CONTINUITY OF LYAPUNOV FUNCTIONS AND OF ENERGY LEVEL
FOR A GENERALIZED GRADIENT SEMIGROUP
E. R. ARAG˜
AO-COSTA1, T. CARABALLO2, A. N. CARVALHO3, AND J. A. LANGA4
Abs ac . The global a ac o o a g adien -like semig oup has a Mo se decomposi ion.
Associa ed o his Mo se decomposi ion he e is a Lyapuno unc ion (di e en iable along
solu ions)-de ined on he whole phase space- which p o es ele an in o ma ion on he s uc-
u e o he a ac o . In his pape we p o e he con inui y o hese Lyapuno unc ions
unde pe u ba ion. On he o he hand, he a ac o o a g adien -like semig oup also has
an ene gy le el decomposi ion which is again a Mo se decomposi ion bu wi h a o al o de
be ween any wo componen s. We claim ha , om a dynamical poin o iew, his is he
op imal decomposi ion o a global a ac o ; ha is, i we s a om he ines Mo se de-
composi ion, he ene gy le el decomposi ion is he coa ses Mo se decomposi ion ha s ill
p oduces a Lyapuno unc ion which gi es he same in o ma ion abou he s uc u e o he
a ac o . We also es ablish su icien condi ions which ensu e he s abili y o his kind o
decomposi ion unde pe u ba ion. In pa icula , i connec ions be ween di e en isola ed
in a ian se s inside he a ac o emain unde pe u ba ion, we show he con inui y o he
ene gy le el Mo se decomposi ion. The class o Mo se-Smale sys ems illus a es ou esul s.
1. In oduc ion
Quali a i e p ope ies o in ini e-dimensional dynamical sys ems has been ecei ing e y
much a en ion h oughou he las ou decades (see, o ins ance, [5], [9], [14] o [2]). The
analysis o compac a ac ing in a ian se s has de eloped a p o ound a ea o esea ch, p o-
iding c ucial in o ma ion o an inc easing numbe o models o phenomena om Physics,
Biology, Economics, Enginee ing and o he s.
The asymp o ic beha iou o a dissipa i e sys em can be desc ibed by a s udy o i s
associa ed global a ac o . Mo eo e , a ca e ul s udy o he geome ical s uc u e -and
i s s abili y unde pe u ba ions- o he global a ac o leads o he unde s anding o i s
in e nal dynamics, which, essen ially, desc ibes he long ime beha iou o he whole sys em.
1Pa ially suppo ed by CAPES/DGU 267/2008 and FAPESP 2008/50248-0, B azil.
2Pa ially suppo ed by Minis e io de Ciencia e Inno aci´on g an # MTM2008-00088, PBH2006-0003-PC,
and Jun a de Andaluc´ıa g an s # P07-FQM-02468, # FQM314 and HF2008-0039, Spain.
3Pa ially suppo ed by CNPq 305447/2005-0 and 451761/2008-1, CAPES/DGU 267/2008 and FAPESP
2008/53094-4, B azil, Minis e io de Ciencia e Inno aci´on g an # MTM2008-00088, Spain, and Jun a de
Andaluc´ıa g an # P07-FQM-02468.
4Pa ially suppo ed by Minis e io de Ciencia e Inno aci´on g an s # MTM2008-00088, # PBH2006-0003-
PC, and Jun a de Andaluc´ıa g an s # P07-FQM-02468, # FQM314 and HF2008-0039, Spain.
1
2 E. R. ARAG ˜
AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA
The mos gene al esul in his line ollows om [4], which desc ibes any low on a compac
me ic space as a decomposi ion o chain ecu en isola ed in a ian se s and connec ions
be ween hem. In he e minology o [4], his is called a Mo se decomposi ion o a compac
in a ian se (see De ini ion 2.10 below), and has been conside ed in di e en amewo ks,
as in he case o lows ([4]) and semi lows on compac spaces ([13]), o e en compac and
non-compac opological spaces ([8, 11, 12]).
Recen ly, i has been in oduced in [3] he so-called g adien -like semig oups wi h espec
o a disjoin amily o isola ed in a ian se s Ξ= (Ξ1,· · · ,Ξn) on he global a ac o (see
De ini ion 2.8 below) in Banach spaces, as an in e media e concep be ween g adien semi-
g oups (i.e., hose possessing a Lyapuno unc ion) and semig oups possessing a g adien -like
a ac o ( ha is, an a ac o ha is cha ac e ized as he union o he uns able se s o as-
socia ed isola ed in a ian se s).
In [1], gi en a g adien -like semig oup in a gene al me ic space, we cons uc a di e en-
iable (along solu ions) gene alized Lyapuno unc ion p o ing ha g adien -like semig oups
a e in ac g adien semig oups. This unc ion is no only cons an on each isola ed in a ian
se as in he classical heo y o [4], bu i also de ec s he poin s in he phase space wi h
o bi s ha ing a single alue o his unc ion, a c ucial p ope y o Lyapuno unc ions (see,
o ins ance, [5]). Indeed, we will say ha a semig oup {T( ) : ≥0}wi h a global a ac o
Aand a disjoin amily o isola ed in a ian se s Ξ={Ξ1,· · · ,Ξn}is a gene alized g adien
semig oup wi h espec o Ξi he e exis s a con inuous unc ion V:X→Rsuch ha , V
is cons an in each in each Ξi, 1 ≤i≤n, [0,∞)3 7→ V(T( )x)∈Ris dec easing o each
x∈X, and V(T( )x) = V(x) o all ≥0 i and only i x∈Sn
i=1 Ξi. Fo he cons uc ion o
he Lyapuno unc ion, i is p o ed in [1] ha he disjoin amily o isola ed in a ian se s o
a g adien -like semig oup on a gene al me ic space can be eo de ed in such a way ha i
becomes a Mo se decomposi ion o he global a ac o . A e inemen o he esul s om [4]
leads o de ine a gene alized Lyapuno unc ion, no only on he a ac o bu on he whole
phase space. In addi ion, he Lyapuno unc ion V:X→Ro a gene alized g adien -like
semig oup can be chosen in such a way ha V(Ξj) = j.
Mo eo e , as g adien -like semig oups a e s able unde pe u ba ion (see [3]), we conclude
ha g adien semig oups a e s able unde pe u ba ion. In o he wo ds, he exis ence o a
con inuous Lyapuno unc ion is obus unde pe u ba ion. In his pape , we a e able o
go u he in his di ec ion, i.e., we p o ide condi ions o which no only a pe u ba ion
o a g adien semig oup is s ill g adien , bu also he associa ed Lyapuno unc ions mo e
con inuously unde he pe u ba ion. A ca e ul s udy o he uppe and lowe semicon inui y
o local a ac o s and epelle s will be c ucial in ou a gumen .
