XX Cong eso de Ecuaciones Di e enciales y Aplicaciones
X Cong eso de Ma em´
a ica Aplicada
Se illa, 24-28 sep iemb e 2007
(pp. 1–8)
Phase po ai s o sepa able Hamil onian sys ems
A. Guillamon1, Ch. Pan azi2,
1Dep . de Ma em`a ica Aplicada I, Uni e si a Poli `ecnica de Ca alunya, D . Ma a´on n.44-50, 08028,
Ba celona, Ca alonia, Spain,E-mail: [email p o ec ed].
2Dep . de Ma em`a ica Aplicada I, Uni e si a Poli `ecnica de Ca alunya, A da. Diagonal 647 08028
Ba celona E-mails: [email p o ec ed].
Resumen
We s udy some gene aliza ions o po en ial Hamil onian sys ems (H(x, y) = y2+
F(x)) wi h one deg ee o eedom. In pa icula , we a e in e es ed in Hamil onian
sys ems wi h Hamil onian unc ions o ype H(x, y) = F(x) + G(y) a ising in applied
mechanical p oblems. We p esen an algo i hm o plo he phase po ai (include he
beha io a in ini y) o any Hamil onian sys em o ype H(x, y) = F(x)+G(y), whe e
Fand Ga e a bi a y polynomials. We a e able o gi e he ull desc ip ion in he
Poinca ´e disk acco ding o he g aphs o Fand G, ex ending he well-known me hod
o he “ ini e”phase po ai o po en ial sys ems.
1. In oduc ion
The algo i hm o plo he ini e phase po ai o po en ial sys ems is a well-known and
classical example o quali a i e heo y o o dina y di e en ial equa ions. The clue o be
able o opologically classi y his ype o Hamil onian sys ems is, ob iously, he simplici y
o i s ene gy unc ion,
H(x, y) = y2
2+F(x),
wi h F∈C1(IR) (see Figu e 1).
The ac ha H“depends”basically on F(x) allows o ela e he phase po ai o
{x0=−Hy(x, y), y0=Hx(x, y)}wi h he g aph o he one- a iable unc ion F(x). This
kind o educ ion was also explo ed in [2] o Hamil onian sys ems o ype H( , θ) =
2/2 + n+1 g(θ).
The aim o his wo k is o ex end his ype o esul s o Hamil onian sys ems de-
pending o mo e han one unc ion. We ha e chosen a na u al gene aliza ion o po en ial
sys ems, hose wi h ene gy unc ion H(x, y) = F(x) + G(y), which co e s a wide ange
1
Ch. Pan azi, A. Guillamon
F
....
.
.
Figu a 1: Fini e ep esen a ion o he phase po ai o a po en ial sys ems; minima o F
coincide wi h cen e poin s while maxima coincide wi h saddles.
o conse a i e physical models. We p o ide an algo i hm o ob ain he global (along he
pape , “global”means ha includes he beha iou a in ini y) phase po ai o polynomial
Hamil onian sys ems o he o m
x0=−G0(y), y0=F0(x),(1)
whe e Fand Ga e a bi a y polynomials. We will e e o his ec o ield as XH. Ou
desc ip ion includes he classical classi ica ion o ini e c i ical poin s o smoo h po en ial
sys ems.
To ix no a ion, we w i e:
F0(x) = anxn+· · · +a1x, an6= 0; G0(y) = bmym+· · · +b1y, bm6= 0.(2)
In he nex wo sec ions we p o ide he esul s ha suppo he algo i hm. Fo he
sake o space, his documen only p esen s he main s eam o apply he algo i hm, bu
does no co e all he p oo s and de ails.
2. Bounded dynamics
Nex wo esul s a e use ul o de e mine he beha iou o bounded o bi s o he sys em
(1).
P oposi ion 1 (Fini e c i ical poin s) Le P= (x0, y0)be a ini e singula poin o
sys em (1). Then,
(a) Pis a saddle i and only i Fhas a maximum ( esp., a minimum) a x0and Ghas
a minimum ( esp., a maximum) a y0.
