Bifurcation of a Unique Stable Periodic Orbit from a Homoclinic Orbit in Infinite-Dimensional
Systems
Author(s): Shui-Nee Chow and Bo Deng
Source: Transactions of the American Mathematical Society, Vol. 312, No. 2 (Apr., 1989), pp. 539587
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/2001001
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TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 312, Number 2, April 1989
BIFURCATION OF A UNIQUE STABLE
PERIODIC ORBIT FROM A HOMOCLINIC ORBIT
IN INFINITE-DIMENSIONAL SYSTEMS
SHUI-NEE CHOW AND BO DENG
we showhowa uniquestablepericonditions,
ABSTRACT. Undersomegeneric
orbitforsemilinear
froma homoclinic
parabolicequaodic orbitcan bifurcate
of
equations.Thisis a generalization
functional
differential
tionsand retarded
a resultof Sil'nikovforordinary
differential
equations.
0. INTRODUCTION
Consideran autonomousdifferential
equationin the plane witha real parametere:
(0.1)
x = f(x ,e),
equilibriumforall small e
wherex E R . Assumethat x = 0 is a hyperbolic
and Dxf(O , e) = A. Supposethat A > 0 and u < 0 are theeigenvaluesof A
and
(0.2)
A+ , < 0.
Assumethatthereexistsa homoclinicorbitof (0.1) to theorigin0 at e = 0 .
conditionson equation(0.1)
It is wellknownthatundercertaintransversality
will
fromthehomoclinicorbit
an exponentially
stableperiodicorbit
bifurcate
as the parametere changes(see Andronov,Leontovich,Gordonand Maier
[1] and Chow and Hale [3], forexample,formoredetails). In [9] Neimark
and Sil'nikovgeneralizedthe above resultto R3 replacingcondition(0.2) by
is nontrivial
an appropriateconditionon the eigenvalues.This generalization
toreducethestudy
sincewearenotabletomakea smoothchangeofcoordinates
of local behaviorof solutionsneara hyperbolic
equilibriumto thatof a linear
on thelongarerelatedto somefineestimates
system.In fact,all thedifficulties
ofa hyperbolic
timebehaviorofsolutionsin a smallneighborhood
equilibrium.
In [10],Sil'nikovconsideredsimilarproblemsin higherdimensions.(For some
interesting
applications,see, forexample,Evans,Fenicheland Feroe [4] and
Feroe [5].)
ReceivedbytheeditorsJune5, 1986and,in revisedform,June8, 1987and October13, 1987.
SubjectClassification
(1985 Revision).Primary35K22, 34K99; Secondary
1980 Mathematics
35B33,35B10.
The first
supported
byDARPA and NSF GrantDMS 8401719.
authorwas partially
?
1989 American Mathematical Society
0002-9947/89 $1.00 + $.25 per page
539
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540
S.-N.CHOW AND BO DENG
The purposeof thispaperis to showthatSil'nikov'stheoremmaybe genersystems.We considertwokindsof infinitealizedto someinfinite-dimensional
equationsand
dimensionalsystems,namely,semilinearparabolicdifferential
bytheseequaequations.Sincetheflowsdefined
differential
retarded
functional
tionsare semiflows
(solutionsmaynotbe extendedbackwardsin time),we are
forourproblems.However,thebasic
notableto use Sil'nikov'smethoddirectly
is
of
idea
Sil'nikov
approach based on an
[10] whichreducesthebifurcation
familyofmaps(see
offixedpointsfora one-parameter
problemto continuation
(iii) in ?2 forsemilinearparabolicequationsor Lemma3.11 in ?3 forretarded
cases,themain
functional
differential
equations).As in thefinite-dimensional
in theproofis to obtainlocal estimatesof solutionsof our equations
difficulty
equilibria.We notethatin manyaspects,ourestimatesare new
nearhyperbolic
cases. Our estimatesare all relatedto linearvarievenin thefinite-dimensional
of the nonlinearequations.By usingthese
ationalequationsalongsemiorbits
of theabovementioned
properties
we are able to obtainsmoothness
estimates,
mapsand to findthefixedpoints.
methods.In [2],
In [2 and 12],similarresultsare obtainedbyusingdifferent
BlazqueggeneralizedSil'nikov'stheoremto semilinearparabolicequationsby
systems.Thisworkseemstobe
usingSil'nikov'swork[10] on finite-dimensional
Sil'nikov'stheoremto retardedfunctional
sketchy.In [12],Walthergeneralized
(inclination
differential
equations.Walther'sapproachis basedon the A-lemma
equations.
differential
lemmaJforretardedfunctional
We also notethatwhilewe requirethevectorfieldsto be at least C3, Walther
differential
needsonlyC2 vectorfieldsand obtainssimilarresultsforfunctional
equations.