On he o he hand, obse e ha any Mo se decomposi ion Ξ= (Ξ1,· · · ,Ξn) o a compac
in a ian se Aleads o a pa ial o de among he isola ed in a ian se s Ξi; ha is, we
can de ine an o de be ween wo isola ed in a ian se s Ξiand Ξji he e is a chain o
global solu ions {ξ`,1≤`≤j−i}, wi h lim →∞ ξ`( ) = Ξi+`−1and lim →−∞ ξ`( ) = Ξi+`,
ENERGY LEVEL DECOMPOSITION 3
1≤`≤j−i. This de ines a pa ial o de and some o he isola ed in a ian se s in Ξmay no
be compa able. In Sec ion 4 we ew i e and expand he cons uc ion in [1] o a new Mo se
decomposi ion o he a ac o o a gene alized g adien -like semig oup which imp o es
he cons uc ion and dynamical p ope ies o i s associa ed Lyapuno unc ion. Indeed, we
show ha , gi en any gene alized g adien -like semig oup wi h espec o he disjoin amily
o isola ed in a ian se s Ξ= (Ξ1,· · · ,Ξn), he e exis s ano he Mo se decomposi ion gi en
by he so-called ene gy le els N= (N1,N2,· · · ,Np), p≤n, which can be o ally o de ed by
he low. Each o he le els Ni, 1 ≤i≤pis made o a ini e union o he isola ed in a ian
se s in Ξand Nis o ally o de ed. The associa ed Lyapuno unc ion akes di e en alues
in any wo di e en se s o Nand any wo elemen s o Ξwhich a e con ained in he same
elemen o N(same ene gy le el) a e no connec ed.
Because o his ene gy le el decomposi ion can be made om any g adien -like semig oup
(i.e., o any Mo se decomposi ion wi h a ini e numbe o componen s), when we s a o m
he ines Mo se decomposi ion o an in a ian se in he sense o [12], we claim ha ou new
dynamical decomposi ion is op imal, since i s associa ed Lyapuno unc ion is he simples
one in o de o desc ibe connec ed isola ed in a ian se s inside he global a ac o .
We ecall ha , gi en a Mo se decomposi ion o an a ac o , i can be con inuous unde
pe u ba ion e en i he connec ions be ween se s a e des oyed (see igu es in Sec ion 3).
This is saying ha when we desc ibe he geome ic s uc u e o he a ac o using he
associa ed isola ed in a ian subse s i may, unde pe u ba ion, change d as ically he way
hese isola ed in a ian subse s a e connec ed. In Sec ion 5 we p o e ha , i connec ions a e
kep unde pe u ba ion, hen he ene gy le el decomposi ion is s able unde pe u ba ion.
The e exis s a gene al class o semig oups sa is ying his las p ope y, being Mo se-Smale
sys ems ([7], [6]) he p o o ype o hem.
2. Mo se decomposi ion o global a ac o s o gene alized
g adien -like semig oups
Le Xbe a me ic space wi h me ic d:X×X→R+, whe e R+= [0,∞). Gi en a subse
A⊂X, he −neighbo hood o Ais he se O(A) = {x∈X:d(x, a)<  o some a∈A}
De ini ion 2.1. A amily o mappings {T( ) : ≥0}is a semig oup in Xi
•T(0) = IX, wi h IXbeing he iden i y map in X,
•T( +s) = T( )T(s), o all , s ∈R+and
•R+×X3( , x)7→ T( )x∈Xis con inuous.
The no ion o in a iance plays a undamen al ole in he s udy o he asymp o ic beha io
o semig oups
De ini ion 2.2. A subse Ao Xis said in a ian unde he ac ion semig oup {T( ) : ≥0}
i T( )A=A o all ≥0.
4 E. R. ARAG ˜
AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA
Gi en A, B ⊂X, he Hausdo semidis ance be ween Aand Bis gi en by
dis (A, B) = sup
a∈A
in
b∈Bd(a, b),
and he Hausdo dis ance by
dH(A, B) := dis (A, B) + dis (B, A).
Fo any subse s A, B and Cin Xi holds
dis (A, C)≤dis (A, B) + dis (B, C).
De ini ion 2.3. Gi en wo subse s A, B o Xwe say ha Aa ac s Bunde he ac ion o
he semig oup {T( ) : ≥0}i dis (T( )B, A) →∞
−→ 0and we say ha Aabso bs Bunde he
ac ion o {T( ) : ≥0}i he e is a B>0such ha T( )B⊂A o all ≥ B.
Wi h his we a e in condi ion o de ine global a ac o s.
De ini ion 2.4. A subse Ao Xis a global a ac o o a semig oup {T( ) : ≥0}i i
is compac , in a ian unde he ac ion o {T( ) : ≥0}and o e e y bounded subse Bo
Xwe ha e ha Aa ac s Bunde he ac ion o {T( ) : ≥0}.
2.1. G adien -like semig oups and Mo se decomposi ion o a ac o s. Nex we
seek o in oduce he no ion o gene alized g adien -like semig oups (see [3]). To ha end
we i s need he de ini ion o isola ed in a ian se .
De ini ion 2.5. Le {T( ) : ≥0}be a semig oup. We say ha an in a ian se Ξ⊂X o
he semig oup {T( ) : ≥0}is an isola ed in a ian se i he e is an  > 0such ha Ξis
he maximal in a ian subse o O(Ξ).
A disjoin amily o isola ed in a ian se s is a amily {Ξ1,· · · ,Ξn}o isola ed in a ian
se s wi h he p ope y ha , o some  > 0,
O(Ξi)∩ O(Ξj) = ∅,1≤i<j≤n.
De ini ion 2.6. A global solu ion o a semig oup {T( ) : ≥0}is a con inuous unc ion
ξ:R→Xwi h he p ope y ha T( )ξ(s) = ξ( +s) o all s∈Rand o all ∈R+. We
say ha ξ:R→Xis a global solu ion h ough x∈Xi i is a global solu ion and ξ(0) = x.
De ini ion 2.7. Le {T( ) : ≥0}be a semig oup which has a disjoin amily o iso-
la ed in a ian se s Ξ={Ξ1,· · · ,Ξn}. A homoclinic s uc u e associa ed o Ξis a subse
{Ξk1,· · · ,Ξkp}o Ξ(p≤n) oge he wi h a se o global solu ions {ξ1,· · · , ξp}such ha
Ξkj
→−∞
←− ξj( ) →∞
−→ Ξkj+1 ,1≤j≤p
whe e Ξkp+1 := Ξk1.
We a e now eady o de ine gene alized g adien -like semig oups ([3])
ENERGY LEVEL DECOMPOSITION 5
De ini ion 2.8. Le {T( ) : ≥0}be a semig oup wi h a global a ac o Aand a dis-
join amily o isola ed in a ian se s Ξ={Ξ1,· · · ,Ξn}. We say ha {T( ) : ≥0}is a
gene alized g adien -like semig oup associa ed o Ξi
•Fo any global solu ion ξ:R→ A he e a e 1≤i, j ≤nsuch ha
Ξi
→−∞
←− ξ( ) →∞
−→ Ξj.
•The e is no homoclinic s uc u e associa ed o Ξ.
Now we will in oduce he no ion o a Mo se decomposi ion o an a ac o Ao a g adien -
like semig oup {T( ) : ≥0}. We s a wi h he no ion o a ac o - epelle pai s.
De ini ion 2.9. Le {T( ) : ≥0}be a semig oup wi h a global a ac o A. We say ha a
non-emp y subse Ao Ais a local a ac o i he e is an  > 0such ha ω(O(A)) = A.
The epelle A∗associa ed o a local a ac o Ais he se de ined by
A∗={x∈ A :ω(x)∩A=∅}.
The pai (A, A∗)is called a ac o - epelle pai o {T( ) : ≥0}.