2
Phase po ai s o sepa able Hamil onian sys ems
(b) Pis a cen e poin i ei he Fhas a maximum a x0and Ghas a maximum a y0
o Fhas a minimum a x0and Ghas a minimum a y0.
(c1) Pis a cusp poin i ei he Fhas an in lec ion poin a x0o Ghas an in lec ion
poin a y0.
(c2) I Fhas an in lec ion poin a x0and Ghas an in lec ion poin a y0, hen Pis a
c i ical poin o med by he union o wo hype bolic sec o s.
Saddle poin s will be deno ed by S; cen e s a ising om wo maxima will be deno ed
by C−, while hose coming om wo minima will be deno ed by C+. All o he ypes o
ini e c i ical poin s will be deno ed by D.
The bounded sepa a ices o a saddle o ganize hemsel es acco ding o he ules o he
ollowing p oposi ion.
P oposi ion 2 (Bounded sepa a ices)
Any cen e o ype C+is emb aced by he p oximal sepa a ices o he neighbou ing
saddle S∗which ene gy le el sa is ies
H(S∗) = m´ın{S:H(S)> H(C+)}.
Any cen e o ype C−is emb aced by he p oximal sepa a ices o he neighbou ing
saddle S∗which ene gy le el sa is ies
H(S∗) = m´ax{S:H(S)< H(C+)}.
I he ou sepa a ices o a saddle Semb ace he pe iod annuli o a se o cen e s
C:= {C1, . . . , Cn}, hen C ∪ S∪Wu,s(S)can be ea ed as a new cen e wi h he
same ene gy le el han S.
3. Unbounded dynamics
In o de o s udy he beha iou o sys em (1) nea in ini y we use Poinca ´e compac-
i ica ion (see o ins ance [3]). I we call {X, Y, Z} he coo dina es in which he sphe e
is exp essed, he equa o lies on Z= 0. Along he pape we a e going o use bo h cha s
(U1, F1) ({X > 0}) and (U2, F2) ({Y > 0}) o he ec o ield (1) ex ended o he sphe e.
Addi ionally, we deno e by (V1, G1) ({X < 0}) and (V2, G2) ({Y < 0}) he wo cha s on
he opposi e side o (U1, F1) and (U2, F2) espec i ely.
Depending on he sign and pa i y o n−mwe will use ei he cha , as i can be
app ecia ed om he ollowing cases:
Case 1: n > m.
In he (U1, F1) cha he ec o ield akes he o m





z1
0=zn
2³z1G0(z1
z2) + F0(1
z2)´
=an+an−1z2+· · · +a1zn−1
2+bmzm+1
1zn−m
2+···+b1z2
1zn−1
2,
z2
0=zn+1
2G0(z1
z2) = zn−m+1
2(bmzm
1+bm−1zm−1
1z2+· · · +b1z1zm−1
2).
(3)
3
Ch. Pan azi, A. Guillamon
We no e ha he e a e no c i ical poin s on {z2= 0}since an6= 0.
In he (U2, F2) cha he ec o ield can be w i en as





z0
1=−zn
2³G0(1
z2) + z1F0(z1
z2)´
=−bmzn−m
2− · · · − b1zn−1
2−anzn+1
1− · · · − a1z2
1zn−1
2,
z0
2=−zn+1
2F0(z1
z2) = −anzn
1z2− · · · − a1z1zn
2,
(4)
which has a unique c i ical poin , z1=z2= 0, on {z2= 0}.
Case 2: n=m
On he (U1, F1) cha , he ec o ield becomes
½z0
1= (an+bnzn+1
1)+(an−1+bn−1zn
1)z2+· · · + (a1+b1z2
1)zn−1
2,
z0
2=z2(bnzn
1+· · · +b1z1).(5)
We no e ha c i ical poin s a in ini y a e gi en by zn+1
1=−an
bn.
