THE PARAMETERIZATION METHOD FOR
INVARIANT MANIFOLDS III:
OVERVIEW AND APPLICATIONS
XAVIER CABR´
E, ERNEST FONTICH, AND RAFAEL DE LA LLAVE
Abs ac . We desc ibe a me hod o es ablish exis ence and egula i y o
in a ian mani olds and, a he same ime o ind simple maps which a e
conjuga ed o he dynamics on hem. The me hod es ablishes se e al in-
a ian mani old heo ems. Fo ins ance, i educes he p oo o he usual
s able mani old heo em nea hype bolic poin s o an applica ion o he
implici unc ion heo em in Banach spaces. We also p esen se e al o he
applica ions o he me hod.
Index
1. In oduc ion
2. The pa ame e iza ion me hod
3. Main esul s
Pa I:
4. Analy ic one-dimensional s able mani olds
5. One-dimensional s able di ec ions a ound pe iodic o bi s o analy ic
di e en ial equa ions
6. A C0in a ian s able mani old heo em
Pa II:
7. C one-dimensional in a ian mani olds
8. A C0slow mani old heo em
9. Non- esonan in a ian mani olds o maps
10. Non- esonan in a ian mani olds o di e en ial equa ions
Appendix A. Rema ks on cohomology equa ions and nonuniqueness o in a ian
mani olds
Appendix B. His o ical ema ks and in o ma ion on he li e a u e on non-
esonan in a ian mani olds
1. In oduc ion
The goal o his pape is o p esen a u o ial on “ he pa ame e iza ion
me hod”, a echnique ecen ly in oduced by he au ho s [CFdlL03a, CFdlL03b]
o s udy in a ian mani olds o dynamical sys ems. As a i s simple applica ion,
he me hod allows o gi e quick p oo s o s able and uns able mani old heo ems.
1
2 X. CABR´
E, E. FONTICH, AND R. DE LA LLAVE
Mo e impo an ly, i leads o new esul s on exis ence o in a ian mani olds, as
well as on hei egula i y and dependence on pa ame e s.
To be mo e p ecise, he pa ame e iza ion me hod allows o es ablish he
exis ence o smoo h in a ian mani olds associa ed o linea subspaces in a ian
by he linea iza ion and which sa is y some non- esonance condi ions. As a
no el y wi h espec o p e ious wo ks, he in a ian linea subspaces need no
be spec al subspaces. E en u he , hey need no ha e an in a ian complemen .
The pa ame e iza ion me hod lends i sel o e y e icien compu e imple-
men a ions since i p o ides a global ep esen a ion o he mani old, and i also
allows a e y e icien discussion o dependence on pa ame e s.
Some ex ensions o he me hod o quasi-pe iodic sys ems and nume ical im-
plemen a ions a e p esen ed in [HdlL03b, HdlL03a]. In his pape we highligh
he main geome ic ideas and in a ian objec s ob ained, as well as he main ech-
nical ools (Banach spaces wi h no ms ailo ed o he p oblem, di e en iabili y
esul s o composi ion ope a o s, implici unc ion and ixed poin heo ems, co-
homology equa ions). Hence, we ha e no included he op imal egula i y esul s,
nei he he echnical ideas needed o ob ain hem. The in e es ed eade may
ind hese in [CFdlL03a, CFdlL03b].
Some a ian s o he me hod seem o ha e appea ed in agmen a y o m
in he wo k o Poinca ´e on au omo phic o ms [Poi90], la e in his esea ch on
dynamics, and also in he wo k on Lyapuno [Lya92]. O cou se, mode n ech-
niques such as implici unc ion heo ems on Banach spaces we e no a ailable a
ha ime, which made hese wo ks qui e agmen a y and ull o es ic ions. In
some pa icula applica ions (specially in ela ion wi h nume ical calcula ions),
he me hod seems o ha e been edisco e ed se e al imes, again unde ex a
es ic ions. In Appendix B we commen on hese his o ical ma e s.
In Sec ion 2 we desc ibe he basic ideas and objec s o he me hod, bo h
o dynamical sys ems gi en by maps and o hose gi en by o dina y di e en ial
equa ions. Sec ion 3 desc ibes he main esul o [CFdlL03a], s a ed bo h o
maps and o di e en ial equa ions.
We ha e ied ha each o he sec ions a e Sec ion 3 could be ead inde-
penden ly o each o he . Each o hem p esen s a ull p oo o one esul ha
illus a es some o he main ideas in ol ed wi h he me hod. The applica ions
in Pa I (sec ions 4,5, and 6) a e simple , while he esul s in Pa II (sec ions
7,8,9, and 10) a e sha pe and mo e delica e.
Finally, we ha e included wo appendices wi h commen s on cohomology
equa ions, nonuniqueness o in a ian mani olds (an impo an poin when doing
nume ical compu a ions), and his o ical ema ks on he li e a u e o he subjec ,
including applica ions.
2. The pa ame e iza ion me hod
2.1. The pa ame e iza ion me hod o maps. Gi en a map F:U⊂Rd→
Rdwi h F(0) = 0, whe e Uis an open se con aining he o igin, a na u al way
o y o ind a mani old in a ian unde Fand modeled on a subspace E⊂Rd,
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 3
is o look o an embedding K:U1⊂E→Rdand a map R:U1⊂E→U1in
such a way ha
F◦K=K◦R. (2.1)
The ac ha he mani old K(U1) passes h ough he o igin is ensu ed by
equi ing
K(0) = 0.(2.2)
The ac ha he mani old is angen a he o igin o Eis gua an eed by equi ing
DK(0)E=E. (2.3)
No e ha (2.1) ensu es ha he ange K(U1) o Kis in a ian unde F. We hink
o Kas gi ing a pa ame e iza ion o he in a ian mani old K(U1). Mo eo e ,
Ris he dynamics o F es ic ed o he in a ian mani old.
No e also ha di e en ia ing (2.1) a he o igin and using (2.2) and (2.3),
we deduce
DF(0)E⊂E.
Thus, Emus be an in a ian subspace unde he linea iza ion DF(0) o Fa
he o igin.
The ac ha Ris a ep esen a ion (in some app op ia e coo dina es) o
he dynamics o he map F es ic ed o he mani old, ells us ha we need o
conside i as pa o he objec s o be de e mined (o a leas o be lexible abou
i s choice) since, depending on he nonlinea e ms, he dynamics on he s able
mani old ( o ins ance on he classical s able mani old) may belong o di e en
equi alence classes unde smoo h conjugacy.
An impo an obse a ion is ha i we conside
T(F, K, R) := F◦K−K◦R, (2.4)
and w i e equa ion (2.1) as T(F, K, R) = 0, hen Tis di e en iable in Kwhen-
e e Kis gi en he opology o C spaces (p o ided ha Fis su icien ly di -
e en iable). Hence, equa ion (2.1) can be s udied ia he s anda d implici
unc ion heo em in Banach spaces, e en in he s anda d C spaces. This leads
e y quickly and painlessly o some esul s on exis ence and di e en iabili y wi h
espec o pa ame e s o ini ely di e en iable maps. This is he app oach unde -
aken in his a icle (wi h he excep ion o Sec ion 5, whe e we use he ixed poin
heo em o con ac ions). In [CFdlL03a] we used ins ead ixed poin heo y in
je spaces in o de o ob ain op imal egula i y esul s.
The ac ha Tis di e en iable is in con as wi h he unc ional equa ions
ha one has o deal wi h in he g aph ans o m me hod, in which he ope a o
whose ixed poin gi es he in a ian g aph is no di e en iable in any o he
classical C spaces, e en i i is di e en iable in spaces o analy ic unc ions (see
[Mey75]).
The linea ized e sion o (2.4) wi h espec o Kis o mally (and igo ously,
unde egula i y cases cha ac e ized below and in mo e gene ali y in [dlLO99]):
D2T(F, K, R)∆ = (DF ◦K)∆ −∆◦R. (2.5)
4 X. CABR´
E, E. FONTICH, AND R. DE LA LLAVE
The equa ions o ∆ ob ained se ing D2T(F, K, R)∆ = ηa e called cohomology
equa ions. We desc ibe some aspec s ela ed o hem in Appendix A. These
equa ions ha e a e y ich his o y (see e.g. [BdlLW96]). Once a heo y o he
linea ized equa ions is es ablished, one can s udy he ull nonlinea equa ion (2.1)
using a a ie y o me hods (con ac ion mappings, implici unc ion heo ems,
de o ma ion me hods, e c.)
2.2. The pa ame e iza ion me hod o lows. Ve y simila ideas o hose
used in he p oo o he esul s o maps can be used o s udy in a ian mani olds
o di e en ial equa ions
x0=X(x).
He e Xis a ec o ield in U⊂Rd, whe e Uis an open se con aining he o igin,
wi h X(0) = 0. I Eis a subspace o Rdin a ian by DX(0), we look o a
pa ame e iza ion K:U1⊂E−→ Rdand a ec o ield Rin U1⊂Esuch ha
X ◦ K=DK ·R. (2.6)
Tha is, we ask he ec o ield Xon he image o K o be he pull o wa d
o he ec o ield Rin E. Equa ion (2.6) exp esses ha a he ange o K,
he ec o ield Xis angen o he ange o K. Hence, he ange o Kis
in a ian unde he low o X. Mo eo e , he ec o ield Ris he ep esen a ion
in pa ame e s o he es ic ion o X o he in a ian mani old. This di ec
s udy, which will be desc ibed in Sec ion 10, is illumina ing and, o p ac ical
calcula ions ad an ageous. Ne e heless hose in e es ed mainly in exis ence and
egula i y esul s may p e e an abs ac a gumen which shows ha he esul s
o di e en ial equa ions ollow om he esul s o maps. This indi ec app oach
is as ollows.
I {ϕ } ∈Ris he low associa ed o Xand Wis a mani old such ha ϕ1(W)⊂
W( ha is, Wis an in a ian mani old o he map ϕ1), hen we ha e
ϕ1(ϕ (W)) = ϕ (ϕ1(W)) ⊂ϕ (W).(2.7)
I he in a ian mani old heo em o he map ϕ1includes local uniqueness unde
hypo hesis ha a e in a ian unde he e olu ion by ϕ , since om (2.7) we ob ain
ha ϕ (W) is also in a ian unde ϕ1, we conclude ha
ϕ (W)⊂W.
Tha is, Wis also in a ian unde ϕ , o all > 0.
We poin ou ha in all discussions in his sec ion i makes no di e ence o
eplace Rdby a Banach space X. In he ollowing sec ions we wo k in Rdexcep
in sec ions 6 and 8, whe e we deal wi h in a ian mani olds in Banach spaces.
3. Main esul s
In his sec ion we p esen he s a emen o wo heo ems on non- esonan
in a ian mani olds associa ed o a ixed poin . We ha e selec ed hem as ep e-
sen a i es o he esul s o he pape . Howe e , in he pape we deal wi h o he
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 5
esul s which ange om simple o mo e di icul si ua ions, and also wi h an
in a ian mani old heo em associa ed o pe iodic o bi s o ec o ields.
To s a e he esul s, we i s ecall some s anda d e minology. The spec um
o a linea ope a o Ain Rdwill be deno ed by Spec(A). We emphasize ha
Spec(A) deno es he spec um o he complex ex ension o A, and hence Spec(A)
is a compac subse o C. Fo j∈Nand S⊂C, we use he no a ion
jS := {a1+· · · +aj|ai∈S}.
We use a simila no a ion o sum, di e ence, and p oduc o se s.
We say ha a unc ion is in Cωi i is analy ic.
3.1. Mani olds associa ed o a ixed poin o a map. The heu is ic idea
is ha , gi en a map Fin Rd, wi h F(0) = 0, o e e y linea subspace E⊂Rd
in a ian unde DF(0), he e should co espond a smoo h mani old in a ian
unde he map F, passing h ough he o igin and angen he e o E. O cou se,
his should be ue i he map Fis smoo hly linea izable bu , as he s able (o
s ong s able) mani old heo ems show, he hypo hesis o linea izabili y is much
s onge han needed. Ne e heless, as shown in examples in [dlL97], some non-
esonance condi ions a e necessa y o he exis ence o an in a ian mani old.
The non- esonance condi ions (hypo hesis 3) in he ollowing heo em) con-
sis o ce ain hypo heses on he spec um o DF(0). They a e au oma ically
sa is ied when dealing wi h he s able o s ong s able mani old heo ems.
The ollowing is he esul conce ning non- esonan in a ian mani olds o
maps. I s p oo is gi en in Sec ion 9.
Theo em 3.1. Le F:U⊂Rd→Rdbe a C +1 map in a neighbo hood Uo
he o igin, wi h F(0) = 0, and ∈N∪ {ω}.
Deno e A=DF(0). Le L∈N,L≥1. Assume ha :
1) The e is a linea subspace Eo Rdsuch ha A(E)⊂E. Hence he e is
a decomposi ion Rd=E⊕Cand, wi h espec o i , Ahas he o m
A=AEB
0AC.(3.1)
2) kAEk<1.
3) Spec(AE)j∩Spec(AC) = ∅ o j= 2, . . . , L.
4) Ais in e ible.
5) (Spec(AE))L+1 Spec(A−1)⊂ {z∈C| |z|<1}.
6) L+ 1 ≤ .
Then, he e exis a C map K:U1⊂E→Rd, whe e U1is an open neighbo hood
o 0in E, and a polynomial R:E→Eo deg ee a mos L, such ha
F◦K=K◦Rin U1,
K(0) = 0, DK(0)E=E,
R(0) = 0, DR(0) = AE.
(3.2)

6 X. CABR´
E, E. FONTICH, AND R. DE LA LLAVE
Theo em 3.1 is a simpli ied, bu qui e close, e sion o he main esul o
[CFdlL03a]. In ha pape , he esul was s a ed in gene al Banach spaces ins ead
o jus in Rd. The egula i y esul s on dependence wi h espec o pa ame e s
we e s udied in [CFdlL03b].