No e ha i Ais a local a ac o , hen A∗is closed and in a ian .
De ini ion 2.10. Gi en an inc easing amily ∅=A0⊂A1⊂ · · · ⊂ An=A, o local
a ac o s, de ine Ξj:= Aj∩A∗
j−1,1≤j≤n. The o de ed n-upla Ξ:= (Ξ1,Ξ2,· · · ,Ξn)is
called a Mo se decomposi ion on A.
Rema k 2.11. Obse e ha Ξis a local a ac o i and only i i is compac , in a ian and
a ac s O(Ξ) o some  > 0. We obse e ha he abo e de ini ion di e s sligh ly om he
usual de ini ion since he local a ac o is equi ed o a ac a neighbo hood o Ξin Xand
no in Aas in [4, 13].
The ollowing esul s a e p o ed in A ag˜ao-Cos a e al. [1]:
Lemma 2.12. Le {T( ) : ≥0}be a semig oup in Xwi h a global a ac o Aand an
a ac o - epelle (A, A∗). A global solu ion ξ:R→Xo {T( ) : ≥0}wi h he p ope y
ha ξ( )∈ Oδ(A∗) o all ≤0 o some δ > 0such ha Oδ(A∗)∩A=∅mus sa is y
d(ξ( ), A∗) →−∞
−→ 0.
Lemma 2.13. Le {T( ) : ≥0}be a semig oup in Xwi h a global a ac o Aand (A, A∗)
an a ac o - epelle o {T( ) : ≥0}. I ξ:R→Xis a global bounded solu ion o
{T( ) : ≥0} h ough x /∈A∪A∗, hen ξ( ) →∞
−→ Aand ξ( ) →−∞
−→ A∗. Fu he mo e, i
x∈X A hen, T( )x →∞
−→ A∪A∗.
Co olla y 2.14. I {T( ) : ≥0}is a semig oup in Xwi h a global a ac o Aand
(A, A∗)is an a ac o - epelle pai o {T( ) : ≥0}, hen {T( ) : ≥0}is a gene alized
g adien -like semig oup associa ed o he disjoin amily o isola ed in a ian se s {A, A∗}.

6 E. R. ARAG ˜
AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA
In [1] we desc ibe he cons uc ion o a Mo se decomposi ion o he a ac o o a g adien -
like semig oup associa ed o he disjoin amily o isola ed in a ian se s {Ξ1,· · · ,Ξn}and o
he associa ed collec ion o inc easing local a ac o s s a ing om he collec ion o isola ed
in a ian se s {Ξ1,· · · ,Ξn}. Fo he sake o comple eness, we ecall such a cons uc ion he e.
Le {T( ) : ≥0}be a gene alized g adien -like semig oup wi h associa ed amily o
isola ed in a ian se s Ξ={Ξ1,· · · ,Ξn}. I (a e possible eo de ing) Ξ1is a local a ac o
o {T( ) : ≥0}and
Ξ∗
1={a∈ A :ω(a)∩Ξ1=∅}
each Ξi,i > 1 is con ained in Ξ∗
1and ha o any a /∈ A {Ξ1∪Ξ∗
1}and global solu ion
φ:R→ A wi h φ(0) = awe ha e ha
Ξ∗
1
→−∞
←− φj( ) →∞
−→ Ξ1.
Conside ing he es ic ion T1( ) o T( ) o Ξ∗
1we ha e ha T1( ) is a gene alized g adien -
like semig oup in Ξ∗
1=: Ξ1,0wi h isola ed in a ian se s {Ξ2,· · · ,Ξn}and we may assume
wi hou loss o gene ali y ha Ξ2is a local a ac o o he semig oup {T1( ) : ≥0}in
Ξ∗
1. I Ξ∗
2,1is he epelle associa ed o he isola ed in a ian se Ξ2 o {T1( ) : ≥0}in Ξ∗
1
we may p oceed and conside he es ic ion {T2( ) : ≥0}o he semig oup {T1( ) : ≥0}
o Ξ∗
2,1and {T2( ) : ≥0}is a gene alized g adien -like semig oup in Ξ∗
2,1wi h associa ed
isola ed in a ian se s {Ξ3,· · · ,Ξn}.
P oceeding wi h his un il all isola ed in a ian se s a e exhaus ed we ob ain a eo de ing
o {Ξ1,· · · ,Ξn}in such a way ha Ξjis a local a ac o o he es ic ion o {T( ) : ≥0}
o Ξ∗
j−1,j−2(Ξ∗
0,−1:= A).
De ini ion 2.15. Le {T( ) : ≥0}be a semig oup. The uns able se o an in a ian se Ξ
is de ined by
Wu(Ξ) = {z∈X: he e is a global solu ion ξ:R→X
such ha ξ(0) = zand lim
→−∞ dis (ξ( ),Ξ) = 0}.
De ine A0=∅,A1= Ξ1and o j= 2,3,· · · , n
Aj=Aj−1∪Wu(Ξj) =
j
[
i=1
Wu(Ξi).(2.1)
I is clea ha An=A.
Theo em 2.16. (A ag˜ao-Cos a e al. [1]) Le {T( ) : ≥0}be a gene alized g adien -like
semig oup wi h associa ed amily o isola ed in a ian se s Ξ={Ξ1,· · · ,Ξn} eo de ed in
such a way ha Ξjis an a ac o o he es ic ion o {T( ) : ≥0} o Ξ∗
j−1,j−2. Then Aj
de ined in (2.1) is a local a ac o o {T( ) : ≥0}in X, and
Ξj=Aj∩A∗
j−1,1≤j≤n.
As a consequence, Ξde ines a Mo se decomposi ion on A.
ENERGY LEVEL DECOMPOSITION 7
2.2. A Lyapuno unc ion o a gene alized g adien -like semig oup. Le us now
ecall some de ini ions and esul s om [1].
De ini ion 2.17. We say ha a semig oup {T( ) : ≥0}wi h a global a ac o Aand a
disjoin amily o isola ed in a ian se s Ξ={Ξ1,· · · ,Ξn}is a gene alized g adien semig oup
wi h espec o Ξi he e is a con inuous unc ion V:X→Rsuch ha , Vis cons an in
Ξi, o each 1≤i≤n,[0,∞)3 7→ V(T( )x)∈Ris dec easing o each x∈Xand
V(T( )x) = V(x) o all ≥0i and only i x∈Sn
i=1 Ξi. A unc ion Vwi h he p ope ies
abo e is called a Lyapuno unc ion o he gene alized g adien semig oup {T( ) : ≥0}
wi h espec o Ξ.
P oposi ion 2.18. Le {T( ) : ≥0}be a semig oup in a me ic space (X, d)wi h global
a ac o A, and le (A, A∗)be an a ac o - epelle pai in A.Then, he e exis s a unc ion
:X→Rsa is ying he ollowing:
(i) :X→Ris con inuous in X.
(ii) :X→Ris non-inc easing along solu ions.
(iii) −1(0) = Aand −1(1) ∩ A =A∗.
(i )Gi en z∈X, i (T( )z) = (z) o all ≥0, hen z∈(A∪A∗).