On he (U2, F2) cha he ec o ield is w i en
½z1
0=−bn−bn−1z2− · · · − b1zn−1
2−anzn+1
1−an−1zn
1z2− · · · − a1z2
1zn−1
2,
z0
2=−z2(anzn
1+···+a1z1zn−1
2),(6)
and he c i ical poin s a in ini y a e gi en by zn+1
1=−bn
an.
When nis odd and bn
an<0, bo h (5) and (6) ha e 4 c i ical poin s a in ini y, which
coincide; i bn
an>0, he e a e no c i ical poin s a in ini y. When nis e en, bo h (5) and
(6) ha e 2 c i ical poin s a in ini y, which coincide.
Case 3: n < m. A e an app op ia e change o a iables, his case is he equi alen o
Case 1. Wi hou loss o gene ali y, hen, we will omi i s s udy and concen a e on sys em
(6), de ined on cha (U2, F2).
One o he c ucial s eps o ob ain he global phase po ai is o de e mine which
sepa a ices end (bo h o posi i e and nega i e imes) o c i ical poin s a in ini y. Fo
his pu pose, we need o dis inguish special maxima and minima o unc ions Fand G,
which apply o he case ha S= (x∗, y∗) is a saddle:
De ini ion 1 Gi en a c i ical poin S= (x∗, y∗)o sys em (1), we say ha :
x∗sa is ies he p ope y l( esp., L) i F(x∗)is a minimum ( esp., maximum) o F
and F(x∗)< F(x)( esp., F(x∗)> F(x)) o each x < x∗.
x∗sa is ies he p ope y ( esp., R) i F(x∗)is a minimum ( esp., maximum) o F
and F(x∗)< F(x)( esp., F(x∗)> F(x)) o each x > x∗.
y∗sa is ies he p ope y d( esp., D) i G(y∗)is a minimum ( esp., maximum) o G
and G(y∗)< G(y)( esp., G(y∗)> G(y)) o each y < y∗.
y∗sa is ies he p ope y u( esp., U) i G(y∗)is a minimum ( esp., maximum) o G
and G(y∗)< G(y)( esp., G(y∗)> G(y)) o each y > y∗.
Addi ionally, we de ine he ollowing p ope ies:
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Phase po ai s o sepa able Hamil onian sys ems
The saddle S∗= (x∗, y∗)sa is ies p ope y (h1) i x∗sa is ies bo h he p ope ies
and l. Analogously, we will say ha i sa is ies p ope y (H1) i x∗sa is ies bo h
he p ope ies Rand L.
The saddle S∗= (x∗, y∗)sa is ies p ope y (h2u) ( esp., (H2U), (H2d) o (H2D))
i y∗sa is ies p ope y u( esp. U,do D) and i s maximal ( esp., maximal, minimal
o minimal) wi h espec o his p ope y.
Finally, we say ha (x( ), y( )) is an uppe ( esp., lowe ) sepa a ix o he saddle
Si he e exis s ² > 0such ha o |y( )−y∗|< ², we ha e y( )> y∗( esp., y( )< y∗).
Nex esul (and he co olla ies no shown he e) gi es he beha io o he main sepa-
a ices ha each o come om a c i ical poin a in ini y.
P oposi ion 3 Le S= (x∗, y∗)be a ini e saddle poin . Then:
I Ssa is ies he p ope ies (h1) and (H2U), o (H1) and (H2u), he uppe sepa a-
ices o S o m an ellip ic sec o Ea +∞( ha is, he c i ical poin o he ec o
ield on cha (U2, F2)on {z2= 0}).
I Ssa is ies he p ope ies (h1) and (H2D), o (H1) and (H2d), he lowe sepa a-
ices o S o m an ellip ic sec o Ea −∞ ( ha is, he c i ical poin o he ec o
ield on cha (V2, G2)on {z2= 0}), see Figu e 4.