We ema k ha he loss o one de i a i e in he egula i y o he mani old
as s a ed in Theo em 3.1 abo e (in which Fis a assumed o be C +1, bu he
mani old ob ained is jus C ), can be imp o ed. In [CFdlL03a] he mani old is
p o ed o ha e he same ini e di e en iabili y as he map, ha is, no de i a i e
is los . To ob ain his sha p esul one needs o conside equa ion (2.1) as a ixed
poin p oblem enjoying special p ope ies, and s udy in de ail he con e gence
o he limi — a he han applying he implici unc ion heo em as we do in he
p esen pape .
3.2. Mani olds associa ed o a ixed poin o a ec o ield. As in he
case o maps, i Xis a ec o ield in U⊂Rd, wi h X(0) = 0, and E⊂Rdis a
linea subspace in a ian unde DX(0), he e may co espond a smoo h mani old
in a ian unde he low o X, passing h ough he o igin and angen o Ea i .
The analogous ema ks o he p e ious subsec ion apply in his case.
The co esponding esul is he ollowing heo em. I s p oo is gi en in
Sec ion 10.
Theo em 3.2. Le Xbe a C +1 ec o ield on an open se Uo Rdwi h 0∈U,
such ha X(0) = 0 and ∈N∪{ω}. Le A=DX(0) and L∈N,L≥1. Suppose
ha :
1) The e is a linea subspace Eo Rdsuch ha A(E)⊂E. Hence he e is
a decomposi ion Rd=E⊕Cand, wi h espec o i , Ahas he o m
A=AEB
0AC.
2) Spec(AE)⊂ {z∈C|Re z < 0}.
3) jSpec(AE)∩Spec(AC) = ∅ o j= 2, . . . , L.
4) Spec(−A)+(L+1) Spec(AE) = {−λ+µ1+· · ·+µL+1 |λ∈Spec(A)and
µ1, . . . , µL+1 ∈Spec(AE)} ⊂ {z∈C|Re z < 0}.
5) L+ 1 ≤ .
Then, he e exis a C map K:U1⊂E−→ Rd, whe e U1is a neighbo hood o
0in E, and a polynomial R:E−→ Eo deg ee a mos L, such ha
X ◦ K=DK ·Rin U1,(3.3)
K(0) = 0, DK(0)E=E, (3.4)
R(0) = 0, DR(0) = AE.(3.5)
No e ha (3.3) ensu es ha he image K(U1) o Kis in a ian by he low
o X. Condi ion (3.4) ensu es ha K(U1) passes h ough he o igin and i is
angen o E he e.
We emphasize ha he subspace Eneed no ha e an in a ian complemen .
In addi ion, he spec um o AEand ha o ACneed no be disjoin , since
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 7
condi ion 3) is only equi ed o indices jbigge o equal han 2. Fo example,
he heo em applies o
A=





−2 1
0−2
−3
−5 1
0−5





.
Then, deno ing by Ei he i h coo dina e axis, we could associa e in a ian man-
i olds o E1, E3, E4,E1⊕E2,E4⊕E5, o o sums o hese spaces, e.g., E1⊕E4,
E1⊕E2⊕E4, e c.
On he o he hand, he esul does no apply o E1⊕E3since he eigen alues
co esponding o E1and E3when added gi e an eigen alue in he complemen .
Indeed, i we conside a non-linea pa e4x1·x3, i is easy o see ha he e is
no C2in a ian mani old angen o E1⊕E3.
PART I
4. Analy ic one-dimensional s able mani olds
In his sec ion we p o e a heo em ha se es as mo i a ion o se e al
esul s la e . Indeed, he main ideas o u u e esul s appea he e.
As a ma e o ac , a e y simila heo em had been p o ed by Poinca ´e
[Poi90], which used a e y di e en me hod ( he majo an me hod) om he
one used he e. Clea ly, he me hods used he e (Banach spaces and implici
unc ion heo em) we e no a ailable a he ime o [Poi90]. The heo em has
been edisco e ed in di e en guises in he li e a u e, specially in ela ion wi h
nume ical calcula ions (see Appendix B).
Theo em 4.1. Le F:U⊂Rd→Rdbe an analy ic map in a neighbo hood
Uo 0, wi h F(0) = 0. Le λ∈Rbe an eigen alue o A:= DF(0), and le
∈Rd {0}sa is y A =λ . Assume:
1) Ais in e ible.
2) 0 <|λ|<1.
3) λn/∈Spec(A) o e e y in ege n≥2.
Then, he e exis s an analy ic map K:U1⊂R→Rd, whe e U1is an open
neighbo hood o 0in R, sa is ying
F(K(x)) = K(λx)in U1,(4.1)
K(0) = 0, and K0(0) = . The e o e, he image o Kis an analy ic one-
dimensional mani old in a ian unde Fand angen o a he o igin. Mo e-
o e , he dynamics on he in a ian mani old is conjuga ed o he linea map
x7→ λx in he space o pa ame e s.
8 X. CABR´
E, E. FONTICH, AND R. DE LA LLAVE
In addi ion, i b
Kis ano he analy ic solu ion o F◦K=K◦λin a neigh-
bo hood o he o igin, wi h b
K(0) = 0 and b
K0(0) = βK0(0) o some β∈R, hen
b
K( ) = K(β ) o small enough.
Rema k 4.2. In he s a emen o he heo em, Spec(A) deno es he spec um
o A. No e ha , since 0 /∈Spec(A) by 1), we ha e ha Spec(A) excludes a ball
o adius ρ. By 2), he e is an in ege n0such ha |λ|n0< ρ. Now, i n≥n0
hen condi ion 3) holds.
Hence, hypo hesis 3), e en i i seems o equi e in ini ely many condi ions,
all bu n0o hem a e always ul illed. No e also ha n0is cons an in open
neighbo hoods o A. This shows ha hypo hesis 3) (which is called a non-
esonance condi ion) ails only on a mani old o maps Fo ini e codimension.
I u ns ou ha condi ion 3) is necessa y o ha ing a mani old as he one
claimed in he s a emen . Indeed, conside he map (x, y)7→ (1
2x, 1
4y+x2). An
in a ian mani old angen o he ec o (1,0) can be pu as a g aph o a unc ion
ϕ:U1⊂R→Rsa is ying he in a iance condi ion ϕ(x/2) = ϕ(x)/4 + x2. Such
ϕcan no be C2because aking wo de i a i es on bo h sides o he p e ious
condi ion we ge a con adic ion.
Rema k 4.3. By conside ing F−1and λ−1in place o Fand λ, hypo hesis 2)
in Theo em 4.1 can be changed o |λ|>1.
Rema k 4.4. No e ha we do no equi e λ o be a simple eigen alue. We
could ha e ha λhas o he eigen ec o s, linea ly independen o , o ha is
an eigen ec o in a non- i ial Jo dan block. In he la e case, no e ha he
in a ian eigenspace gene a ed by does no ha e an in a ian complemen .
Rema k 4.5. No e ha i F−1is en i e ( o ins ance, i F−1is a polynomial
as i happens o he H´enon map), hen Kis an en i e unc ion. Indeed, when
F−1is en i e, i Kis de ined on a ball Bρ, (4.1) shows ha K=F−1◦K(λ·) is
de ined on λ−1Bρ=Bλ−1ρ. Repea ing he a gumen , he domain o de ini ion
o Kbecomes he whole plane.
This was he mo i a ion in [Poi90], namely, o cons uc en i e unc ions
which sa is ied polynomial “duplica ion o angle o mulas” simila o he amilia
o mulas o sin, cos, o o ellip ic unc ions. The a gumen we p esen he e leads
o he cons uc ion o unc ions whose “double angle” alues can be exp essed
— h ough (4.1)— as a gi en unc ion o hose o he “single angle”.
One pa icula ly in e es ing case o he si ua ions co e ed by Theo em 4.1 is
when kAk<1, λis simple and i is he eigen alue o Acloses o he uni ci cle.
No e ha , upon i e a ion o A, he componen along is he one ha decays
mo e slowly and hence, he one which con ols he asymp o ic beha io . The
in a ian mani old associa ed o his eigen alue is a nonlinea analogue and can
also be used o s udy he asymp o ic beha io o he i e a es o F. I is usually
called a slow mani old.
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 9
P oo o Theo em 4.1. Using powe se ies, Fcan be conside ed as an analy ic
unc ion in a neighbo hood o 0 in Cd. Following he main idea o he pa ame-
e iza ion me hod, we y o ind K:D⊂C→Cdsuch ha
F◦K(z)−K(λz) = 0 (4.2)
o z∈D, whe e Dis he uni disk o C. We w i e F(ζ) = Aζ +N(ζ) wi h
A=DF(0).
I we y o sol e (4.2) equa ing powe s o zon bo h sides, we ob ain ha
K(z) = Pn≥1Knznshould sa is y
AK1=λK1
AKn+Rn(K1, . . . , Kn−1) = λnKn, n ≥2,(4.3)
whe e Rnis a polynomial exp ession ob ained expanding he composi ion in (4.2).
The i s equa ion in (4.3) does no de e mine K1comple ely, only ells us
ha K1is an eigen ec o o Awi h eigen alue λ. We ake K1 o be a mul iple o
such ha |K1|=δ, whe e δis small enough (we will gi e he p ecise smallness
condi ions on δla e , as hey appea in he p oo ).
Once we ha e chosen K1, (4.3) allows o de e mine in a unique ashion all
he o he Kn’s, as
Kn=−(A−λn)−1Rn(K1, . . . , Kn−1), n ≥2.(4.4)
The in e se used in (4.4) exis s by assump ion 3). One can indeed show, s udying
di ec ly he ecu sion in (4.4), ha he Kn hus de ined lead o an analy ic
unc ion. This was he app oach used in [Poi90].
We will, howe e , ollow ano he ou e o s udy (4.2). We use echniques o
unc ional analysis which can be adap ed o o he se ings, such as dealing wi h
a ini ely di e en iable F, o ea ing maps Fde ined in Banach spaces.
We w i e K(z) = K1z+K>(z), and ecall ha we ha e al eady picked K1
and ha i is small. The equa ion o K>(z) eads
AK>(z) + N(K1z+K>(z)) −K>(λz) = 0.(4.5)
We conside K>belonging o he Banach space Ho analy ic unc ions in he
uni disk, anishing a he o igin along wi h hei i s de i a i e, and wi h he
ollowing no m being ini e:
H=K>:D⊂C→Cd|K>(z) =
∞
X
n=2
Knzn,kK>k:=
∞
X
n=2
|Kn|<∞.
We ecall ha he analy ic unc ions :D⊂C−→ Csuch ha k k:=
Pn≥0| n|<∞ o m a Banach algeb a wi h he p e ious no m (see, e.g., [Ca 95]
o a s aigh o wa d p oo ). Indeed, i su ices o apply he iangle inequali y in
he exp ession o he coe icien s o he p oduc ( g)n=Pi+j=n igjand hen
sum in n. O cou se, he ideal H={ | 0= 1= 0}is also a Banach algeb a.
We can e o mula e (4.5) as an ope a o equa ion
T(K1, K>) = 0,(4.6)
16 X. CABR´
E, E. FONTICH, AND R. DE LA LLAVE
pa ame e iza ion o he pe iodic o bi and an eigen ec o o Φ1o eigen alue
eλ, hen he i s and all highe o de e ms a e uniquely de e mined.
We can now s a e he esul on in a ian mani olds a ound pe iodic o bi s.
Theo em 5.4. Le Xbe an analy ic ec o ield on Rdand assume ha i admi s
a pe iodic o bi γo pe iod T. Le γ=γ(θ),0≤θ≤1, be pa ame e ized
acco ding o (5.1). Le Φ1be he monod omy ma ix associa ed o γ. Tha is,
Φ1= Φ(1), whe e Φθ= Φ(θ)is he undamen al solu ion o (5.8).
Assume ha λ∈Rsa is ies:
1) eλ<1and eλ∈Spec(Φ1). Le ∈Rd {0}be a solu ion o Φ1 =eλ .
2) enλ /∈Spec(Φ1) o e e y in ege n≥2.
Then, he e exis s an analy ic wo-dimensional mani old in a ian unde he
low o X, con aining he pe iodic o bi γ, and angen o he space gene a ed by
{˙γ(θ), K1(θ)}a γ(θ),0≤θ≤1, whe e K1(θ) = eµ(1−θ)ΦθΦ−1
1 . In addi ion,
he mani old can be ep esen ed as he image o an analy ic unc ion Ksa is ying
(5.2),K(θ, 0) = γ(θ)and ∂K
∂σ (θ, 0) = K1(θ). Consequen ly, he mo ion in he
space o pa ame e s is gi en by (5.3).
In nume ical applica ions, i is o en enough o ollow he p ocedu e indi-
ca ed p e iously in his sec ion o ob ain e y app oxima e ep esen a ions o he
mani old. In he nex subsec ion, we gi e a comple e p oo o Theo em 5.4. In
pa icula , we de elop es ima es o he solu ions o equa ions o he o m (5.16)
and we s udy he con e gence o he powe se ies, which will gi e con idence in
he nume ical analysis. Rema k 5.5 explains how he main exis ence esul is an
‘‘a pos e io i” es ima e which jus i ies he esul s o nume ical calcula ions.
5.2. Con e gence o he o mal solu ions. In his subsec ion, we show ha
he solu ion p e iously ob ained is no jus a o mal solu ion, bu ha i con-
e ges and de ines an analy ic unc ion.
5.2.1. Fo mula ion as a ixed poin p oblem. The p oo o Theo em 5.4 consis s
in ew i ing (5.2) as a well-posed ixed poin p oblem. As in many o he p oo s
p esen ed in he pape , we accomplish his by sepa a ing a low pa om a high
pa o he solu ion. He e we sepa a e he linea pa , ha we al eady ha e ound.
Then, once we show ha he e is a ue analy ic solu ion, by he uniqueness o
he e ms o o de bigge o equal han 2, we see ha he o mal o de by o de
calcula ion has o p oduce his analy ic solu ion.