Theo em 2.19. (A agao-Cos a e al. [1]) Le {T( ) : ≥0}be a semig oup wi h global
a ac o Aand a disjoin amily o isola ed in a ian se s Ξ={Ξ1,· · · ,Ξn}. Then, {T( ) :
≥0}is a gene alized g adien semig oup wi h espec o Ξi and only i i is a gene alized
g adien -like semig oup wi h espec o Ξ. Mo eo e , [0,∞)3 7→ V(T( )z)is di e en iable
o all z∈X. Finally, he Lyapuno unc ion V:X→Ro a gene alized g adien -like
semig oup may be chosen in such a way ha V(Ξk) = k,k= 1,· · · , n.
2.3. S abili y unde pe u ba ions o gene alized g adien semig oups. We in o-
duce he no ions o con inui y and asymp o ic compac ness o a pa ame e dependen amily.
We s a wi h he no ion o con inui y o a amily o semig oups.
De ini ion 2.20. A amily o semig oups {Tη( ) : ≥0}η∈[0,1] is said o be con inuous a
η= 0 i Tη( )xη→0
−→ T0( )xuni o mly o ( , x)in compac subse s o R+×X.
De ini ion 2.21. A amily o semig oups {Tη( ) : ≥0}η∈[0,1] is said o be collec i ely
asymp o ically compac a η= 0 i , gi en a sequence (ηk)k∈Nwi h ηk
k→∞
−→ 0, a bounded
sequence (xk)k∈Nin Xand a sequence ( k)k∈Nin R+wi h k
k→∞
−→ ∞, hen (Tηk( k)xk)is
ela i ely compac .
We a e now eady o s a e he ollowing esul om [3].
Theo em 2.22 (Ca alho-Langa).Le {Tη( ) : ≥0}η∈[0,1] be a collec i ely compac amily
o semig oups which is con inuous a η= 0. Assume ha
8 E. R. ARAG ˜
AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA
a) {Tη( ) : ≥0}possesses a global a ac o Aη o each η∈[0,1] and ∪η∈[0,1]Aηis
bounded.
b) The e exis s n∈Nsuch ha Aηhas nisola ed in a ian se s Ξη={Ξ1,η,· · · ,Ξn,η}
o all η∈[0,1], and sup1⩽i⩽ndH(Ξi,η,Ξi,0)η→0
−→ 0.
c) {T0( ) : ≥0}is a gene alized g adien -like semig oup.
Then, he e exis s η0>0such ha , o all η⩽η0,{Tη( ) : ≥0}is a gene alized
g adien -like semig oup associa ed o Ξηand consequen ly
Aη=∪n
i=1Wu(Ξi,η),∀η∈[0, η0].
As an immedia e consequence o his esul and he ones in Sec ion 2.2 we ha e he
ollowing esul .
Co olla y 2.23. Unde he assump ion o Theo em 2.22, he e exis s η0>0such ha , o
all η⩽η0,{Tη( ) : ≥0}is a gene alized g adien semig oup.
Co olla y 2.24. Unde he assump ion o Theo em 2.22, suppose he e exis s n∈Nsuch
ha Aηhas ns a iona y solu ions Sη={ξ1,η,· · ·, ξn,η} o all η∈[0,1] and sup1⩽i⩽nd(ξi,η, ξi,0)
η→0
−→ 0. Then, he e exis s η0>0such ha , o all η⩽η0,{Tη( ) : ≥0}is a g adien
semig oup in he sense o [5].
Rema k 2.25. The p e ious heo em supposes he con inui y o he isola ed in a ian se s
in o de o p o e he s abili y o he gene alized g adien -like semig oups unde pe u ba ion.
No e (c . [3]) ha om a pe u ba ion o a g adien -like semig oup i could eme ge a g adien -
like semig oup wi h a di e en collec ion o isola ed in a ian se s.
On he o he hand, o a gene alized g adien -like semig oup, e en when he isola ed se s
beha e con inuously unde pe u ba ion, some o he connec ions be ween hem may change.
So he dynamics unde pe u ba ion could su e d as ic changes. This ac allows ha he
Lyapuno unc ions ha we ha e cons uc ed beha e discon inuously unde pe u ba ion.
3. Con inui y o he Lyapuno unc ion unde pe u ba ion
Now, we will analyze he con inui y o he Lyapuno unc ion unde sui able pe u ba ions.
De ini ion 3.1. Le (Aη)η∈[0,1] be a amily o se s in a me ic space Xwi h dis ance d:
X×X→R+.We say ha his amily is uppe semicon inuous (u.s.c.) a η= 0 i
lim
η→0+dis (Aη, A0) = 0.
We say ha his amily is lowe semicon inuous (l.s.c.) a η= 0 i
lim
η→0+dis (A0, Aη) = 0.
ENERGY LEVEL DECOMPOSITION 9
Finally, he amily is said o be con inuous a η= 0 i i is uppe and lowe semicon inuous,
i.e., when i holds
lim
η→0+dH(Aη, A0)=0.
Lemma 3.2. Le {Tη( ) : ≥0}η∈[0,1] be a amily o collec i ely asymp o ically compac semi-
g oups in a me ic space Xwhich is con inuous a η= 0 (see De ini ion 2.20). Assume
ha each {Tη( ) : ≥0}has a global a ac o Aηand ha S
η∈[0,1]
Aηis bounded in X. Le
(Aη)η∈[0,1] be a amily o subse s in Xsuch ha Aη⊂ Aηand A0is a local a ac o o
{T0( ) : ≥0}wi h ω(Oε(A0)) = A0, o some ε > 0.
Then, i (Aη)η∈[0,1] is con inuous a η= 0,gi en δ∈(0, ε) he e exis δ0∈(0, δ)and η0>0
such ha o all η∈[0, η0]i holds
γ+
η(Oδ0(Aη)) ⊂ Oδ(Aη),
whe e γ+
η(Oδ0(Aη)) deno es he posi i e o bi o he se Oδ0(Aη)associa ed o {Tη( ) : ≥0}.
P oo . Suppose no , hen he e exis δ∈(0, ) and sequences (zj)j∈Nin X, (ηj)j∈Nin [0,1]
and ( j)j∈Nin Rsuch ha ηj
j→∞
→0+, j
j→∞
→ ∞, dis zj, Aηj<1
j o all j,
dis Tηj( )zj, Aηj< δ o all ∈[0, j) and all j∈N
and
dis Tηj( j)zj, Aηj=δ o all j∈N.
I , o each j, we now de ine ξj: [− j,∞)→Xby ξj( ) := Tηj( + j)zj, hen, by he
collec i e asymp o ic compac ness and he uni o m con e gence in compac se s, i is no
di icul o see ha he e exis a bounded global solu ion ξ0:R→X o {T0( ) : ≥0}and
a subsequence o (ξj)j∈N,deno ed he same, such ha o all , ξ0( ) = lim
j→∞ξj( ).
On he o he hand, gi en < 0, o all jbig enough i holds
dis (ξj( ), A0)≤dis ξj( ), Aηj+ dis Aηj, A0,
om whe e, by he u.s.c. o (Aη)η∈[0,1] ,we ob ain ha o all < 0
dis (ξ0( ), A0)≤δ,
and om δ= dis ξj(0) , Aηj≤dis (ξj(0) , A0) + dis A0, Aηj,by he l.s.c. o (Aη)η∈[0,1],
i ollows ha dis (ξ0(0) , A0) = δ.