Finally, o he in ini e c i ical poin s we can use Ha mann Theo em in case hey a e
hype bolic. Howe e , since he deg ees n,mand he coe icien s anand bmgi e all he
in o ma ion abou he index o he ec o ield on R2, we can deduce he index o he
in ini e c i ical poin s om Poinca ´e-Hop Theo em. Then, wi h he help o P oposi ion 3
and addi ional easonings, we can gi e he opological classi ica ion o all in ini e c i ical
poin s.
Mo e p ecisely he in o ma ion abou he ini e index allows o es ablish he ollowing
esul :
P oposi ion 4 Suppose i s ha n−mis odd and le q∞a c i ical poin on he equa o
o he Poinca ´e sphe e. Then,
I q∞does no ha e ellip ic sec o s, q∞is a node.
I q∞has one ellip ic sec o s, q∞is he union o his ellip ic sec o and a hype bolic
one, each o hem lying on a di e en hemisphe e o he Poinca ´e sphe e (see Figu e
2)(a).
Suppose now ha n−mis e en. Then:
I iR2=−1,q∞is he union o wo ellip ic sec o s (see Figu e 2)(b).
I iR2= 1,q∞is he union o wo hype bolic sec o s (see Figu e 2)(c).
5

Ch. Pan azi, A. Guillamon
.
.
Ellip ic and Two ellip ic hype bolic
hype bolic sec o s sec o s sec o s
Figu a 2: Di e en ypes o c i ical poin s on he equa o o he Poinca ´e sphe e.
4. Algo i hm
Combining p oposi ions 1, 2, 3 and 4 one can desc ibe he comple e phase po ai on
he Poinca ´e sphe e. In Figu es 3 and 4 we gi e a couple o examples, explaining he whole
p ocedu e o build up he global phase po ai .
Ag adecimien os
Pa ially suppo ed by he DGES g an numbe MTM2005-06098-C02-1, he CONA-
CIT g an numbe 2005SGR-986 and he MCyT/FEDER g an MTM2006-00478.
Re e encias
[1] S. Dilibe o,On sys ems o o dina y di e en ial equa ions, Con ibu ions o he Theo y o Nonlinea
Oscilla o s 1(1950), 1–38.
[2] A. Gasull, A. Guillamon, V. Ma˜
nosa Phase po ai o Hamil onian sys ems wi h homogeneous
nonlinea i ies, Nonlinea Analysis: Theo y, Me hods & Applica ions 42 (2000), 679–707.
[3] J. So omayo ,Li¸coes de equa¸co´es di e enciais o din´a ias, IMPA, Rio de Jane io, 1979.
6
Phase po ai s o sepa able Hamil onian sys ems
F’=(x-6)(x-1)x(x-3)(x+1)
2
G’=y(y-9)(y-4)
..
..
.
.
.
.
.
.
.
.
C
C
S
S
S
S
C
C
SC
CS
+
+
+
+
-
-
461
10
7
12
9
11
8
3
5
2
..
.
.
.
.
.
.
.
.
.
.
..
.
.
..
.
.
S8
S9
S5
S6
S3
+
S10
-
.
.
.
.
S8S9
S5S6
.
.
Figu a 3: The saddles S10 and S3become S−
10 and S+
3 espec i ely, see P oposi ion 2.
The e is a connec ion be ween each o he saddles S9,S8,S6,S5and a c i ical poin a
in ini y. C i ical poin s a in ini y a e nodes, see P oposi ion 4.
7
Ch. Pan azi, A. Guillamon
F‘=(x-4)(x-3)(x+5)
G’=(y-1)(y+2)
...
...
CSC
SCS
++
-
143
265
.
.
.
..
CS
C
++
...
SC
S
-
Figu a 4: The saddle S2sa is ies he p ope ies (h1) and (H2D). F om P oposi ion 3, he
lowe sepa a ices o he saddle S2 o m an ellip ic sec o a −∞ and so S2becomes S+
2.
Then, he saddle S4emb aces he “closed he e oclinic loop”S+
2and so S4becomes S+
4. A
he end, he saddle S5emb aces S+
4. Addi ionally, he singula poin a in ini y is a union
o ellip ic and hype bolic sec o s, see also Figu e 2(a).
8