We w i e
K(θ, σ) = K0(θ) + K1(θ)σ+K>(θ, σ)
K≤(θ, σ) = K0(θ) + K1(θ)σ.
We assume ha K0and K1a e al eady de e mined (since we compu ed hem
jus in he i s wo s eps o he i e a i e p ocedu e), so ha he only unknown
is K>.

THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 17
Equa ion (5.2) becomes
1
T
∂
∂θ +λσ
T
∂
∂σ (K≤+K>) = X ◦ (K≤+K>),
ha we w i e as
1
T
∂
∂θ +λσ
T
∂
∂σ (K0+K1σ+K>) = X ◦ K0+DX ◦ K0(K1σ+K>) + H(K>)
(5.17)
whe e His he emainde o Taylo ’s heo em:
H(K>) := X ◦ (K0+K1σ+K>)− X ◦ K0−DX ◦ K0(K1σ+K>).(5.18)
Reg ouping e ms in (5.17) and using (5.5) and (5.6), (5.17) becomes
L+λσ ∂
∂σ K>=TH(K>).(5.19)
The plan o he p oo is o show ha unde he non- esonance condi ions o
Theo em 5.4, he ope a o L+λσ ∂
∂σ is boundedly in e ible in some app op ia ely
de ined spaces. Hence, (5.19) will become
K>=L+λσ ∂
∂σ −1
TH(K>) =: N(K>).(5.20)
Equa ion (5.20) will be analyzed by ixed poin me hods. We in oduce he
no a ion N o deno e he ope a o o which we will be seeking ixed poin s. We
will show ha he Lipschi z cons an o H, and hence o N, can be aken o be
small i we ake K1 o be small. This is easonable since His he second o de
emainde o he Taylo expansion.
I a he han applying he con ac ion mapping heo em, we apply o (5.19)
he implici unc ion heo em in Banach spaces, we ob ain au oma ically smoo h
dependence on pa ame e s o he in a ian mani olds.
Rema k 5.5. No e ha one o he conclusions o he ixed poin me hod is ha
i we ob ain an app oxima e solu ion o (5.20), ha is, a unc ion K>
ap such ha
kK>
ap − N(K>
ap)k ≤ δ, (5.21)
hen he e exis s a ue solu ion K>o he equa ion such ha
kK>−K>
apk ≤ Cδ,
whe e he cons an Cwill be a he explici om he p oo .
This can be used o jus i y nume ical calcula ions. A ca e ul nume ical im-
plemen a ion o he algo i hm discussed a he beginning o he sec ion p oduces
a unc ion K>
ap o which δin (5.21) is jus he ound o e o plus he unca ion
e o , which in se e al p ac ical applica ions —e.g. [Cap04]— can be made o
be a ew housand imes he ound-o uni . Hence, he p oo p esen ed he e will
ensu e ha he nume ically compu ed solu ions a e close o he ue one. This
p ocedu e is wha is usually called “a pos e io i es ima es” in nume ical analysis.
18 X. CABR´
E, E. FONTICH, AND R. DE LA LLAVE
Mo eo e , we es ablish ha he ue solu ion is close o he compu ed one in
an analy ic no m de ined below. Hence, he compu ed solu ion gi es in o ma ion
no only abou he loca ion o he s able mani old, bu also abou i s de i a i es.
This makes possible o discuss bi u ca ions, angencies, e c.
5.2.2. No ms. In his sec ion we in oduce no ms ha a e con enien o ca y
ou he con ac ion mapping a gumen .
Le αand βbe posi i e numbe s. We conside he se s
Sα={θ∈C/Z| |Im θ|< α}
Dβ={σ∈C| |σ|< β}
Uα,β =Sα×Dβ.
No e ha Sαcan be conside ed as a complex ex ension o he o us T1. Func ions
in Sαcan be iden i ied wi h unc ions o pe iod 1 de ined in a complex s ip.
We conside he space o unc ions
Γα={ :Sα→Cd| con inuous and analy ic in Sα}.
We endow Γαwi h he no m k kΓα= supθ∈Sα| (θ)|. As i is well known, his
no m makes Γαa Banach space.
Gi en a unc ion Kde ined on Uα,β which is analy ic in bo h a iables, we
w i e i as
K(θ, σ) =
∞
X
n=0
Kn(θ)σn,
and we deno e by Hα,β he space o analy ic unc ions o which
kKkHα,β :=
∞
X
n=0
kKnkΓαβn<∞.
I is an easy exe cise o check ha Hα,β is a Banach space wi h he no m k·kHα,β .
We will use he same no a ion o no ms o unc ions om hese domains
aking alues in o he spaces (e.g., ma ices). When we conside unc ions which
ake alues in spaces o which he e is a mul iplica ion (e.g., ma ix alued
unc ions and ec o alued unc ions), he spaces Γαand Hα,β inhe i Banach
algeb a p ope ies. Fo example, i mand Ma e d×dma ix alued unc ions
de ined in Sαand Uα,β espec i ely, and i and Va e d-dimensional ec o
alued unc ions de ined in Sαand Uα,β espec i ely, hen we ha e
km kΓα≤ kmkΓαk kΓα
kMV kHα,β ≤ kMkHα,β kVkHα,β .(5.22)
The i s inequali y in (5.22) is jus ha he sup emum o he p oduc is less
han he p oduc o he sup ema. The second inequali y is a consequence o he
well known p oduc o mula o powe se ies:
MV =
∞
X
n=0
σn
n
X
k=0
MkVn−k.
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 19
The e o e,
kMV kHα,β =
∞
X
n=0
βn
n
X
k=0
MkVn−kΓα
≤
∞
X
n=0
βkβn−k
n
X
k=0
kMkkΓαkVn−kkΓα
=∞
X
k=0
βkkMkkΓα∞
X
i=0
βikVikΓα.
Rema k 5.6. The e a e o he no ms ha we could ha e used in he p oo . Fo
example, we could ake he sup ema on bo h a iables, which in some espec s
is mo e na u al. The choice we ha e made is based on he obse a ion ha since
he a gumen is based in sol ing equa ions o each coe icien in σ, i is na u al
o use a no m ha emphasizes he ole o he coe icien s in powe s o σ. A he
same ime, he sup emum no m in he angle a iables makes simple o ob ain
es ima es o exp essions such as (5.11).
The inequali y
sup
(θ,σ)∈Uα,β
|K(θ, σ)| ≤ kKkHα,β (5.23)
is ob ious om he iangle inequali y. On he o he hand, using Cauchy in eg al
o mula
Kn(θ) = 1
2πi Z|z|=β
z−(n+1)K(θ, z)dz,
we ob ain
kKnkΓα≤β−nsup
(θ,σ)∈Uα,β
|K(θ, σ)|.(5.24)
This immedia ely gi es
kKkHα,β−δ≤βδ−1sup
(θ,σ)∈Uα,β
|K(θ, σ)|.(5.25)
Inequali ies (5.23) and (5.25) allow us o use sup emum o es ima e de i a i es.
I is also con enien o hink o unc ions in Hα,β as analy ic unc ions on Dβ
wi h alues in he space Γα.
Recall ha K(θ, σ) = K0(θ) + K1(θ)σ+K>(θ, σ) and ha we conside K
in he space Hα,β. We he e o e ake he unc ion K>in he space
H2
α,β ={K>∈Hα,β |K>(θ, 0) ≡∂σK>(θ, 0) ≡0}.
Equi alen ly, we a e equi ing ha he wo i s coe icien s (K>)0= (K>)1≡0
in he expansion o K>in powe s o σ. We endow H2
α,β wi h he no m k·kH2
α,β =
k·kHα,β inhe i ed om Hα,β. Finally, we conside he ope a o s Hand N, de ined
in (5.18) and (5.20), ac ing on unc ions K>in he space H2
α,β.
20 X. CABR´
E, E. FONTICH, AND R. DE LA LLAVE
5.2.3. Es ima es o he linea ized equa ion. We s a es ablishing es ima es in
he space H2
α,β o solu ions o he equa ion
L+λσ ∂
∂σ Ψ(θ, σ) = η(θ, σ).(5.26)
This will be an easy ask since (5.26) is equi alen , ma ching coe icien s in σn,
o
(L+nλ)Ψn(θ) = ηn(θ), n ≥2.(5.27)
The es ima es o each o hese coe icien s can be eadily ob ained om P opo-
si ion 5.2.
Lemma 5.7. Assume ha , o some α > 0and some cons an s C1and C2,
1) kΦkΓα≤C1and kΦ−1kΓα≤C1, whe e Φθis he undamen al solu ion
o he linea ized equa ion.
2) µ < 0and eµis no an eigen alue o he monod omy ma ix Φ1.
3) k(Φ1−eµId)−1k ≤ C2.
Then, he solu ion ∆o he linea ized equa ion (5.10) sa is ies
k∆kΓα≤CkRkΓα,(5.28)
whe e C=C2
1(1 + C2).
The p oo ollows om o mulas (5.11) and (5.12), using ha µ < 0.
F om now on, we ake α > 0 small enough so ha γ=K0, Φ and Φ−1all
belong o he space Γα, as in he p e ious p oposi ion.
Lemma 5.8. Le α > 0be small as indica ed abo e, and le β > 0. Assume
ha λ < 0is such ha enλ is no an eigen alue o he monod omy ma ix Φ1 o
e e y in ege n≥2.
Le η(θ, σ) = P∞
n=2 η(θ)σnbe a unc ion in H2
α,β. Then, he e is one and
only one Ψ∈H2
α,β sol ing (5.26). In addi ion, i sa is ies
kΨkH2
α,β ≤CkηkH2
α,β ,
o some cons an C.
P oo . We use ha (5.26) is equi alen o he sys em o equa ions (5.27), n≥2.
Nex , unde he assump ions ha λ < 0 and eλn is no an eigen alue o Φ1
o n≥2, we can bound k(Φ1−eλn Id)−1kuni o mly in n. He e we ha e used he
non- esonance condi ions oge he wi h he ac ha Φ1−eλn Id →Φ1, which is
in e ible, as n→+∞.
Hence, he bounds we ob ain applying Lemma 5.7 o each o he coe i-
cien s Ψna e uni o m in n. The e o e kΨnkΓα≤CkηnkΓα o some cons an C
independen o n, om which he desi ed esul ollows. 
Now we u n o show ha he maps Hand Nde ined in (5.18) and (5.20)
a e indeed well de ined in he space H2
α,β. Mo eo e , we will show ha i K1is
chosen small enough, we can ge a ball cen e ed a 0 mapped by Nin o i sel ,
and on which he Lipschi z cons an o Nis small.
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 21
P oposi ion 5.9. Le K0and K1be chosen as h oughou his sec ion. Le α > 0
be small as indica ed abo e, and le β, and ρbe posi i e numbe s. Assume ha :
1) The ec o ield Xis analy ic and bounded in a domain ha includes he
complex ball o adius a ound each poin K0(θ),θ∈Sα. Le
a:= sup
θ∈Sα
sup
|z−K0(θ)|≤
|X(z)|.
2) kK1(θ)σkHα,β ≤ρ.
3) ρ≤ /4.
Then:
a) The map K>∈Bρ(0) ⊂H2
α,β 7→ X(K0+K1σ+K>)∈Hα,β is well
de ined, whe e Bρ(0) is he ball in H2
α,β o adius ρcen e ed a he o igin.
b) I K>∈Bρ(0) ⊂H2
α,β hen
H(K>) = X(K0+K1σ+K>)− X ◦ K0−DX ◦ K0(K1σ+K>) (5.29)
belongs o H2
α,β.
c) Fo e e y K>and ˜
K>in Bρ(0) ⊂H2
α,β, we ha e
kH(K>)kH2
α,β ≤8a −2ρ2,(5.30)
kH(K>)− H(˜
K>)kH2
α,β ≤6a −2ρkK>−˜
K>kH2
α,β .(5.31)
P oo . By he analy ici y p ope ies o Xand hypo hesis 1), we see ha i
X(K0(θ) + z) = X
n≥0
Xn(θ)zn,
we hen ha e
|Xn(θ)| ≤ a −n;
his is p o ed using Cauchy in eg al o mula as in (5.24) be o e. Since he abo e
inequali y is ue o any θ∈Sα, we ha e kXnkΓα≤a −n.
Using he Banach algeb a p ope ies o ou spaces o unc ions, we see ha
X(K0(θ) + K1(θ)σ+K>(θ, σ)) =
∞
X
n=0
Xn(θ)K1(θ)σ+K>(θ, σ)n(5.32)
is well de ined and con e ges uni o mly, o K>∈Bρ(0). Indeed, he Hα,β-no m
o each e m in he se ies is bounded by kXnkHα,β (kK1(θ)σkHα,β +kK>kHα,β )n≤
a −n(2ρ)n, which con e ges by assump ion 3).
Nex , om (5.32) and he de ini ion (5.29) o H, we see ha
H(K>)(θ, σ) =
∞
X
n=2
Xn(θ)K1(θ)σ+K>(θ, σ)n.

22 X. CABR´
E, E. FONTICH, AND R. DE LA LLAVE
The same a gumen as be o e es ablishes ha his se ies con e ges, and hence
H(K>) is well de ined in H2
α,β. Mo eo e , using assump ions 2) and 3), we ob ain
kH(K>)kH2
α,β ≤
∞
X
n=2
kXnkHα,β kK1(θ)σkHα,β +kK>kHα,β n
≤
∞
X
n=2
a −n(2ρ)n=a(2 −1ρ)21
1−2 −1ρ
≤8a( −1ρ)2,
as claimed in (5.30).
Finally, we also ha e
kH(K>)− H(˜
K>)kH2
α,β
≤
∞
X
n=2
kXnkHα,β (K1(θ)σ+K>(θ, σ))n−(K1(θ)σ+˜
K>(θ, σ))nHα,β
≤
∞
X
n=2
a −nkK>−˜
K>kH2
α,β n(2ρ)n−1
=a −1kK>−˜
K>kH2
α,β
∞
X
n=2
n(2 −1ρ)n−1≤6a −1 −1ρkK>−˜
K>kH2
α,β .