Bu , as δ < ε, hen A0a ac s K={ξ0( ) : ≤0},which con adic s he ac ha
dis (ξ0(0) , A0) = δ. 
We also ha e he ollowing lemma:
16 E. R. ARAG ˜
AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA
On he o he hand, by he lowe semicon inui y o (Aη)η∈[0,1] a η= 0,le η1∈(0, η0] such
ha A0⊂ Oε0
2(Aη) i η∈[0, η1],and hen, Oε0
2(A0)⊂ Oε0(Aη) i η∈[0, η1],and hus
(3.17) implies ha
γ+
ηOε0
2(A0)⊂ Oε
2(Aη) i η∈[0, η1],
om which hη(w) = sup
≥0
dis (Tη( )w, Aη)≤ε
2 o all η∈[0, η1] and w∈ Oε0
2(A0), so ha
we conclude
sup
w∈O ε0
2
(A0)
|hη(w)−h0(w)| ≤ ε o all η∈[0, η1].
In hese condi ions, gi en ε > 0,each z∈Xpossesses a neighbou hood Oσ(z),wi h
σ=σ(ε, z)>0,and he e exis s an index η0=η0(ε, z)>0 such ha
sup
w∈Oσ(z)
|hη(w)−h0(w)| ≤ εi η∈[0, η0],
so ha we conclude he con e gence o hη
η→0+
→h0uni o mly in compac se s o Xby a
simila a gumen o he one in S ep 2. This comple es he p oo . 
Rema k 3.5. Le {Tη( ) : ≥0}η∈[0,1] be a amily o semig oups in a me ic space X
sa is ying hypo heses o Theo em 2.22 wi h isola ed in a ian se s Ξη:= {Ξ1,η,· · · ,Ξn,η}
eo de ed in such a way ha Ξjis a local a ac o o he es ic ion o {Tη( ) : ≥0} o
(Ξη)∗
j−1,j−2. Fo sui ably small η,{Tη( ) : ≥0}is a g adien -like semig oup associa ed o
Ξη,and such ha Ξηis a Mo se decomposi ion wi h associa ed local a ac o s
A0,η := ∅and o each j= 1,2,· · · , n
Aj,η :=
j
[
i=1
Wu
η(Ξi,η).
Then, he epelle s a e gi en by:
A∗
n,η := ∅and o each j= 0,1,· · · , n −1
A∗
j,η :=
n
[
i=j+1
Ws
η(Ξi,η),
whe e, o each η∈[0,1] and all i= 1,2,· · · , n
Ws
η(Ξi,η) := nz∈ Aη:Tη( )z →∞
→Ξi,ηo
and Ws
loc,η (Ξi,η)o Wu
loc,η (Ξi,η)(in he con ex o gene alized g adien -like semig oups) is he
in e sec ion o Ws
η(Ξi,η)o Wu
η(Ξi,η)wi h a neighbo hood o Ξi,η.
Thus, we mus look o su icien condi ions o ob ain con inui y o s able ( es ic ed o he
a ac o s) and uns able mani olds in o de o ob ain he con inui y o Lyapuno unc ions.
This las ema k leads us o he ollowing esul :

ENERGY LEVEL DECOMPOSITION 17
Co olla y 3.6. Le {Tη( ) : ≥0}η∈[0,1] be a amily o semig oups in a me ic space (X, d)
sa is ying he hypo heses o Theo em 2.22 wi h isola ed in a ian se s Ξη:= {Ξ1,η,· · · ,Ξn,η}.
I he s able Ws
η(Ξj,η)η∈[0,1] and uns able Wu
η(Ξj,η)η∈[0,1] mani olds a e con inuous a
η= 0, o all j= 1,2,· · · , n, hen he Lyapuno unc ions associa ed o {Tη( ) : ≥0},
gi en wi h he aid o P oposi ion 3.4, o ηsmall enough beha e con inuously a η= 0.
4. Ene gy le el decomposi ion o a gene alized g adien -like semig oup
We now gi e a dynamical desc ip ion o a gene alized g adien -like semig oup by eo de ing
and eg ouping he co esponding isola ed in a ian subse s o ob ain a o ally o de ed amily
o isola ed in a ian se s ha we will e e o as ene gy le els. This new amily o isola ed
in a ian se s is a Mo se decomposi ion o Awi h ewe in a ian se s bu in such a way ha
i s ill gi es us a Lyapuno unc ion ha is cons an only in he solu ions lying in he o iginal
isola ed in a ian se s. In a ce ain sense, his decomposi ion is he coa ses decomposi ion
which s ill gi es us a Lyapuno unc ion which is cons an only in he solu ions lying in he
o iginal isola ed in a ian se s.
Assume ha {T( ) : ≥0}is a gene alized g adien -like semig oup wi h espec o he
disjoin amily o isola ed in a ian se s Ξ={Ξ1,Ξ2,· · · ,Ξn}.
(a) Gi en Ξl1and Ξl2∈Ξ, we say ha Ξl1p ecedes Ξl2(we w i e Ξl1≺Ξl2), i he e
exis s a global solu ion ξ:R→Xo {T( ) : ≥0}such ha ξ(R)*Ξl1∪Ξl2and
lim
→−∞ d(ξ( ),Ξl2) = 0 and lim
→∞ d(ξ( ),Ξl1) = 0.
(b) Le us conside
M1:= {Ξ`∈Ξ: he e is no elemen Ξ ∈Ξ ha p eceeds Ξ`}
and, o any in ege k≥2
Mk:= {Ξ`∈Ξ: i Ξ ∈Ξand Ξ ≺Ξ` hen Ξ ∈ Mk−1}.
No e ha , by de ini ion, Mk⊂ Mk+1.
(c) We now de ine he se s
N1:= [
Ξ∈M1
Ξ,and Nk:= [
Ξ∈Mk Mk−1
Ξ, o all k≥2.
Since Ξis ini e, he e exis s a posi i e in ege qsuch ha Mk=Mq o each
k > q, so ha , Nk=∅ o all k > q. Thus, le N1,N2,· · · ,Np, he le el se s wi h
p:= min{q∈N:Mk=Mq o each k > q}.
We ha e he ollowing i s esul ela ed o his amily o se s:
18 E. R. ARAG ˜
AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA
Lemma 4.1. (see [1]) Le {T( ) : ≥0}be a semig oup wi h global a ac o A. Assume
ha {T( ) : ≥0}is a gene alized g adien -like semig oup wi h espec o he disjoin amily
o isola ed in a ian se s Ξ={Ξ1,Ξ2,· · · ,Ξn}. Then each elemen o Ξis con ained in Nk,
o some k≤p.
The ollowing esul will show ha N= (N1,N2,· · · ,Np) is a Mo se decomposi ion o
A.
Theo em 4.2. Le {T( ) : ≥0}be a semig oup wi h global a ac o A.I {T( ) :
≥0}is a gene alized g adien -like semig oup wi h espec o Ξ={Ξ1,Ξ2,· · · ,Ξn}, hen
(N1,N2,· · · ,Np)is a Mo se decomposi ion o A.
P oo . Clea ly {T( ) : ≥0}is a gene alized g adien -like semig oup wi h espec o N.
The p oo o he esul now ollows om Theo em 2.16 (see [1]). 
In o de o see ha he con inui y o local uns able mani olds is no su icien o ob ain
he con inui y o Lyapuno unc ions one may conside he example in he ollowing pic u e
- 
?
