he unc ions g1(ψ) = ∂ψ1G(ψ) and g2(ψ) = ∂ψ2G(ψ) ha e bounds ob ained om
(6.30)
|g1| ∼ ε−1/2,|∂ψ1g1| ∼ ε−1,|∂ψ2g1| ∼ ε−1,
|g2| ∼ ε−1/2,|∂ψ1g2| ∼ ε−1,|∂ψ2g2| ∼ ε−1.
Thus, η= max{η, η00}and he condi ions o Lemma B.2 becomes
(η (ψ1) + η0g1(ψ))2+ (|η 0(ψ1) + η0∂ψ1g1(ψ)|+|η0∂ψ2g1(ψ)|)2<1
(η (ψ1) + η00g2(ψ))2+ (|η 0(ψ1) + η00∂ψ1g2(ψ)|+|η00∂ψ2g2(ψ)|)2<1
Due o η0< η00, he second inequali y implies he i s one. We ecall also ha | | ≤ M
and | 0| ≤ N,Mand Na e he cons an s de ined in (6.29). Then he condi ions a e
sa is ied i
η < ε
pε(Mε1/2+ 1)2+ (Nε + 2)2,
142 EXPONENTIALLY SMALL SPLITTING OF SEPARATRICES
and his inequali y is ue o ηε, hus, K(ψ) has 4 nondegene a e c i ical poin s.
Mo eo e , since he le pa s o (6.31) a e small in η, he c i ical poin s a e a pe u -
ba ion o he poin s ψ(j),0,j= 1,2,3,4.
The de e minan is
de (D2K(ψ)) = (−LS1cos ψ1+ηLS1 0+η0LS1∂ψ1g1)(−LS0cos ψ2+η00LS0∂ψ2g2)
−η0η00LS1LS0∂ψ2g1∂ψ1g2=LS1LS0(cos ψ1cos ψ2+O(η/ε)) .
(6.32)
I we choose ηε2, we ha e a small e m O(η/ε) in (6.32). Thus, subs i u ing he
c i ical poin s ψ(j)
∗,j= 1,2,3,4, we will ge he expec ed es ima es and comple e he
p oo o Lemma.
Rema k 6.7. In (6.32), we can choose ηεawi h any a > 1 ( o ins ance, ηε1.01
o ηε2013) o he alidi y o Lemma 6.4. Jus o simplici y, we chose ηε2. No e
ha ηis exponen ially small in εand, hence, can be bounded by a powe o ε.
We ansla e he esul s o Lemma 6.4 om he unc ion K(ψ) o he spli ing
po en ial L(θ). I is well known ha each c i ical poin o K(ψ) gi es ise o κc i ical
poin s o L(θ), whe e κ:= |de An|. We ha e checked ha o all he equencies
(6.15), κ= 1. Thus, he linea change (6.27) is one- o-one, and applying he in e se
change o i , we ob ain 4 nondegene a e c i ical poin s o L(m+1)
θ(j)
∗=A−1
n(ψ(j)
∗+b), j = 1,2,3,4.(6.33)
We also ind an es ima e o he minimal eigen alue (in modulus) m(j)o D2L(m+1) a
each c i ical poin .
Lemma 6.5. Assume η= max η, LSm+2 /LS0ε2,ηis gi en in (6.28). Then he
spli ing po en ial Lhas exac ly 4 c i ical poin s θ(j)
∗, gi en by (6.33), all nondegene a e
and sa is ying m(j)∼√εLS0.
P oo . We can p esen he minimal (in modulus) eigen alue o D2L(θ(j)
∗) in o m
m(j)=2|D|
|T|+√T2−4D,
whe e D= de D2L(θ(j)
∗) and T= D2L(θ(j)
∗). Thus, we need o ind es ima es o D
and T.
One can p o e ha D2L(θ) = (An)>D2K(ψ)An. The e o e, i D2K=k11 k12
k12 k22 ,
we see ha
D2L=k11S1·S>
1+k12[S1·(S0)>+S0·S>
1] + k22S0·(S0)>,
whe e k11 =LS1(−cos ψ1+η 0+η0∂ψ1g1),
k12 =LSm+2 ∂2
ψ1ψ2G=η0LS1∂ψ2g1=η00LS0∂ψ1g2,
k22 =LS0(−cos ψ2+η00∂ψ2g2).
ASYMPTOTIC ESTIMATES 143
A he c i ical poin s ψ(j)
∗,|k11|∼LS1,|k12|∼LSm+2  LS0,k22 =LS0 LS1. We ge
ha
T=k11|S1|2
2+ 2k12hS1, S0i+k22|S0|2
2,
he e we use he 2-no m |x|2=p(x1)2+ (x2)2, bu since he 1-no m and 2-no m a e
equi alen (|x|1/√2≤ |x|2≤ |x|1), we can apply he es ima es (6.23) o ob ain
|T| ∼ 1
√εLS1.
Mo eo e , since |de An|= 1, we ge om Lemma 6.4
|D|=|de D2K(ψ(j)
∗)|=LS1LS0(1 + O(η/ε)) ∼ LS1LS0.
Thus, ha ing |D|  T2, we conclude
m(j)∼|D|
|T|∼√εLS0.
6.2.6 P oo o Theo ems 6.1 and 6.2
Theo em 6.1, p. 125, is a consequence o he ac ha M(θ) = ∂θL(θ) and he es ima e
o he mos dominan ha monic LS1gi en in Lemma 6.2, p. 138. We conside he
app oxima ion L(m+1) by 2 essen ial dominan ha monics and deduce he ollowing
es ima es:
|∂θL(m+1)| ∼ |S1|LS1∼1
ε1/4LS1,|∂L−∂θL(m+1)| ∼ 1
ε1/4LSm+2 .
Since LSm+2  LS1, we ge he es ima e
|M| =|∂θL| ∼ µ
√εexp −C0h1(ε)
ε1/4.
Theo em 6.2, p. 130, ollows om Lemma 6.5 and ha he nondegene a e c i ical
poin s o L(θ) co espond o simple ze os o M(θ). Applying he es ima e o he
second essen ial dominan ha monics LS0 om Lemma 6.2, we ob ain he expec ed
es ima e o he minimal (in modulus) eigen alue o he spli ing ma ix ∂θM=D2L:
m∗∼√εLS0.
No e ha he esul o Lemma 6.5 applies only o η= max LS2/LS1,LSm+2 /LS0ε,
and his condi ion excludes om conside a ion some (dec easing as n→ ∞) neighbo -
hoods o he poin s εn(gi en in (6.13)), ε00
nand ε000
n(gi en in (6.21)), whe e he second
and he hi d essen ial dominan ha monics o Ma e o he same magni ude.