This es ablishes (5.31). 
No e ha assump ions 2) and 3) in he p e ious p oposi ion can be ac-
complished ei he by aking K1small enough ( ecall ha K1is de ined up o a
mul iplica i e cons an ), o by aking βsmall enough.
Since N= (L+λσ ∂
∂σ )−1TH, we see om es ima es (5.30), (5.31) and
Lemma 5.8 ha i we choose ρsmall enough, we ob ain a ball ha ge s mapped
in o i sel by Nand on which he map Nis a con ac ion. The e o e, Nhas
a unique ixed poin in such ball. This inishes he p oo o he esul o his
sec ion.
Rema k 5.10. In [Mey75] one can ind ha he ope a o Nis ac ually analy ic
in he indica ed spaces. Hence, we could use he implici unc ion heo em and
ob ain au oma ically smoo h dependence on pa ame e s.
Rema k 5.11. When one conside s a disc e e ime dynamical sys em (i.e., a
map) whose in e se is en i e (e.g. polynomial), we a gued in Rema k 4.5 ha
he solu ions Kis en i e.
In con as , he solu ions o polynomial di e en ial equa ions usually a e no
en i e, and e y o en hey p esen essen ial singula i ies. Hence, when wo king
wi h di e en ial equa ions, one should no expec he coe icien s o decay as .
Choosing he bas indica ed in Rema k 5.3 is qui e impo an o nume ical
applica ions.
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 23
6. AC0in a ian s able mani old heo em
In his sec ion we p o e a e sion o he s able mani old heo em. I is no
op imal in e ms o he egula i y ob ained, bu i s p oo is simple and ob ains
di e en iabili y wi h espec o pa ame e s. In con as o o he sec ions in his
pape , we o mula e he esul s o maps in gene al Banach spaces. We also call
a en ion o some a he sub le echnicali ies such as he ac ha , in in ini e
dimensional spaces, he exis ence o smoo h cu -o unc ions canno be aken o
g an ed.
To s a e i , we use he ollowing e minology. We say ha a map is C1
ui i
is o class C1and has uni o mly con inuous de i a i e. Recall ha o a unc ion
de ined in a ini e dimensional space, i he unc ion is C1in a neighbo hood o
a poin hen i is C1
uin a smalle neighbo hood —since con inuous unc ions in
compac se s a e uni o mly con inuous. Recall also ha i a unc ion is C1+ε o
some ε > 0, hen i is C1
u, e en in in ini e dimensions.
Theo em 6.1. Le Xbe a Banach space, and F:U⊂X→Xbe a C1map in
a neighbo hood Uo 0, wi h F(0) = 0. Le A:= DF(0). Assume:
1) Ais an in e ible ope a o .
2) The e exis s a decomposi ion
X=Xs⊕Xu
such ha :
2.1) I is in a ian unde A. Tha is,
AXs⊂Xs, AXu⊂Xu.
2.2) Le As:= πsA|Xsand Au:= πuA|Xu, whe e πsand πua e he
p ojec ions on o Xsand Xu, espec i ely. Suppose ha kAsk<1
and kA−1
uk<1.
3) I Xis in ini e dimensional, assume ha Xadmi s smoo h cu -o unc-
ions, and ha Fis C1
u(U)( ha is, DF is uni o mly con inuous in U).
Then, he e exis s a con inuous map K:U1⊂Xs→X, whe e U1is a neighbo -
hood o 0, such ha
a) K(0) = 0.
b) F◦K=K◦Asin U1.
Mo eo e , assume ha Fλis a C1 amily o C1
umaps (i.e., he map λ∈
V7→ Fλ∈C1
u, whe e Vis a neighbo hood o 0in Λ, is C1when he maps Fλ
a e gi en he C1 opology) wi h Fλ(0) = 0, and ha F0sa is ies he hypo heses
abo e. Then, o λsmall enough, he e exis s a con inuous map Kλsa is ying
Kλ(0) = 0 and
Fλ◦Kλ=Kλ◦A0,s in a neighbo hood o he o igin,
whe e A0,s =πsA0|Xsand A0=DF0(0). In addi ion, he map λ7→ Kλis C1in
a neighbo hood o 0when he maps Kλa e gi en he C0 opology.
24 X. CABR´
E, E. FONTICH, AND R. DE LA LLAVE
Rema k 6.2. The hypo hesis on cu -o unc ions, made in 3) abo e, means
ha he e exis s a C∞ unc ion ξ:X→Rwhich is iden ically 1 in he uni
ball cen e ed a 0 and iden ically 0 ou side o he ball o adius 2 cen e ed a
0. Such unc ion always exis s i Xis ini e dimensional o a Hilbe space. I
su ices o ake Ψ(|x|), whe e Ψ is a C∞ eal alued unc ion o e he eals wi h
he indica ed p ope ies.
Pe haps su p isingly, he exis ence o smoo h cu -o unc ions is no ue
o a bi a y Banach spaces. Fo example, C0[0,1] does no admi a C2cu -o
unc ion. We e e o [DGZ93].
The p e ious esul , Theo em 6.1, is a om op imal in se e al espec s. Fo
example, he egula i y can be imp o ed, and he exis ence o cu -o unc ions
can be elimina ed (see he ollowing ema k). Ne e heless, we poin ou he
na u alness and he speed o he p oo .
Rema k 6.3. Fo some ρ > 0 small enough, he local s able in a ian mani old
K(Bρ(0)) can be cha ac e ized by he ollowing dynamical p ope y:
K(Bρ(0)) ∩V={x∈V|F(i)(x)∈V o all i≥0}
o e e y su icien ly small neighbo hood Vo he o igin; see [PdM82]. This ac
au oma ically implies he uniqueness o he local in a ian mani old. The abo e
cha ac e iza ion (and some ex a wo k) also allows o es ablish ha Kis indeed
di e en iable a 0 and ha DK(0) = (Id,0). Tha is, K(Bρ(0)) is angen o Xs
a he o igin. We e e o [PdM82] o all hese ma e s.
P oo o Theo em 6.1. We w i e A:= DF(0) and N(x) := F(x)−Ax. A p e-
limina y educ ion, s anda d in he ield, is o conside he unc ion
Fδ(x) := Ax +Nδ(x),whe e Nδ(x) := ξ(x)N(δx)
δ.
o δ > 0 small enough. He e ξ:X→Ris a smoo h cu -o unc ion (ξ≡1 in
he uni ball cen e ed a 0 and ξ≡0 ou side o he ball o adius 2 cen e ed a 0).
E en i Nis only de ined in a neighbo hood o he o igin, o δsmall enough
we may conside he nonlinea i y Nδ o be de ined and be C1
uin he whole X,
since ξhas bounded suppo . We use he e ha , in ini e dimensions, e e y C1
unc ion in a neighbo hood o 0 is au oma ically C1
uin a smalle neighbo hood
(since con inuous unc ions in compac se s a e uni o mly con inuous).
I is impo an o no e ha i Wδis a mani old h ough 0 in a ian unde
Fδ, hen δWδis in a ian unde Fin a neighbo hood o 0. I su ices he e o e o
ind an in a ian mani old o Fδ o δsmall enough. This will be accomplished
using he implici unc ion heo em. Fo his, we conside he nonlinea i y Nδ
as belonging o he ollowing Banach space.
We wo k in he space C1
0,u o bounded C1maps M:X→Xwi h bounded
and uni o mly con inuous de i a i e DM in he whole X, and such ha M(0) =
DM(0) = 0. We equip his space wi h he s anda d C1no m.
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 25
No e ha Nδ∈C1
0,u o all δsmall enough. Using ha N(0) = DN(0) = 0,
i is easy o check ha , by choosing δsmall enough, we can assume ha kNδkC1
is as small as we wan .
Abusing o no a ion, we wo k wi h maps F=A+Nwi h N∈C1
0,u —whe e
in eali y, all wha ollows is applied o Fδ=A+Nδ, o which Nδdoes uly
belong o C1
0,u.
We w i e he pa ame e iza ion as
K= (Id,0) + K>,
and look o K>in he Banach space C0
0=C0
0(Xs;X) o bounded con inuous
maps K>:Xs→Xwi h K>(0) = 0. We ha e
F◦K−K◦As= (A+N)◦((Id,0) + K>)−((Id,0) + K>)◦As
=A◦K>+N◦((Id,0) + K>)−K>◦As.
Theo em 6.1 ollows by applica ion o he implici unc ion heo em in Ba-
nach spaces o he ope a o
T(N, K>):=A◦K>+N◦((Id,0) + K>)−K>◦As,
conside ed as an ope a o om C1
0,u ×C0
0 o C0
0.
We wan o sol e T(N, K>) = 0 and ob ain K>as a unc ion o N, o N
nea 0.
No e ha T(0,0) = 0 — his co esponds o he linea map F=A, o which
he in a ian mani old is Xs. Recall also ha , by aking δsmall, we may assume
ha N=Nδis as small in C1
0,u as needed.
I is easy o e i y (see [dlLO99]) ha he ope a o Tis C1, and ha
D2T(N, K>)∆ = A∆ + DN ◦((Id,0) + K>)∆ −∆◦As.
I su ices o e i y ha he exp ession abo e sa is ies he de ini ion o de i a i e.
The e i ica ion o he de ini ion o de i a i e is whe e we use ha DN is uni-
o mly con inuous. Mo e de ails on he e i ica ion and examples ha show ha
uni o m con inui y o DN is needed o ge di e en iabili y o Tcan be ound in
[dlLO99].
To comple e he p oo , we only need o show ha he ope a o
S:= D2T(0,0),
gi en by
S∆ = A∆−∆◦As,
is in e ible om C0
0 o C0
0. This amoun s o, gi en η∈C0
0, ind ∆ ∈C0
0such
ha
A∆−∆◦As=η(6.1)
and show ha k∆ukC0≤CkηkC0.
32 X. CABR´
E, E. FONTICH, AND R. DE LA LLAVE
P oo o Lemma 8.3. Le ξ:R+−→ R+be a C1cu -o unc ion such ha
0≤ξ( )≤1, ξ( ) = 1 i 0 ≤ ≤1, and ξ( ) = 0 i ≥2.
To p o e he exis ence o a igh in e se S−1, o any gi en η= (η1, η2)∈Γ
we ha e o ind ∆ = (∆1,∆2)∈Γ such ha
S∆ = η.
The o mulas
∆1=
∞
X
i=0
A−(i+1)
1η1◦Ai
1(8.4)
∆2=−
∞
X
i=0
Ai
2η2◦A−(i+1)
1(8.5)
gi e a o mal solu ion. I will be a ue solu ion p o ided he se ies con e ge
uni o mly and η2is globally de ined in o de ha he o mula o ∆2makes sense.
To deal wi h he second di icul y we only ha e o ex end η2in a con inuous way,
which is always possible in a Banach space as ollows. We subs i u e η2by he
ex ension ˜η2de ined by
˜η2(x) = η2(x),i |x|<1
˜η2(x) = ξ(|x|)η2(x/|x|),i |x| ≥ 1.
No e ha i |A−(i+1)
1x|<1 hen |˜η2(A−(i+1)
1x)| ≤ kηkΓ(|A−(i+1)
1x|)1+µ.
Also, i |A−(i+1)
1x| ≥ 1 hen |˜η2(A−(i+1)
1x)| ≤ ξ(|A−(i+1)
1x|) sup|y|=1 |η(y)| ≤
kηkΓ≤ kηkΓ(|A−(i+1)
1x|)1+µ.
By hypo hesis 3) we ha e
|A−(i+1)
1η1(Ai
1(x))| ≤ kA−1
1ki+1kηkΓ|Ai
1x|1+ν
≤ kηkΓkA−1
1kkA−1
1k kA1k1+νi|x|1+ν
and
|Ai
2η2(A−(i+1)
1(x))| ≤ kA2kikηkΓ(kA−1
1ki+1|x|)1+µ
≤ kηkΓkA−1
1k1+µkA2k kA−1
1k1+µi|x|1+µ,
which p o e ha he se ies in (8.4) and (8.5) a e uni o mly con e gen and he e-
o e hei sums de ine con inuous unc ions. F om hese bounds i is also clea
ha ∆ so ob ained indeed belongs o Γ and k∆kΓ≤CkηkΓ.
We conside he ope a o N:C1+ε
0×Γ−→ Γ de ined by
N(N, K>) = N◦((Id,0) + K>).
We ecall ha C1+ε
0is he subspace o C1+εconsis ing o he unc ions anishing
a he o igin oge he wi h hei de i a i e. E en ha he e a e se e al a ailable
esul s on he di e en iabili y o he composi ion ope a o , o he gi en opology
o Γ we need o p o ide a p oo .

THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 33
P oposi ion 8.5. Unde he abo e condi ions, we ha e ha Nis C1and
D2N(N, K>)∆ = DN ◦((Id,0) + K>) ∆.
P oo . Le α= min(ν, µ) and β= max(ν, µ). No e ha (1+α)(1+ε)≥1+α+ε≥
1 + ε≥1 + β. To sho en he no a ion we will w i e K= (Id,0) + K>.Nis
linea wi h espec o N. We claim ha N7→ N (N, K>) is con inuous. Indeed,
since
|N(K(x))| ≤ Z1
0
|DN(sK(x))K(x)|ds ≤kNkC1+ε
1 + ε|x|+kK>kΓ|x|1+α1+ε
we see ha kT (N, K>)kΓ≤CkNkC1+ε.
Now we p o e ha Tis C1wi h espec o K>. We i s show ha i is
di e en iable. This ollows easily om he bound
|N(K(x) + ∆(x)) −N(K(x)) −DN(K(x))∆(x)|
=Z1
0hDN(K(x) + s∆(x)) −DN(K(x))i∆(x)ds
≤Z1
0
kNkC1+εsε|∆(x)|1+εds ≤1
1 + εkNkC1+εk∆k1+ε
Γ|x|(1+α)(1+ε).