6
q
d1
q
d2
q
c
q
b
q
a-
i

^
i
Y
Y


Figu e 01
^
)
K

KKK)


R
^
-
?
6
q
c1
q
c2
q
b2q
b1
q
a-
-
i

^
i
Y
Y


Figu e 02
^
)
K

KKK)


R
^
No e ha , in bo h cases he semig oup associa ed a e g adien -like. Also, he semig oup
associa ed o Figu e 01 has ene gy le els N1={a},N2={b},N3={c},N4={d1, d2},
while he semig oup associa ed o Figu e 02 has ene gy le els N1={a},N2={b1, b2},
N3={c1, c2}. This clea ly shows ha , e en i all equilib ia a e hype bolic, i he connec ions
be ween hem a e no s able unde pe u ba ions, he le el se s may be discon inuous.
ENERGY LEVEL DECOMPOSITION 19
Rema k 4.3. All he concep s and esul s in he p e ious sec ion can be w i en in he
pa icula case in which we ha e a ini e se o equilib ia. Indeed, le {T( ) : ≥0}be a
g adien -like semig oup in Xwi h global a ac o Awi h equilib ium poin s E={ζ1,· · · , ζn}.
Then, he e exis s an ene gy le el decomposi ion in Amade o equilib ium poin s.
5. Ene gy le els o a gene alized g adien -like semig oup unde
pe u ba ion
Again, o each η∈[0,1], le {Tη( ) : ≥0}a semig oup on a me ic space X, wi h global
a ac o Aη,and a ini e amily o isola ed bounded se s Ξη={Ξ1,η,Ξ2,η,· · · ,Ξp,η},such
ha each {Tη( ) : ≥0}is a gene alized g adien -like semig oup wi h espec o Ξη.We
suppose
sup
1≤i≤p
dH(Ξi,η,Ξi,0)η→0+
→0.
Unde hese condi ions, we gi e su icien condi ions so ha he ene gy le els a e con inuous
unde pe u ba ion.
Lemma 5.1. Fo each η∈[0,1], le {Tη( ) : ≥0}be a semig oup on a me ic space X, wi h
global a ac o Aηand a ini e amily o isola ed bounded se s Ξη={Ξ1,η,Ξ2,η,· · · ,Ξn,η},
such ha each {Tη( ) : ≥0}is a gene alized g adien -like semig oup wi h espec o Ξη,and
le Nη= (N1,η, ..., Np(η),η)be he co esponding Mo se decomposi ion o med by he ene gy
le els. Assume he hypo heses o Theo em 2.22. Le Ξ0∈ N1,0and (Ξη)η∈(0,1], wi h Ξη∈Ξη
o each η∈(0,1], he unique amily such ha dH(Ξη,Ξ0)η→0+
→0.
Then he e exis δ > 0and η1∈(0,1] such ha , o any η∈(0, η1],i z∈Xis such ha
dis (z, Ξη)< δ hen dis (Tη( )z, Ξη) →∞
→0.Mo eo e , o i≥2i Ξ0∈ Ni,0and (Ξη)η∈(0,1],
wi h Ξη∈Ξη o each η∈(0,1],is he unique amily such ha dH(Ξη,Ξ0)η→0+
→0, hen he e
exis δ > 0and ηi∈(0,1] such ha o any η∈(0, ηi],i z∈Xsa is ies dis (z, Ξη)< δ, hen,
ei he dis (Tη( )z, Ξη) →∞
→0,o dis Tη( )z, M(i−1),η →∞
→, whe e M(i−1),η :=
i−1
S
j=1
Nj,η, in
pa icula dis (Tη( )z, Mi,η) →0
−→ 0.
P oo . Fo he case Ξ0∈ N1,0, suppose no . Then, he e exis ηk→0+in (0,1],(zk)k∈Nin
Xand (Ξηk)k∈Nsub amily o (Ξη)η∈(0,1] wi h d(zk,Ξηk)<1
ksuch ha Tηk( )zkdoes no
con e ge o Ξηkwhen → ∞, o all k. Fix δ0>0 such ha Oδ0(Ξi,η)∩ Oδ0(Ξj,η) = ∅
o i6=jand ηsmall enough. Then, as each {Tηk( ) : ≥0}is a gene alized g adien -like
semig oup wi h espec o Ξηk, so ha , o each k, dis Tηk( )zk,Ξ(k)
ηk →∞
→0 o some
Ξ(k)
ηk∈Ξηk {Ξηk}and so o kbig enough, we can ind τk>0 such ha
dis (Tηk( )zk,Ξηk)< δ0 o ∈[0, τk) and (5.1)
dis (Tηk(τk)zk,Ξηk) = δ0.(5.2)
20 E. R. ARAG ˜
AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA
By he uni o m con e gence Tη
η→0+
→T0on compac s o [0,∞)×X, om dH(Ξηk,Ξ0)k→∞
→0
and by (5.2) we ha e ha τk
k→∞
→ ∞.Thus, conside , o each kbig enough, he map ξk:
[−τk,∞)→Xgi en by ξk( ) := Tηk( +τk)zk ∈[−τk,∞).By he collec i e compac ness
and om (5.1) he e exis s a global bounded solu ion ξ0:R→X o {T0( ) : ≥0}such
ha lim
k→∞ξk( ) = ξ0( ) o all ∈Rand lim
→−∞dis (ξ0( ),Ξ0) = 0.Bu , as Ξ0in N1,0wi h
{T0( ) : ≥0}gene alized g adien -like, we ha e ξ0( )∈Ξ0 o all ∈R,which con adic s
d(ξ0(0) ,Ξ0) = δ0,which comes om (5.2) as k→ ∞.
Fo i= 2, we also a gue by con adic ion. Then we ob ain ηk→0+in (0,1],(zk)k∈Nin
Xand (Ξηk)k∈Nsub amily o (Ξη)η∈(0,1] wi h dis (zk,Ξηk)<1
ksuch ha Tηk( )zkdoes no
con e ge o Ξηkwhen → ∞ and Tηk( )zkdoes no con e ge o N1,ηkwhen → ∞.Now,
le δ > 0 such ha he conclusion o he p e ious case is sa is ied in Oδ(N1,ηk) o all kbig
enough and wi h Oδ(Ξi,η)∩ Oδ(Ξj,η) = ∅ o i6=jand ηsmall enough.
Thus, o all ≥0 and kwe ha e
dis (Tηk( )zk,N1,ηk)≥δ. (5.3)
On he o he hand, as each {Tηk( ) : ≥0}is a gene alized g adien -like semig oup, o
each kwe ha e ha dis Tηk( )z, Ξ(k)
ηk →∞
→0 o some Ξ(k)
ηk∈Ξηk {Ξηk}and, consequen ly,
o each kla ge enough, he e exis s τk>0 sa is ying
dis (Tηk( )zk,Ξηk)< δ o ∈[0, τk) and (5.4)
dis (Tηk(τk)zk,Ξηk) = δ. (5.5)
Again, by (5.5) ,i holds τk
k→∞
→ ∞ and hen, i we de ine ξk: [−τk,∞)→Xgi en
by ξk( ) := Tηk( +τk)zk ∈[−τk,∞), om he collec i e compac ness and (5.4) ,we
ob ain he exis ence o a global bounded solu ion o {T0( ) : ≥0}, ξ0:R→X, such
ha lim
k→∞ξk( ) = ξ0( ) o all ∈Rwi h lim
→−∞d(ξ0( ),Ξ0)=0.Since {T0( ) : ≥0}
is a gene alized g adien -like semig oup wi h espec o Ξ0, he e exis s Ξl1∈Ξ0wi h
lim
→∞dis (ξ0( ),Ξl1) = 0.As Ξ0∈ N2,0,i holds ha Ξl1∈ N1,0, om whe e, o τ > 0 wi h
dis (ξ0(τ),Ξl1)<δ
2,we deduce, o kbig enough, dis (Tηk(τ+τk)zk,N1,ηk)<δ
2,which
con adic s (5.3). A simila a gumen o he emaining cases inishes he p oo . 