Finally we show ha D2Nis con inuous. We ha e o bound
kD2N(N, K>)−D2N(N, K>)kL(Γ,Γ) = sup
k∆kΓ≤1
k[DN ◦K−DN ◦K]∆kΓ.
The con inui y ollows om
hkDN(K(x)) −DN(K(x))k+kDN(K(x)) −DN(K(x))ki|∆(x)|
≤[kDN −DNkCε|K(x)|ε+kDNkCε|K>(x)−K>(x)|ε]k∆kΓ|x|1+α
≤hkN−NkC1+ε|x|+kK>kΓ|x|1+αε
+kNkC1+εkK>−K>kε
Γ|x|(1+α)εik∆kΓ|x|1+α
≤h1 + kK>kΓ|x|αεkN−NkC1+ε+kNkC1+εkK>−K>kε
Γi·
·k∆kΓ|x|1+α+ε.

Now we can easily inish he p oo o Theo em 8.1. We de ine T:C1+ε
0×
Γ−→ Γ by
T(N, K>) = K>+S−1N◦((Id,0) + K>).
We ha e ha T(0,0) = 0. By Lemma 8.3 and P oposi ion 8.5 he ope a o T
is C1and D2T(0,0) = Id. Then we can apply he implici unc ion heo em o
T(N, K>) = 0 and ob ain a neighbo hood Vo 0 in C1+ε
0and a C1 unc ion
K>,∗= (K>,∗
1, K>,∗
2) : V⊂C1+ε
0−→ Γ
such ha T(N, K>,∗(N)) = 0 o all N∈V.
34 X. CABR´
E, E. FONTICH, AND R. DE LA LLAVE
We no e ha he ac ha |K>,∗
1(x)| ≤ C|x|1+νand |K>,∗
2(x)| ≤ C|x|1+µ
implies ha Kis di e en iable a 0, DK>,∗
1(0) = 0, DK>,∗
2(0) = 0. The e o e,
he in a ian mani old ob ained is angen o X1a 0, simply because K=
(Id +K>,∗
1, K>,∗
2). 
9. Non- esonan in a ian mani olds o maps
The goal o his sec ion is o s udy some non- esonan in a ian mani olds
o maps. The non- esonan in a ian mani olds we e in oduced in [dlL97], and
hey include as pa icula cases he s able o s ong s able mani olds. Fo op imal
esul s conce ning di e en iabili y and also in he se ing o Banach spaces we
e e o [CFdlL03a, CFdlL03b]. An exposi ion o esul s ha can be ob ained
using he g aph ans o m is in [dlL03].
The conc e e esul ha we p o e in his sec ion is Theo em 3.1, s a ed in
Sec ion 3. We will p esen i s a p oo in he analy ic case, which is simple ,
and hen a p oo in he ini ely di e en iable case.
9.1. O e iew o he p oo . We look o Ko he o m
K=K≤+K>
whe e K≤(x) = PL
i=1 Kix⊗iis a polynomial o deg ee L. We ecall ha Kiis a
symme ic i-linea ope a o in E⊗i aking alues in Rd. Simila ly, we will w i e
R(x) = PL
i=1 Rix⊗iwhe e Riis a symme ic i-linea ope a o in E⊗i aking
alues in E.
In Subsec ion 9.2, we will show ha , unde app op ia e non- esonance con-
di ions, i is possible o ind K≤and Rjus ma ching powe s o x. Then, he
sea ch o a K> ha leads o an in a ian pa ame e iza ion will be educed o
sol ing a nonlinea equa ion in a Banach space, which will be discussed in Sub-
sec ion 9.3 o he analy ic case, and in Subsec ion 9.4 o he ini e di e en iable
case.
9.2. Solu ion o he o mal p oblem.
Lemma 9.1. Assume ha (Spec(AE))i∩Spec(AC) = ∅ o i= 2, . . . , L and
ha ≥L. Then, we can ind polynomials K≤, R as be o e in such a way ha
Dj(F◦K≤−K≤◦R)(0) = 0, j = 0, . . . , L, (9.1)
K≤(0) = 0, DK≤(0) = (Id,0),(9.2)
R(0) = 0, DR(0) = AE.(9.3)
Mo eo e , i we assume ha N=F−Ais su icien ly small, hen K≤−(Id,0)
and R−AEwill be a bi a ily small.
This lemma is a simpli ied e sion o he no mal o m calcula ions which o -
en appea in dynamical sys ems. A good e e ence o ela ed esul s is [Nel69].
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 35
O undamen al impo ance in he p oo o he lemma a e he ope a o s (some-
imes called Syl es e ope a o s) de ined on he space Si=Si(X, Y ) o symme ic
i-mul ilinea ope a o s om X o Y(X,Ybeing ec o spaces) by
Li
A,BM=AM −MB⊗i,(9.4)
whe e A,Ba e linea maps.
A key esul in he s udy o hese ope a o s is he ollowing.
P oposi ion 9.2.
Spec(Li
A,B) = Spec(A)−(Spec(B))i
={λ−µ1·. . . ·µi|λ∈Spec(A), µ1, . . . , µi∈Spec(B)}.(9.5)
A p oo o his p oposi ion can be ound in [Nel69]. In [CFdlL03a] (and
also in [BK98]), one can ind ano he p oo and an analogue o Banach spaces.
We no e ha in he gene ali y o Banach spaces one has he inclusion ⊂ins ead
equali y in (9.5).
P oo o P oposi ion 9.2. No e ha since he le and he igh hand sides o
(9.5) a e con inuous wi h espec o he ma ices Aand B, i su ices o p o e
(9.5) o a dense se o ma ices Aand B. Hence, i su ices o es ablish he esul
when Aand Ba e diagonalizable o e he complex.
In such a case, we see ha i (λj, ej), (µj, j) a e eigen alues and eigen ec o s
o A, B espec i ely, gi en a se o indices σ1, . . . , σi, and jwe can conside he
mul ilinea ope a o de ined by
Γj
σ1,...,σi( α1⊗ · · · ⊗ αi) = (eji he se s {α1, . . . , αi},{σ1, . . . , σi}a e equal,
0 o he wise.
Clea ly, Γi
σ1,...,σiis symme ic and we ha e
Li
A,BΓj
σ1,...,σi= (λj−µσ1·. . . ·µσi)Γj
σ1,...,σi.
Hence, Γj
σ1,...,σiis an eigen ec o o Li
A,B o eigen alue λj−µσ1·. . .·µσi. The e o e
we ha e he inclusion Spec(Li
A,B)⊃Spec(A)−(Spec(B))i.
To p o e he opposi e inclusion, we no e ha he Γ’s ob ained o di e en
(σ, j)’s a e linea ly independen . Hence, since hei numbe equals he dimension
o he space o symme ic i-linea ope a o s, we see ha hey a e a comple e se
o eigen ec o s. Hence, we ha e ound all he spec um o Li
A,B.
Rema k 9.3. No e ha , as a pa icula case o P oposi ion 9.2, we ob ain ha ,
i iis such ha (max |µj|)i<min |λj|(which happens o all
i > log min |λj|
log max |µj|
when max |µj|<1), hen Li
A,B is in e ible.
36 X. CABR´
E, E. FONTICH, AND R. DE LA LLAVE
P oo o Lemma 9.1. The case j= 0 o he conclusions is sa is ied when K≤(0) =
0, R(0) = 0. Hence, we pick K0= 0, R0= 0. The case j= 1 amoun s o
AK1=K1R1. Hence, we ake R1=AEand K1=IE, whe e IE= (Id,0) is he
imme sion o Ein o Rd. In his way, we ensu e (9.2) and (9.3).
The p e ious choice o R1and K1will only a ec he smallness condi ions
ha we ha e o impose on he nonlinea e ms, which as in p e ious sec ions, can
be adjus ed by scaling. This is, o cou se, analogous o ou choice o a mul iple o
he eigen ec o o he one dimensional in a ian mani olds o p e ious sec ions.
Fo j≥2, equa ing e ms o o de jin F◦K−K◦R= 0 we ob ain
AKj−KjA⊗j
E−K1Rj+Pj(K1, . . . , Kj−1, R1, . . . , Rj−1) = 0,(9.6)
whe e Pjis a polynomial exp ession in i s a gumen s. Taking p ojec ions o e
he spaces Eand C, deno ing by KE
j= ΠEKj, e c., and using he block no a ion
o Aas in (3.1), o mula (9.6) becomes
AEKE
j+BKC
j−KE
jA⊗j
E−Rj+PE
j= 0
ACKC
j−KC
jA⊗j
E+PC
j= 0,(9.7)
whe e we ha e used ha ΠCK1= 0.
Using he ope a o s Lwe can w i e (9.7) as
Lj
AE,AEKE
j=Rj−BKC
j−PE
j(9.8)
Lj
AC,AEKC
j=−PC
j.(9.9)
No e ha , by P oposi ion 9.2, he hypo heses o Lemma 9.1 imply ha Lj
AC,AE
is in e ible. Now we ollow he nex i e a i e algo i hm o sol e (9.7). Assuming
we al eady know Ki,Ri o 1 ≤i < j,
1) Since a his s age, he igh hand side o (9.9) is known and Lj
AC,AEis
in e ible, we can ob ain one and only one solu ion KC
jo i .
2) We choose Rjin such a way ha he igh hand side o (9.8) is in he
ange o Lj
AE,AE.
3) We sol e equa ion (9.8) o KE
j.
A pa icula way o do 2) and 3) abo e, is o choose Rj:= BKC
j+PE
jand
hen KE
j= 0. O cou se o he p ocedu es a e possible. We can add o Rj e ms
in he ange o Lj
AE,AE, and we can add o KE
j e ms in he ke nel o Lj
AE,AE.
The las s a emen o he lemma jus ollows om obse ing ha he polyno-
mials PE
jand PC
J anish when N≡0 and a e con inuous in he Taylo coe icien s
o N. This inishes he p oo o Lemma 9.1. 
Rema k 9.4. Hypo heses 2), 3) and 5) o Theo em 3.1 imply ha i j > L he
ope a o Lj
AE,AEis in e ible. In pa icula , when j > L we can ake Rj= 0.
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 37
9.3. P oo o Theo em 3.1 when =ω.We discuss i s he case =ω, ha
is when all he conside ed unc ions a e analy ic.
We ake a no m in Rdsuch ha he associa ed ope a o no ms kAEkand
kA−1k e i y kAEk< ρ(AE) + εand kA−1k< ρ(A−1) + ε o some εsmall
enough, whe e ρs ands o he spec al adius (see P oposi ion A1 in [CFdlL03a]
o de ails). Then, by hypo hesis 5),
kA−1k kAEkL+1 <1.(9.10)
No e ha he equa ion F◦K=K◦Rcan be w i en as
T(N, K>) := AK≤+AK>+N◦(K≤+K>)−K≤◦R−K>◦R= 0,(9.11)
whe e we need o conside K≤and Ras unc ionals o Nwhich a e compu ed
p ecisely h ough he algo i hm ha we ha e indica ed in he p oo o Lemma 9.1.
To p o e Theo em 3.1 i su ices o conside small N, since we can always educe
o his case by scaling. This scaling echnique will be conside ed in de ail la e .
Le
Hk
δ={G:¯
Bδ(0) ⊂E→Rd|G=
∞
X
i=k
Gix⊗i,
∞
X
i=k
|Gi|δi<∞}
endowed wi h he no m kGk:= P∞
i=k|Gi|δi. We conside T:H2
3×HL+1
2→
HL+1
2(he e he space H2
3co esponds o maps N om Rd o Rd, ins ead o
maps om E o Rd). No e ha i N= 0 hen K≤=IEand R=AE, and hence
T(0,0) = 0.
P oposi ion 9.5. We ha e:
1) The ope a o T:V⊂H2
3×HL+1
2→HL+1
2is analy ic in a neighbo hood
Vo (0,0).
2) D2T(0,0)∆ = A∆−∆◦AE.
P oo . The ac ha K≤, R a e analy ic in Nis a consequence o he ac ha
hey a e algeb aic exp essions in a ini e numbe o coe icien s o N.
Also om Lemma 9.1, i Nis small enough, Ris a con ac ion. The e o e
he ope a o Tis well de ined.
Now, we use ha on he se Hk
3× {G∈H`
δ| kGk<2}, he map (H, G)7→
H◦Gis analy ic (see [Mey75] o mo e de ails). The idea is ha i we in e p e
he se ies H◦G=PiHiGias a se ies in he Banach algeb a o analy ic unc ions,
we can bound he no m o he gene al e m by kHiGik ≤ |Hi|kGki, and hence
he se ies ob ained summing HiGicon e ges as a se ies o elemen s in he Banach
algeb a.
Applying his ac o H=N, G =K≤+K>, we ob ain ha he hi d e m
in (9.11) is analy ic. Applying i when H=K>, G =Rwe ob ain he analy ici y
o he las e m in (9.11). 
Lemma 9.6. Unde hypo hesis 5) o Theo em 3.1, we ha e ha D2T(0,0) om
HL+1
2 o HL+1
2is boundedly in e ible.

38 X. CABR´
E, E. FONTICH, AND R. DE LA LLAVE
P oo . I we conside he equa ion
A∆−∆◦AE=η(9.12)
wi h η=P∞
i=L+1 ηix⊗i, we see ha
∆i−A−1∆iA⊗i
E=A−1ηi, i ≥L+ 1.(9.13)
By hypo hesis 2) and (9.10) we ha e ha kA−1k kAEki<1, o i≥L+ 1.
The e o e (9.13) de ines ∆ uniquely and we ha e
|∆i| ≤ kA−1k
1− kA−1k kAEki|ηi| ≤ kA−1k
1− kA−1k kAEkL+1 |ηi|,
and hence ∞
X
i≥L+1
|∆i|2i≤C
∞
X
i≥L+1
|ηi|2i.