Theo em 5.2. Suppose he hypo heses o he p e ious lemma. Le N1,η,· · · ,Np(η),η he
ene gy le els associa ed o he amily Ξη={Ξ1,η,· · · ,Ξn,η} o η∈(0,1],and suppose ha ,
(H)i (ηk)k∈Nis a sequence in (0,1] wi h ηk
k→∞
→0+and (Ξηk)k∈Nsa is y ha , o some
i∈T
k∈N
{1,2,· · · , p(ηk)},Ξηk∈ Ni,ηkand dH(Ξηk,Ξ0)k→∞
→0 hen Ξ0∈ Ni,0.
Then, i pdeno es he numbe o ene gy le els o {T0( ) : ≥0},w i en as N1,0,· · ·,Np,0,
he e exis s η∗∈(0,1] such ha o all η∈(0, η∗] he semig oup {Tη( ) : ≥0}possesses
ENERGY LEVEL DECOMPOSITION 21
also pene gy le els, N1,η,· · · ,Np,η (i.e., p(η) = p o all η∈(0, η∗]),and
dH(Ni,η,Ni,0)η→0+
→0 o all i= 1,2,· · · , p.
P oo . Le us w i e he ene gy le els o he limi case.
Fo i= 1,2,· · · , p,Ni,0=nΞ(i)
l1,0,· · · ,Ξ(i)
lk(i),0o.
I we de ine, o each η∈(0,1] and i= 1,2,· · · , p he se s Hi,η := nΞ(i)
l1,η,· · · ,Ξ(i)
lk(i),ηoand
H0
i,η :=
k(i)
S
j=1
Ξ(i)
lj,η, hen, Theo em 2.22 implies dHH0
i,η,Ni,0η→0+
→0, o all i= 1,2,· · · , p.
The se s Hi,η’s a e he na u al candida es o be he ene gy le els o Tη(·).Indeed,
le us p o e ha i holds ha Hi,η =Ni,η, o i= 1,2,· · · , p and ηsmall enough, i.e.,
H1,η,H2,η,· · · ,Hp,η a e he ene gy le els o {Tη( ) : ≥0} o ηsmall enough.
Fo i= 1 le Ξ(1)
l1,ηη∈(0,1] be he amily wi hin he se H1,η.Then, he e exis s θ1∈(0,1]
such ha Ξ(1)
l1,η ∈ N1,η o all η∈(0, θ1].Indeed, i no , we can ind a sequence ηk→0+and
global solu ions ξk:R→X o {Tηk( ) : ≥0}such ha lim
→−∞dis ξk( ),Ξ(1)
lj,ηk= 0 and
lim
→∞dis (ξk( ),Ξηk) = 0, o some isola ed in a ian se Ξηk∈Ξηk, bu wi h Ξηk6= Ξ(1)
l1,ηk o
all k.
Choose now, o each kbig enough, τksuch ha dis ξk( ),Ξ(1)
lj,ηk< δ0 o all <τkand
dis ξk(τk),Ξ(1)
l1,ηk=δ0,(5.6)
whe e δ0>0 sa is ies Oδ0(Ξi,η)∩ Oδ0(Ξj,η) = ∅ o i6=jand ηsmall enough.
I we de ine o kbig enough, ζk:R→Xby ζk( ) := ξk( +τk) ∈R,we ge ζ0:R→X
a global solu ion o {T0( ) : ≥0}and a subsequence o (ζk)k∈N,w i en he same, sa is ying
lim
k→∞dis (ζk( ), ζ0( )) = 0 o all ∈R,wi h lim
→−∞dis ζ0( ),Ξ(1)
l1,0= 0.Since Ξ(1)
l1,0∈ N1,0,
om he de ini ion o N1,0i ollows ha ζ0( )∈Ξ(1)
l1,0 o all ∈R. Bu his ac con adic s
ha dis ζ0(0) ,Ξ(1)
l1,0=δ0,which comes om (5.6) as k→ ∞. The same a gumen o
j= 2,· · · , k (1) leads o η1∈(0,1] such ha Ξ(1)
lj,η ∈ N1,η o j= 1,· · · , k (1) y η∈(0, η1],
ha is, H1,η ⊂ N1,η o all η∈(0, η1].
On he o he hand, he e exis s η0
1∈(0, η1] such ha N1,η ⊂ H1,η when η∈(0, η0
1].I no ,
he e exis a sequence ηk→0+and, o each k, an isola ed in a ian se Ξηk∈ N1,ηk H1,ηk
such ha he sequence (Ξηk)k∈Nsa is ies dH(Ξηk,Ξ0)k→∞
→0.Howe e , om (H), we ha e
Ξ0∈ N1,0,which con adic s ha Ξηk/∈ H1,ηk o any k. Thus, we conclude ha H1,η =N1,η
o η∈(0, η0
1].
We now show ha he e exis s η2∈(0, η0
1] such ha H2,η ⊂ M2,η i η∈(0, η2].No e ha ,
i his claim holds, om he p oo o he abo e case we ha e ha H2,η ⊂ M2,η N1,η =N2,η,
once H2,η is disjoin o H1,η =N1,η o all η∈(0, η2].

22 E. R. ARAG ˜
AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA
To ge he exis ence o η2, ake he amily Ξ(2)
l1,ηη∈(0,η1]o he elemen s in H2,η’s. Then
he e exis s θ2∈(0, η1] such ha Ξ(2)
l1,η ∈ M2,η o all η∈(0, θ2].I no , by he same a gumen
abo e, we ge a subsequence ηk→0+and co esponding global solu ions ξk:R→X o
{Tηk( ) : ≥0}such ha lim
→−∞dis ξk( ),Ξ(2)
l1,ηk= 0 and lim
→∞d(ξk( ),Ξηk) = 0, o some
isola ed in a ian se s Ξηk∈Ξηkwi h Ξηk/∈ M1,ηk=N1,ηkand Ξηk6= Ξ(2)
l1,ηk o all k. As
abo e and o he same δ0le , o each k, τk∈Rsuch ha dis ξk( ),Ξ(2)
l1,ηk< δ0 o all
< τkand dis ξk(τk),Ξ(2)
l1,ηk=δ0.
F om lemma 5.1, le δ > 0 and ¯η1∈(0, η0
1] such ha he asymp o ic s abili y o he
elemen s in N1,η a e sa is ied in Oδ(N1,η) i η∈[0,¯η1].Then,
dis (ξk( ),N1,ηk)≥δ, o all ∈Rand all k∈N.(5.7)
I we de ine he solu ions o ζk:R→Xby ζk( ) = ξk( +τk) ∈R,we ge again a
global solu ion ζ0:R→Xde {T0( ) : ≥0}such ha lim
k→∞ζk( ) = ζ0( ) o all ∈Rwi h
lim dis
→−∞ ζ0( ),Ξ(2)
l1,0= 0. As {T0( ) : ≥0}is a gene alized g adien -like semig oup, he e
exis s Ξ0∈Ξ0such ha lim dis
→∞ (ζ0( ),Ξ0) = 0 and since Ξ(2)
l1,0∈ N2,0we ge Ξ0∈ N1,0.
Thus, le τ > 0 such ha dis (ζ0(τ),Ξ0)<δ
2, om which i ollows he exis ence o k0∈N
such ha dis (ξk(τ+τk),N1,ηk)< δ o all k≥k0,which con adic s (5.7) .
The same a gumen can be used o all j= 2,· · · , k (2) and so we ob ain η2∈(0,1] such
ha H2,η ⊂ M2,η i η∈(0, η2].
Again, we ge η0
2∈(0, η2] such ha N2,η ⊂ H2,η i η∈(0, η0
2], om which we conclude
ha H2,η =N2,η o all η∈(0, η0
2].
Finally, epea ing he easoning o i= 3,· · · , p and ecalling ha Ξη=H1,η ∪ · · · ∪ Hp,η
o each η, he p oo is comple ed. 
In he ollowing heo em we s a e a su icien condi ion o he hypo heses in he p e ious
esul . In pa icula , we p o e ha he s abili y o connec ing o bi s unde pe u ba ion
gi es he desi ed esul on he con inui y o he ene gy le el se s.
Conside {Tη( ) : ≥0}η∈[0,1] a amily o gene alized g adien -like semig oups wi h espec
o Ξη={Ξ1,η,· · · ,Ξn,η} o each η∈[0,1].Suppose ha :
(HG) Fo each Ξl1,0,Ξl2,0∈Ξ0such ha Ξl1,0≺Ξl2,0,i Ξl1,η,Ξl2,η a e in Ξη o η∈(0,1]
and sa is y dH(Ξl1,η,Ξl1,0)η→0+
→0 and dH(Ξl2,η,Ξl2,0)η→0+
→0, hen Ξl1,η ≺Ξl2,η o all ηsmall
enough.
Theo em 5.3. Suppose hypo heses in Theo em 2.22, and ha (HG)is sa is ied o {Tη( ) :
≥0}η∈[0,1].
ENERGY LEVEL DECOMPOSITION 23
Then, i (ηk)k∈Nis a sequence in (0,1] wi h ηk
k→∞
→0+and o some i∈T
k∈N
{1,2,· · · , p(ηk)}
(Ξηk)k∈Nis a sequence wi h Ξηk∈ Ni,ηk o all kand dH(Ξηk,Ξ0)η→0+
→0 o some Ξ0∈Ξ0,
hen Ξ0∈ Ni,0.
P oo . Indeed, i o i= 1 and Ξ0does no belong o N1,0 he e exi s Ξl1∈Ξ0wi h Ξl1≺Ξ0
bu wi h Ξl16= Ξ0.Then, le Ξl1,ηkk∈N he sequence wi h Ξl1,ηk∈Ξηk, o all k, such ha
dH(Ξl1,η,Ξl1)η→0+
→0.By (HG) we ha e Ξl1,ηk≺Ξηk o all kbig enough, which con adic s
ha Ξηk∈ N1,ηk. Thus he esul is ue o i= 1 and om i and he i s pa o he p oo
in Theo em 5.2, we ge η1∈(0,1] such ha H1,η,a e N1,η o η∈(0, η1].
Fo i= 2, i Ξ0does no belong o N2,0=M2,0 N1,0we ha e, on he one hand, ha Ξ0is
no in N1,0,since i Ξ0∈ N1,0,as we ha e seen abo e N1,η =H1,η and so dH(N1,η,N1,0)η→0+
→0,
so ha Ξηk∈ N1,ηk o all kbig enough, which con adic s ha Ξηk∈ N2,ηk o all k.
Thus, Ξ0∈ N3,0∪ N4,0∪ · · · ∪ Nn,0and so we can ind Ξl1∈Ξ0wi h Ξl1≺Ξ0such
ha Ξl1is no in N1,0.Le (Ξl1,ηk)k∈N he sequence wi h Ξl1,ηk∈Ξηk, o all k, such ha
dH(Ξl1,ηk,Ξl1)k→∞
→0.By (HG) we ha e Ξl1,ηk≺Ξηk o all kbig enough, bu , as Ξηk∈ N2,ηk
o each k, hen Ξl1,ηk∈ N1,ηk o each k, bu hen we ge ha Ξl1∈ N1,0,which is a
con adic ion, so ha he case i= 2 is also p o ed.
Thus, by he second pa in he p oo o Theo em 5.2 we ge η2∈(0, η1] such ha he
se s H2,η,de ined as in he p e ious heo em, a e he se s N2,η o η∈(0, η1], om which, in
pa icula , dH(N2,η,N2,0)η→0+
→0.
Fo i= 3,suppose Ξ0/∈ N3,0.Again, we hen ha e ha Ξ0/∈ N1,0∪ N2,0=M2,0,since as
dH(Ni,η,Ni,0)η→0+
→0 o i= 1 and 2,i Ξ0∈ N1,0∪N2,0we would ha e ha Ξηk∈ N1,ηk∪N2,ηk
o all kbig enough, which con adic s ha Ξηk∈ N3,ηk o all k.
Thus, Ξ0∈ N4,0∪ · · · ∪ Nn,0and hen we can ind Ξl1∈Ξ0wi h Ξl1≺Ξ0such ha
Ξl1/∈ M2,0.As in he abo e cases, le (Ξl1,ηk)k∈N he sequence wi h Ξl1,ηk∈Ξ0, o all k,
such ha dH(Ξl1,ηk,Ξl1)k→∞
→0.F om (HG) we ha e ha Ξl1,ηk≺Ξηk o all kbig enough,
bu since Ξηk∈ N3,ηk o each k, hen Ξl1,ηk∈ M2,ηk o each k, bu hen Ξl1mus be in
N1,0∪ N2,0=M2,0,which con adic s he way i was chosen.
The a gumen mus s op in a ini e numbe o s eps and so he p oo is inished. 
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(T. Ca aballo and J. A. Langa) Depa amen o de Ecuaciones Di e enciales y An´
alisis Num´
e ico,
Uni e sidad de Se illa, Apdo. de Co eos 1160, 41080-Se illa Spain
E-mail add ess, T. Ca aballo: [email p o ec ed]
E-mail add ess, J. A. Langa: [email p o ec ed]
(A. N. Ca alho and E. R. A ag˜ao-Cos a) Ins i u o de Ciˆ
encias Ma em´
a icas e de Compu ac¸ao,
Uni e sidade de S˜
ao Paulo-Campus de S˜
ao Ca los, Caixa Pos al 668, 13560-970 S˜
ao Ca los
SP, B azil
E-mail add ess, E. R. A ag˜ao–Cos a: [email p o ec ed]
E-mail add ess, A. N. Ca alho: [email p o ec ed]