We can now inish he p oo o Theo em 3.1 when =ω. P oposi ion 9.5
and Lemma 9.6 gi e ha he hypo heses o he implici unc ion heo em o
T(N, K>) = 0 nea (0,0) a e sa is ied, and he e o e we ha e es ablished Theo-
em 3.1 o Nsmall enough.
Theo em 3.1 o a gene al Ncan be ob ained by obse ing ha i we conside
Fδ= (1/δ)F(δx) o δsmall enough, hen Nδis small in he sense needed by
he esul hi he o p o ed. Hence, we ob ain
Fδ◦K=K◦R
o some K, R analy ic. I is immedia e o e i y ha K1/δ(x) = δK(1
δx),
R1/δ(x) = δR(1
δx) e i y
F◦K1/δ =K1/δ ◦R1/δ
in a neighbo hood o 0, as well as he o he claims.
9.4. P oo o Theo em 3.1 when ∈N.The me hod used in he p e ious
subsec ion does no di ec ly wo k in his case because he e m (K>, R)7→ K>◦R
ceases o be di e en iable wi h espec o Rwhen we gi e K>and K>◦R he
C opology wi h ∈N. Simply no e ha when di e en ia ing K>◦R o mally
wi h espec o R, he e m DK>appea s and does no belong o C , bu jus
o C −1.
We w i e N=N≤+N>, whe e N≤is he Taylo polynomial o Nup o
o de L, and we deno e by K0,R0 he solu ion o (A+N≤)◦K0=K0◦R0which
is ob ained applying he analy ic esul —al eady es ablished— o F≤=A+N≤
in place o F=A+N. Then we look o Kin he o m K=K0+K>.
Le Σ be he space o polynomials Q:Rd→Rdo deg ee less han o equal
o Lsuch ha Q(0) = 0 and DQ(0) = 0 and, o ≥L+ 1, le
C
L(¯
Bρ(0)) = { ∈C (¯
Bρ(0)) |Di (0) = 0,0≤i≤L}
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 39
endowed wi h he opology gi en by
||| ||| := max
L+1≤i≤ sup
|x|≤ρ
kDi (x)k.
I is clea ly a Banach space. We conside he ope a o
N:V⊂Σ×C +1
L(¯
B2(0)) ×C
L(¯
B1(0)) −→ C
L(¯
B1(0))
in a neighbo hood Vo (0,0,0) de ined by
N(N≤, N>, K>) =AK0+AK>+N≤◦(K0+K>)
+N>◦(K0+K>)−K0◦R0−K>◦R0,(9.14)
whe e K0and R0a e conside ed as analy ic unc ions o N≤.
Wi h hese no a ions, he o iginal p oblem becomes N(N≤, N>, K>) = 0.
Since N(0,0,0) = 0, again we y o cons uc he solu ion by using he implici
unc ion heo em.
P oposi ion 9.7. The ope a o Nde ined in (9.14) is con inuous wi h espec
o all h ee a iables and C1wi h espec o K>. Mo eo e ,
D3N(0,0,0)∆ = A∆−∆◦AE.(9.15)
P oo . We no e ha he linea ope a o H1
2→C +1
0(¯
B1(0)) sending a map o
i sel is con inuous and hus di e en iable. Then we can conside Nas being he
composi ion o he maps
V⊂Σ×C +1
L(¯
B1(0)) ×C
L(¯
B1(0)) →Σ×H1
2×H1
2×C +1
L(¯
B1(0)) ×C
L(¯
B1(0))
sending (N≤, N>, K>) o (N≤, K0, R0, N>, K>), whe e K0and R0a e he maps
ob ained applying he analy ic esul o F≤=A+N≤, which depend analy ically
on N≤, he injec ion
Σ×H1
2×H1
2×C +1
L(¯
B1(0)) ×C
L(¯
B1(0))
−→ Σ×C +1
0(¯
B1(0)) ×C +1
0(¯
B1(0)) ×C +1
L(¯
B1(0)) ×C
L(¯
B1(0))
and inally he map
˜
V⊂Σ×C +1
0(¯
B1(0)) ×C +1
0(¯
B1(0)) ×C +1
L(¯
B1(0)) ×C
L(¯
B1(0)) →C (¯
B1(0))
(9.16)
sending (N≤, K0, R0, N>, K>) o N(N≤, N>, K>). We es ic Vand ˜
Vin
o de o ha e he maps well de ined. Since he space C
Lis a closed linea
subspace o C and he no m he e is he es ic ion o he C no m, we ob ain
ha he di e en iabili y esul s in [dlLO99] also hold o C
L. Then he map
(9.16) is con inuous (we a e wo king wi h se s o maps de ined in spaces o ini e
dimension) and i is C1wi h espec o K>. Hence Nis con inuous wi h espec
o i s h ee a iables and C1wi h espec o K>. Mo eo e , by he de ini ions
o K0and R0 he ange o Nis con ained in C
L(¯
B1(0)). Ac ually we ha e ha
D3N(N≤, N>, K>)∆ = [A+DN≤◦(K0+K>)+DN>◦(K0+K>)]∆−∆◦R0.

40 X. CABR´
E, E. FONTICH, AND R. DE LA LLAVE
The ollowing is sligh ly weake han Lemma 5 o [BdlLW96]. The p oo ,
howe e is simple .
Lemma 9.8. The ope a o D3N(0,0,0) is boundedly in e ible as an ope a o
om C
L(¯
B1(0)) o i sel .
P oo . We ha e o sol e A∆−∆◦AE=ηwhich is equi alen o
∆−A−1∆◦AE=A−1η.
We in oduce he ope a o Ade ined by A∆ = A−1∆◦AE. We claim ha he
no m o Aconside ed as an ope a o om C
L(¯
B1(0)) o i sel is s ic ly smalle
han 1. Indeed, i L+ 1 ≤i≤
kDi(A−1∆◦AE)kC0≤ kA−1k kDi∆kC0kAEki≤ kA−1k kDi∆kC0kAEkL+1
(9.17)
and hence |||A∆||| ≤ kA−1k kAEkL+1|||∆|||. The e o e, we ha e ha ∆ =
(Id −A)−1(A−1η). 
P oposi ion 9.7 and Lemma 9.8 es ablish he hypo heses o he gene alized
e sion o he implici unc ion heo em, which assumes con inui y o he map and
being C1wi h espec o he a iable ha one wan s o isola e, bu only p o ides
con inui y o he implici unc ion ha i de ines (see [Ni 01]). Applying his o
N(N≤, N>, K>) = 0 nea (0,0,0) we ge a con inuous map K>=M(N≤, N>)
de ined in a neighbo hood o (0,0).
Now, gi en a map Fsa is ying he hypo heses o Theo em 3.1, we scale
i o Fδ=A+N≤,δ +N>,δ wi h δso small ha (N≤,δ, N>,δ) belongs o
he domain o M. The pa ame e iza ion K=K0+K> hus ob ained is he
solu ion o Fδ◦K=K◦R0, whe e K0and R0a e he analy ic maps depending
on F≤,δ =A+N≤,δ p o ided by he p oo in he analy ic case.
10. Non- esonan in a ian mani olds o di e en ial equa ions
The esul s we ha e p o ed in he p e ious sec ion ansla e o esul s o
lows using he a gumen men ioned a he end o Sec ion 2. Ne e heless, i is
in e es ing, specially om he poin o iew o implemen ing algo i hms, o gi e
di ec p oo s o he esul s o di e en ial equa ions. We will see ha he leading
ideas and me hods a e e y simila o hose o maps.
The esul we deal wi hin his sec ion is Theo em 3.2, which we p o e ol-
lowing he pa ame e iza ion me hod.
Fo di e en ial equa ions x0=X(x) such ha X(0) = 0, i we ha e an in a i-
an subspace Eby DX(0), we look o a pa ame e iza ion Kand a polynomial
Rde ined in Esuch ha
X ◦ K=DK ·R, (10.1)
ha is, we ask he ec o ield Xon he image o K o be he pull o wa d o a
ec o ield Rin E.
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 41
Following he same s a egy as o he case o maps we look o Ko he
o m
K=K≤+K>
whe e K≤is a polynomial o deg ee Land K>is a unc ion anishing a he
o igin oge he wi h i s i s Lde i a i es.
The polynomials K≤and Rwill be ound ma ching powe s in (10.1). Then,
we will w i e a unc ional equa ion o K>whose solu ion will be ound by
applying he implici unc ion heo em in an app op ia e Banach space.
10.1. Fo mal solu ion. We summa ize he o mal calcula ions needed o ind
K≤and Rin he ollowing esul .
Lemma 10.1. Gi en X:U⊂Rd−→ Rd,X(0) = 0,X ∈ CL, sa is ying
hypo heses 3) o Theo em 3.2, we can ind polynomials K≤and Ro deg ee no
bigge han Lsuch ha
Dj(X ◦ K≤−DK≤·R)(0) = 0,0≤j≤L, (10.2)
K≤(0) = 0, DK≤(0)E=E, (10.3)
R(0) = 0, DR(0) = AE.(10.4)
Mo eo e , i we assume ha N=F−Aand Ba e su icien ly small, hen
K≤−(Id,0) and R−AEa e a bi a ily small.
Fo he calcula ions in Lemma 10.1 we will use he ope a o s e
Li
A,B om he
space Sio symme ic i-linea ope a o s in Ewi h alues in E(o Rd), de ined
by
(e
Li
A,BK)(x) = AK(x)−DK(x)Bx, (10.5)
o x∈E. These ope a o s a e simila o he ones used in Sec ion 9, bu hei
spec um is di e en .
P oposi ion 10.2.
Spec( e
Li
A,B) = Spec(A)−iSpec(B) (10.6)
:= {λ−(µ1+µ2+· · · +µi)|λ∈Spec(A), µ1, . . . , µi∈Spec(B)}.
P oo . The p oo is comple ely analogous o he one o P oposi ion 9.2. Because
o he con inui y o he objec s o (10.6) wi h espec o Aand B, i is su icien
o p o e (10.6) o he dense subse o diagonalizable ma ices Aand B.
Assuming ha Aand Ba e diagonalizable, le (λj, ej) and (µj, j) be he
eigen alues and eigen ec o s o Aand B espec i ely. Gi en indices σ1, . . . , σi
and `, le he i-linea symme ic o m Γ`
σ1,...,σi∈Sibe de ined by
Γ`
σ1,...,σi( α1⊗ · · · ⊗ αi) = (e`i he se s {α1, . . . , αi} {σ1, . . . , σi}a e equal,
0 o he wise.
No e ha he Γ0sso de ined a e linea ly independen and o m a basis o Si. An
easy calcula ion gi es ha e
Li
A,BΓ`
σ1,...,σi= (λ`−(µσ1+· · · +µσi))Γ`
σ1,...,σi.
48 X. CABR´
E, E. FONTICH, AND R. DE LA LLAVE
P oposi ion 10.8. Unde he p e ious condi ions, we ha e:
a) The ope a o V:V⊂Σ×C +1
L(¯
B2(0)) ×C
L(¯
B1(0)) −→ C
L(¯
B1(0))
de ined by (10.30) is con inuous wi h espec o all he h ee a iables
and C1wi h espec o K>in a neighbo hood o Vo (0,0,0).
b) We ha e V(0,0,0) = 0 and
D3V(0,0,0)∆ = ∆.
P oo . The ope a o Vis he iden i y plus W. The la e can be w i en as he
composi ion W1(W2(N≤),W3(N≤, N>, K>)) whe e
W3(N≤, N>, K>) = (A+N≤+N>)◦(K0+K>)−(A+N≤)◦K0−AK>,
W2sends N≤ o he low ϕ( , x) o x0=AEx+N≤(x), and
W1(ϕ, g) = Z∞
0
ψ(s)g(ϕ(s, x)) ds. (10.31)
We ecall ha , as in he p e ious sec ion, Σ →H1
3sending N≤ o K0is analy ic,
H1
3→C
0(¯
B1(0)) sending K0 o i sel is C∞, and Σ →C +1
0(¯
B2(0)) sending N≤
o i sel is C∞. By he esul s o [dlLO99] and he a gumen in P oposi ion 9.7
he ope a o
W3:V3⊂C +1
1(¯
B2(0)) ×C +1
L(¯
B2(0)) ×C
0(¯
B1(0)) ×C
L(¯
B1(0)) →C (¯
B1(0))
de ined by
W3(N≤, N>, K0, K>) = (A+N≤+N>)◦(K0+K>)−(A+N≤)◦
K0−AK>is o class C1. Then W3is C1, akes alues in C
L(¯
B1(0)) and
D3W3(N≤, N>, K>)∆ = DX(K0+K>)∆ −A∆.
In pa icula , D3W3(0,0,0) = 0.
The p oposi ion will be p o ed once we es ablish he egula i y o W1and
W2, ha we do in he nex lemmas. 
To s udy he egula i y o W1and W2we in oduce he space
Γ ={ϕ: [0,∞)ׯ
B1(0) →E|ϕ∈C0, ϕ( , ·)∈C ,
max
0≤j≤ sup
,x
e−(µ++ε) |Dj
xϕ( , x)|<∞}
wi h he no m kϕkΓ:= max0≤j≤ sup ,x e−(µ++ε) kDj
xϕ( , x)k. I is a Banach
space.
Lemma 10.9. The map W2:V2⊂Σ→Γ which sends RN o ϕ, whe e ϕ( , x)
is he solu ion o x0=AEx+RN(x)is well de ined in a neighbo hood V2o 0
and i is con inuous.
F om he basic heo y o o dina y di e en ial equa ions we know ha ϕ( , x)
depends con inuously on RN, in he sense ha i ais he ec o o he coe icien s
o RN,ϕ( , x, a) is con inuous. Howe e he lemma s a es ha he con inui y
holds wi h espec o he no m in Γ . We will use he ollowing e sion o

THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 49
G onwall’s lemma: i u: [0, b)→Ris con inuous : [0, b)→Ris di e en iable,
wi h (0) = 0, α≥0 and u( )≤ ( ) + αR
0u(s)ds o ∈[0, b) hen
u( )≤Z
0
0(s)eα( −s)ds, ∈[0, b).
P oo o Lemma 10.9. Fi s we no e ha he map Σ →Σ which sends N≤ o RN
is analy ic. F om Lemma 10.7 we know ha i kRNkC +1 ≤ε/2 hen |ϕ( , x)| ≤
e(µ++ε) |x|,kDxϕ( , x)k ≤ e(µ++ε) and, o j≥2, kDj
xϕ( , x)k ≤ Mje(µ++ε) ,
o all x∈¯
B1(0) and ≥0.
Le R, ˜
R∈¯
Bε/2(0) ⊂C +1
1(B1(0)) and le ϕ, ˜ϕbe he associa ed lows.
Assume ha kR−˜
RkC +1 ≤δ. We w i e
ϕ( , x) = eAE x+Z
0
eAE( −s)R(ϕ(s, x)) ds
and he analogous o mula o ˜ϕ. Sub ac ing bo h equa ions we ha e
˜ϕ( , x)−ϕ( , x) = Z
0
eAE( −s)[˜
R( ˜ϕ)−R( ˜ϕ) + R( ˜ϕ)−R(ϕ)] ds.
Mul iplying bo h sides by e−(µ++ε/2) , using ha |˜
R( ˜ϕ(s, x)) −R( ˜ϕ(s, x))| ≤
ck˜
R−RkC2|ϕ(s, x)|2and aking no ms we ge
e−(µ++ε/2) |˜ϕ( , x)−ϕ( , x)|
≤Z
0
cδe(µ++(3/2)ε)sds + (ε/2) Z
0
e−(µ++ε/2)s|˜ϕ(s, x)−ϕ(s, x)|ds
and by G onwall’s lemma, e−(µ++ε/2) |˜ϕ( , x)−ϕ( , x)| ≤ cδe(ε/2) (−1/(µ++ε)).
Hence
|˜ϕ( , x)−ϕ( , x)| ≤ cδ
−(µ++ε)e(µ++ε) .
P oceeding in he same way om
Dxϕ( , x) = eAE +Z
0
eAE( −s)DR(ϕ(s, x))Dxϕ(s, x)ds
and he analogous o mula o Dx˜ϕ( , x) we a i e a
kDx˜ϕ( , x)−Dxϕ( , x)k ≤ δ+ (ε/2)k˜ϕ( , x)−ϕ( , x)kΓ
−(µ++ε)e(µ++ε) .
I k≥2 we p oceed induc i ely s a ing om
Dk
xϕ( , x) = Z
0
eA( −s)
k
X
j=1 X
i0s
cDjR(ϕ(s, x))Di1
xϕ(s, x). . . Dij
xϕ(s, x)ds
and using he analogous manipula ions as in he cases k= 0,1 we ge
kDk
x˜ϕ( , x)−Dk
xϕ( , x)k ≤ δcke(µ++ε) ,
whe e cka e posi i e cons an s independen o ϕand ˜ϕ.
50 X. CABR´
E, E. FONTICH, AND R. DE LA LLAVE
Lemma 10.10. The map W1:V1⊂Γ ×C
L(¯
B1(0)) →C
L(¯
B1(0)) de ined by
(10.31) is well de ined in a neighbo hood V1o (0,0), is con inuous wi h espec
o bo h a iables, is C1wi h espec o g, and
D2W1(ϕ, g)∆ = Z∞
0
ψ(s)∆(ϕ(s, x)) ds. (10.32)
P oo . Th oughou he p oo Cwill mean a cons an independen on he unc-
ions, which may ake di e en alues in di e en places. We w i e
W1( ˜ϕ, ˜g)− W1(ϕ, g)
=Z∞
0
ψ(s)[˜g( ˜ϕ(s, x)) −g( ˜ϕ(s, x))] ds +Z∞
0
ψ(s)[g( ˜ϕ(s, x)) −g(ϕ(s, x))] ds.
Le L+ 1 ≤k≤ . The Dk
xde i a i e o ψ(s)[˜g( ˜ϕ(s, x)) −g( ˜ϕ(s, x))] is
bounded by
ψ(s)
k
X
i=1 X
1≤`1,...,`i≤k
`1+···+`i=k
C[Di˜g( ˜ϕ(s, x)) −Dig( ˜ϕ(s, x))]·
·D`1
x˜ϕ(s, x)· · · D`i
x˜ϕ(s, x)
≤Me(−λ−+ε)s
k
X
i=1 X
`0s
Ck˜g−gkCk|˜ϕ(s, x)|(L−i+1)+·
·kD`1
x˜ϕ(s, x)k · · · kD`i
x˜ϕ(s, x)k
≤Me(−λ−+ε)s
L
X
i=1 X
`0s
Ck˜g−gkCke(µ++ε)(L−i+1)+s|x|(L−i+1)+·
·M`1e(µ++ε)s· · · M`ie(µ++ε)s
≤Ce[−λ−+ε+(µ++ε)(L+1)]sk˜g−gkCk,
since (L−i+ 1)++i≥L+ 1.
The Dk
xde i a i e o ψ(s)[g( ˜ϕ(s, x)) −g(ϕ(s, x))] is bounded by
kψ(s)
k
X
i=1 X
1≤`1,...,`i≤k
`1+···+`i=k
C[Dig( ˜ϕ(s, x))D`1
x˜ϕ(s, x)· · · D`i
x˜ϕ(s, x)
−Dig(ϕ(s, x))D`1
xϕ(s, x)· · · D`i
xϕ(s, x)]k.
Now, by adding and sub ac ing app op ia e e ms, we ge he desi ed bounds.
We ha e o deal wi h e ms [Dig( ˜ϕ(s, x))−Dig(ϕ(s, x))]D`1
x˜ϕ(s, x)· · · D`i
x˜ϕ(s, x)
which a e bounded by
kgkCL+1 e(µ++ε)(L+1−i−1)s|˜ϕ(s, x)−ϕ(s, x)| kD`1
x˜ϕ(s, x)k. . . kD`i
x˜ϕ(s, x)k
≤Ce(µ++ε)(L+1)sk˜ϕ−ϕkΓ
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 51
i i < L, by
kgkCi+1 |˜ϕ(s, x)−ϕ(s, x)| kD`1
x˜ϕ(s, x)k. . . kD`i
x˜ϕ(s, x)k
≤Ck˜ϕ−ϕkΓe(µ++ε)se(µ++ε)is
i L≤i < , and by Ce(µ++ε) sω(k˜ϕ−ϕkC0), whe e ωis he modulus o
con inui y o D g, by he uni o m con inui y o D gon ¯
B1(0).
We also need o con ol he e ms o he o m
Djg(ϕ)D`1
x˜ϕ· · · [D`m
x˜ϕ−D`m
xϕ]· · · D`i
xϕ,
which a e bounded by
kgkCi+1 |ϕ(s, x)|L+1−i−1Ce(µ++ε)(i−1)skD`m
x˜ϕ−D`m
xϕkΓe(µ++ε)s
≤CkgkCi+1 e(µ++ε)(L+1)sk˜ϕ−ϕkΓ,
i i < L and analogous bounds in he o he cases, as we ha e go o he p e ious
e ms.
When we in eg a e om 0 o ∞, we ake he sup emum o e xand he
maximum o e k, we ob ain he con inui y in he opologies we a e wo king
wi h.
To p o e ha W1is di e en iable wi h espec o gwe only ha e o check
ha i is a bounded linea ope a o in g. This ollows immedia ely aking ˜g= 0
and ˜ϕ= 0 in he p e ious es ima es. In such a way we ge
kW1(g)kC
L≤CkgkC
L.
To s udy he con inui y o D2W1(ϕ, g), in iew o o mula (10.32) we ha e o
do he same kind o es ima es as we ha e done when dealing wi h ψ(s)[g( ˜ϕ(s, x))−
g( ˜ϕ(s, x))] bu changing gby ∆ ∈Γ .
The end o he p oo o Theo em 3.2 in he di e en iable case ollows in
a comple ely analogous way as in he end o he p oo o Theo em 3.1. Ap-
plying he gene alized e sion o he implici unc ion heo em (see [Ni 01]) o
V(N≤, N>, K>) = 0 nea (0,0,0) we ge a con inuous map K>=V∗(N≤, N>)
de ined in a neighbo hood o (0,0).
Gi en a ec o ield Xsa is ying he hypo heses o Theo em 3.2, we scale
i o Xδ=A+N≤,δ +N>,δ wi h δso small ha (N≤,δ, N>,δ) belongs o he
domain o V∗. The pa ame e iza ion K=K0+K> hus ob ained is he solu ion
we a e looking o .
Rema k 10.11. No ice he ema kable simila i ies be ween he p oo s o The-
o em 3.1 and Theo em 3.2. Also he analogy o he a gumen in P oposi ion 9.2
which compu es he spec um o he ope a o s Li
A,B de ined in (9.4) and he
one in P oposi ion 10.2 which compu es he spec um o he ope a o s e
Li
A,B de-
ined in (10.5). Simila ly, hei use o he ecu si e solu ion o he hie a chy o
equa ions o he low o de e ms is comple ely analogous.
52 X. CABR´
E, E. FONTICH, AND R. DE LA LLAVE
Appendix A. Rema ks on cohomology equa ions and nonuniqueness
o in a ian mani olds
Since cohomology equa ions play an impo an ole in his heo y, i is in-
e es ing o gi e a heu is ic guide o hei solu ion.
A cohomology equa ion is an equa ion o ∆ o he o m
M(x)∆(x)−∆◦R(x) = η(x) (A.1)
whe e ∆ and ηa e ec o alued unc ions, Mis a unc ion aking alues on he
space o linea ope a o s and Ris a di eomo phism. When acing an equa ion
o he o m (A.1), i is na u al o y o isola e ∆ explici ly om one e m and
i e a e he esul ing exp ession.
I we isola e ∆ om he second e m in (A.1), we a e le wi h
∆(x) = −η◦R−1(x) + M◦R−1(x) ∆ ◦R−1(x) (A.2)
which, upon i e a ion, leads o
∆ = −η◦R−1−M◦R−1η◦R−2−M◦R−1M◦R−2η◦R−3− · · ·
−[M◦R−1M◦R−2· · · M◦R−n]η◦R−n−1
+ [M◦R−1M◦R−2· · · M◦R−n−1]∆ ◦R−n−1.
(A.3)
I we isola e ∆ om he i s e m in (A.1), we a e le wi h
∆(x) = M−1(x)η(x) + M−1(x)∆ ◦R(x) (A.4)
which, upon i e a ion, leads o
∆ = M−1η+M−1M−1◦R η ◦R+· · ·
+ [M−1M−1◦R· · · M−1◦Rn]η◦Rn
+ [M−1M−1◦R· · · M−1◦Rn]∆ ◦Rn+1.
(A.5)
No e ha he gene al e m in bo h (A.3) and (A.5) consis s o he mul ipli-
ca ion by a la ge numbe o linea ope a o s applied o ηcomposed by he igh
wi h a high i e a ed o Ro o R−1. These sums can be shown o con e ge as
n→ ∞ in wo di e en cases by wo di e en a gumen s.
In he i s a gumen we use ha i Mis a con ac ion, hen (A.3) will
con e ge in he k·kC0no m. In he p oblems conside ed in his pape he poin
is ha i M=DF ◦Kis a con ac ion hen R, which ag ees a i s o de wi h
i close o he ixed poin , will also be a con ac ion. Hence, R−1is expansi e.
Since (A.3) in ol es composing wi h R−1, his will equi e ha he unc ions
a e de ined e e ywhe e. I i is no he case i equi es pe o ming ex ensions,
e c. We no e ha pe o ming ex ensions o Rcauses ha he esul ing mani old
may depend on he ex ension p ocedu e, so ha he local esul s will be qui e
non-unique. This s udy is he basis o he esul s in Sec ion 6.
The second a gumen uses ha he Rn(x) con e ges o a poin as n→+∞.
This happens when kDRkC0<1 in a neighbo hood o he o igin. In his case,
M−1will be an expansion. To ha e con e gence o he igh hand side o (A.5)
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 53
we will need ha he ope a o ∆ 7→ ∆◦Ris a s ong enough con ac ion o
o e come he expansion caused by M−1. The basic idea o ob ain con ac ion
ha gi es sense o (A.5) is o wo k in a space o unc ions ∆ de ined on Usuch
ha supx∈U|x|−L|∆(x)|<∞and o use he weigh ed no m
k∆k:= sup
x∈U
|x|−L|∆(x)|.
Wi h his no m, we ha e
k∆◦Rk= sup
x∈U
|x|−L|R(x)|L|R(x)|−L|∆◦R(x)|
≤sup
x∈U
|x|−L|R(x)|L·sup
x∈U
|R(x)|−L|∆◦R(x)|
≤ kDRkL
C0k∆k
(A.6)
Fo low L, o wo k in such spaces is qui e na u al and we easily ha e he
con ac i e p ope y. This is he basis o he esul s in Sec ion 8.
Fo high L, we ha e o eso o o he me hods. We obse e ha , unde ap-
p op ia e condi ions, one can ob ain he low o de e ms o he p oblem ma ching
de i a i es and hen, ob ain he emainde using his me hod. This is he basis
o he esul s in Sec ion 9. No e ha sol ing he p oblem o he low o de e ms
equi es non- esonance condi ions.
We no e ha in bo h cases, we ob ain uniqueness o he solu ion in he
co esponding space (a e ha ing ixed an ex ension o R−1in he i s case).
Ne e heless, i is qui e impo an o no e ha , e en in he case ha bo h
solu ions (A.3), (A.5) make sense, hey may ail o be he same. No e ha , e en
i one had equal solu ions o a ce ain η, adding an small bump o ηcauses
pe u ba ions ha go owa ds he o igin in (A.3), and ha go owa ds in ini y
in (A.5) and hence, o his pe u bed η he solu ions gi en by (A.3) and (A.5)
will be di e en .
One impo an di e ence be ween he me hods o solu ion is ha , i we
ake de i a i es o he gene al e m in (A.3), we pick ac o s DR−n, which a e
g owing. Howe e in (A.5), we ob ain ac o s DRnwhich a e dec easing. Hence,
i he se ies o he k-de i a i es o he e ms in (A.5) con e ge, he se ies o he
j-de i a i es also con e ge o alues o j om k o he deg ee o di e en iabili y
o η. The solu ions p oduced by (A.5), as soon as hey s a con e ging, hey
ha e all he de i a i es ha ηhas. On he o he hand, hose p oduced by (A.3)
only ha e a ini e numbe o de i a i es ha canno be imp o ed by assuming
mo e di e en iabili y o η.
Un o una ely, he egula i y ha can be p oduced au oma ically by (A.3)
is always smalle han ha allowed by boo s ap using (A.5).
All he abo e phenomena ha e a co espondence in he heo y o in a ian
mani olds. The ac ha slow mani olds wi h low egula i y a e no unique has
been in he li e a u e o a ce ain ime. I one makes hypo heses ha imply
ha he e is ce ain g ow h a in ini y, one can eadily show uniqueness. This

54 X. CABR´
E, E. FONTICH, AND R. DE LA LLAVE
was also known using I win’s me hod [dlLW95]. On he o he hand, unde non-
esonance assump ions, one can ge uniqueness unde mode a e di e en iabili y
assump ions.
Tha is, among all he ough in a ian mani olds angen o he space, he e
is one which is mode a ely di e en iable, and his mode a ely di e en iable man-
i old is as smoo h as he map.
Examples ha show ha he mani olds wi h good beha io a in ini y do
no ag ee wi h he mode a ely smoo h ones ha e been cons uc ed in [dlL97].
Appendix B. His o ical ema ks and in o ma ion on he li e a u e
on non- esonan in a ian mani olds
In his sec ion we ha e collec ed some e e ences on he p oblem o in a ian
mani olds associa ed o subspaces in he s able pa o he spec um.
In con as o he e y as li e a u e on s able mani olds — o which a
simila a emp would be beyond he capaci y o he au ho s—, he li e a u e
on in a ian mani olds associa ed o smalle se s o he spec um is much mo e
limi ed. O cou se, his a emp canno be conside ed a de ini i e e o ( o
ins ance, we ha e no been able o ace he wo k o Da boux, which is men ioned
by Poinca ´e and Lyapuno ). We can only hope ha ou modes sea ch can inspi e
o he s o do a mo e ho ough job.
B.1. Ea ly his o y. I seems o us ha one-dimensional in a ian submani olds
we e mo e o less known in he analy ic case, and wi h esonance condi ions
somewha s onge han hose conside ed in he p esen pape .
I seems well accep ed ha some e sions o in a ian mani old heo y, a
leas o he analy ic case, we e known o Da boux, Poinca ´e and Lyapuno .
Un o una ely, we ha e no been able o loca e he wo ks o Da boux, bu we
will commen on some wo ks o Poinca ´e and Lyapuno .
B.2. Two esul s o Poinca ´e. One o us (R.L.) lea ned abou he exis ence
o [Poi90] ( ep oduced in [Poi50]) om con e sa ions wi h D. Ruelle in he ea ly
80’s.
The mo i a ion o [Poi90] was he heo y o special unc ions.
When Fis a polynomial and E=C, he equa ion
F◦K( ) = K(λ ) (B.1)
can be in e p e ed as saying ha he sys em o unc ions gi en by he componen s
o Kadmi s a mul iplica ion ule ( h´eo eme de mul iplica ion). Examples o such
sys ems o unc ions (o sys ems sa is ying he closely ela ed addi ion ules) a e
he igonome ic unc ions and he ellip ic unc ions. Fo ins ance, K(θ) =
(sin θ, cos θ) sa is ies K(2θ) = F(K(θ)), whe e F(x, y) = (2xy, y2−x2). No e
ha F(0,1) = (0,1). Simila o mulas o he duplica ion o he a gumen a e
known o ellip ic in eg als. The ac ha he e a e duplica ion o mulas is ela ed
o he sol abili y o he quin ic using ellip ic unc ions and hei in e ses.
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 55
The pape [Poi90] shows ha , gi en a map Fand p o ided ha λ,|λ|>1,
is a simple eigen alue o DF(0) and ha he e a e no eigen alues o DF(0) which
a e powe s o λ, one can ind a o mal se ies o K. Mo eo e , using he majo an
me hod, one can show ha he o mal se ies o Kcon e ges.
The pape also con ains he in e es ing obse a ion (see page 541 in [Poi50])
ha when Fis a polynomial, e e y unc ion Ksa is ying (B.1) is en i e. The
eason is ha , when Fis a polynomial, he unc ional equa ion (B.1) o ces he
domain o de ini ion o K o be in a ian unde mul iplica ion by λ. Hence, i
i con ains a ball, i is he whole complex plane. We no e ha his obse a ion
gene alizes wi hou di icul y o he si ua ion when Fis an en i e unc ion and
we a e wo king on a Banach space.
In [Poi90], Poinca ´e also s udies he case when F−1is a a ional ans o ma-
ion, ha he calls C emona. In his case, he makes some dynamical obse a ions.
Fo ins ance, in he bo om hal o page 561 o [Poi50], he ela es he ques ion o
exis ence o solu ion o whe he he i e a es o he ans o ma ion con e ge o a
ixed poin — his is indeed he dynamical cha ac e iza ion o in a ian mani old.
F om a mo e dynamical poin o iew, simila se ies we e conside ed in
[Poi87], whe e all chap e VII is de o ed o asymp o ic expansions a ound pe i-
odic solu ions o pe iodic ec o ields. Taking ime-Tmaps, his p oblem educes
o he se ing abou maps ha we ha e conside ed in his pape . The loga i hms
o he eigen alues o he ime-Tmap a e called exposan s cha ac ´e is iques. In
mode n language, hey a e he Floque exponen s. No e ha wha we would
call oday Lyapuno exponen s (which can be conside ed in mo e gene al se -
ings han pe iodic sys ems) a e, in he case o pe iodic sys ems, he eal pa
o Poinca ´e’s exposan s cha ac ´e is iques. Mo e con usingly, in he ansla ion
o Lyapuno ha we ha e used, he name cha ac e is ic exponen e e s o he
nega i e o wha we call now Lyapuno exponen . This would be, o cou se, he
nega i e o he eal pa o he exposan cha ac ´e is ique o he pa icula case
o pe iodic sys ems.
In [Poi87] he c ucial pa ag aphs dealing wi h s able and uns able mani olds
a e 104 and 105. In pa ag aph 104, unde he assump ion ha he e a e no
esonances ( he non- esonance condi ion is he las o mula o pa ag aph 104),
i is shown ha one can ob ain a o mal powe se ies expansion o exponen ials
wi h a bi a y cons an s.
The con e gence o he se ies is s udied in 105. The i s pa ag aph asse s
he con e gence o he se ies o expansions in powe s o he exponen ial unde
he assump ion ha he eigen alues belong o wha we now call he Poinca ´e
domain (i.e., when he con ex hull o he eigen alues does no include ze o). O
cou se, he eason why his condi ion en e s is ha , o eigen alues sa is ying
hese condi ions, he small di iso s ha appea a e bounded away om ze o.
We no e ha , e en i i is no said explici ly, he condi ion ha he eigen al-
ues a e di e en is indeed assumed. The p oo o con e gence is a he succinc .
Ne e heless, i should ha e been qui e clea o Poinca ´e and his con empo a ies
56 X. CABR´
E, E. FONTICH, AND R. DE LA LLAVE
since i is e y simila o a gumen s ha had been done in de ail in his hesis
[Poi79] ( ep oduced in [Poi16]).
F om he poin o iew o in a ian mani old heo y, he las pa ag aph o
page 339 is qui e in e es ing. He e, Poinca ´e discusses he case when he e a e
s able and uns able cha ac e is ic exponen s a he same ime. He obse es ha
he se ies o K emains con e gen i one se s o ze o he cons an s co esponding
o coo dina es along he expanding o neu al eigendi ec ions. The a gumen s
he e a e somewha skimpy, bu a mode n ma hema ician can supply he missing
de ails wi hou oo much ouble. One is le wi h a se o solu ions which
end o ze o pa ame e ized by as many cons an s as s able di ec ions. This is,
o cou se, ou mode n s able mani old. A simila cons uc ion wo ks o he
uns able solu ions. Poinca ´e called hese solu ions solu ions asymp o iques.
The es o chap e VII con ains a a ie y o expansions o hese se s o
solu ions. I includes, qui e no ably, he expansions in e ms o a slow pa ame e ,
which a e hen shown o be di e gen . O cou se, much mode n wo k is s ill being
done in hese slow pe u ba ions and ela ed a eas.
B.3. The wo k o Lyapuno . In chap e s 11-33 [Lya92] (see also he summa y
in chap e 3 and he p oo s o con e gence in chap e 23), Lyapuno in oduces
he me hod o a bi a y cons an s, which consis s in inding exponen ial solu ions
wi h a bi a y cons an s. Since he cons an s do no e ol e in ime, his is closely
ela ed o he p oblem o linea iza ion; (compa e he expansions o he sys em
s udied and hose o he linea sys ems). In a ian mani olds can be ob ained by
se ing some o he cons an s o ze o.
One impo an di e ence be ween [Lya92] and [Poi90], [Poi87] is ha [Lya92]
conside s sys ems which a e egula ( oughly, he de ini ion is ha he o wa d
and backwa d Lyapuno exponen s ag ee). This is a mo e gene al se ing han
ha o pe iodic sys ems. In he case o egula sys ems, [Lya92] con ains ex-
pansions o he solu ions in e ms o a bi a y cons an s. The de i a ion o he
o mal expansions in [Lya92] does no need non- esonance condi ions.
In chap e 23 o [Lya92], he ques ion o con e gence o hese o mal expan-
sions is s udied. This is done unde he condi ion ha he e a e no esonances
and no epea ed eigen alues, and ha all he eigen alues a e s able o uns able.
He e one can ind a no e gi ing c edi o [Poi79] o dealing wi h he mo e gene al
case o he Poinca ´e domain.
In pa icula , we call a en ion o Theo em II o sec ion 24, which is a
comple e s a emen o he s ong s able mani old heo em o analy ic sys ems
(see also Theo em II o chap e 13).
One in e es ing ema k o Lyapuno in chap e 11 is ha one can conside
amilies ha co espond o any subse o eigen alues. This amoun s o se ing
o ze o a subse o he a bi a y cons an s used in he expansion. This is ha d o
in e p e om he dynamical poin o iew since he a bi a y cons an s do no
ha e a dynamical in e p e a ion. In pa icula , he se ob ained se ing hem
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 57
o ze o does no need o be in a ian . O cou se, se ing o ze o all he non-
dec easing modes is an in a ian se , as poin ed ou by Poinca ´e. Wi h mode n
insigh , se ing o ze o all he modes ha a e non-dec easing o dec easing mo e
slowly han a ce ain a e is in a ian . Indeed, i is he s ongly s able mani old.
As i was shown in examples in [dlL97], in gene al one canno ge in a ian
mani olds angen o a subspace i he e a e esonances o he ype we ha e
excluded in he p esen pape .
O e all, one canno be bu su p ised by he eno mous simila i ies in he
p oblems and in he esul s be ween he con en s o hese chap e s and he co -
esponding ones o he book by Poinca ´e, which appea ed in he same yea . O
cou se, he e a e big di e ences in s yle and in he me hods as well as in he way
ha p oo s a e p esen ed.
A mode n exposi ion o some o he con e gence esul s o Lyapuno can
be ound in [Le 77] V.4. I con ains a s a emen and a p oo o he expansion
in a bi a y cons an s unde non- esonance assump ions and p o ided ha all
eigen alues a e s able, and ha he linea iza ion is a cons an (we ema k ha
using Floque heo y, one can educe he pe iodic case o he cons an case). We
ha e no been able o loca e in any o hese classical wo ks he conside a ion o
esonan e ms.
B.4. Mode n wo k. I seems ha he pa icula case o one dimensional s able
in a ian mani olds (when he e a e no esonances) has appea ed se e al imes
in he mode n li e a u e.
The pape s [FR81] and [FG92] use he pa ame e iza ion me hod o one
dimensional mani olds, specially in conjunc ion wi h nume ical analysis. They
es ablish no only con e gence o he se ies in ol ed, bu hey also es ima e he
e o s incu ed when using a nume ical app oxima ion. Indeed, bo h pape s ha e
aken ca e o es ima ing ac ually he oundo e o so ha a ini e calcula ion
can es ablish ac s abou ans e sali y o in e sec ions, e c.
I seems o us ha simila esul s could be ob ained using he unc ional
equa ions (2.1) and he heo y de eloped in he p esen a icle.
Nume ical wo k o highe dimensional maps has been s udied in [BK98],
which unde ook he ask o sys ema ically compu ing Taylo expansions o in-
a ian mani olds. This could be conside ed one implemen a ion o ou esul in
Lemma 9.1 o ini e dimensional sys ems. The au ho s o [BK98] indeed made
he obse a ion ha he calcula ions can be ca ied ou o any o de p o ided
ha he e a e no esonances bu hey lea e open he issue o whe he hese o mal
calcula ions a e he je o an in a ian objec .
We no e ha ou o malism could be used o p o ide an a pos e io i es ima e
o he e o o hese nume ical calcula ions. Once a polynomial sa is ies (2.1)
qui e accu a ely, hen i is close o being a ixed poin o a map N(which is a
solu ion o T= 0). Since Nis a con ac ion, he e is a ixed poin a a dis ance
ha can be es ima ed by he e o o he nume ical app oxima ion. This is he
usual a pos e io i es ima es o nume ical analysis.