In ?1,we considersemilinear
parabolicequations.We deriveall thenecessary
theoneestimatesrequiredin theproofof ourmainresult.In ?2,we construct
familyof mapsand proveourmainresult(Theorem2.3). Retarded
parameter
differential
functional
equationsaretreatedin ?3. An applicationis givenin ?4.
In thispaper,we willgivedetailedproofsforsemilinearparabolicequations.
For retardedfunctionaldifferential
equations,we will onlygiveproofswhen
fromthoseforsemilinearparabolicequations.
theyare different
We thankthe refereeof this paper formanyveryhelpful
Acknowledgment.
version.
revisedfromitsfirst
This paperhas been substantially
improvements.
are due to thecarefulreadingof thereferee.
Mostof theimprovements
1. LOCAL
ANALYSIS
In thissection,we considerthebehaviorofsolutionsofa semilinear
equation
on
estimates
are
based
in a Banachspace neara hyperbolic
equilibrium.Our
theworkof Henry[8] and a modifiedGronwall'sinequality(Lemma 1.1). The
main estimateis statedin Lemma 1.3 whichgivesthe exponentialboundsin
finitetimeof solutionsof thevariationalequationalonga solutionin a small
of theequilibrium.We also givetheupperand lowerboundfor
neighborhood
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BIFURCATIONOF A UNIQUE STABLE PERIODIC ORBIT
541
ofthehyperbolic
thetimeforwhicha solutionwillstayin a fixedneighborhood
equilibrium.These estimatesseem to be new even forordinarydifferential
equationsin Rd. The mainpurposeofthissectionis to definea map in a small
whichis closelyrelatedto theusual
equilibrium
ofthehyperbolic
neighborhood
In Lemma 1.7,we showthatthismap
Poincaremap butis somewhatdifferent.
of the map is
is Lipschitzwitha small Lipschitzconstant.The construction
based on Sil'nikov'sideas [10] and thesmoothness
is based on theestimates.
Let X be a Banach space withnorm I - I and A: X -? X be a linearsecan analyticsemigroup
toraloperatorwithdensedomain92(A) whichgenerates
>
<
on
Aa
t
X.
For
0
a
let
be
the
<
I,
{e A,
0}
a-fractional
powerof A
the
spectrum
Let
a(A)
be
.2r(Aa)
.
of
with
domain
[8,
pp.
24-30])
(see Henry
1,
A and assumethat Re av(A)> 0 . Defineforeach 0 < a <
Ixla = lAaxl.
Xa= O(Aa),
It is well knownthat X' is a Banach space withnorm j Ij, and X0 = X.
for a > fi> 0, Xa is a densesubspaceof Xfl withcontinuous
Furthermore,
inclusion.Let Y1 and Y2 be Banach spaces. We denoteby L(Y1 , Y2) the
Banach space of all boundedlinearoperatorsfrom Yj to Y2 equippedwith
theoperatornorminducedby thenormsof Y, and Y2.
of the originand e6 > 0 be
Let U0 c X' x R be an open neighborhood
autonomousequationwitha parameter
semilinear
fixed.Considerthefollowing
re[-o
(11)~
ix=-Ax
=
Ay +
+-f (X,y,g),
g(x,y
e),
(X,Iy) EUo I8E
(X ,y)
Y
E Uo
X I g: Uo X [-%,eo]o
where f: Uo x [-%oIe o
[-6o'80],
e e [-eo
0
]
R and A > 0 is fixed.
hypotheses:
Let a:(A) be thespectrum
of A. Considerthefollowing
(HI) Rea(A) >? > A> 0, where u is a fixedconstant;
(H2) f E CN[Uox [-e0,e0],X], g E C3[Uox [-e0,e0] ,R] and
f(0 ?0,')
=
0,
D(X Y)f(0
g(0,0,?)
=
0,
D(XY)g(0,0,0)
' ? 0)'
=
0, '
0,
[= ?
E
ccE
[-eo I60] I
o
S
are takenin theBanachspace Xa x R.
wherethederivatives
B(J1)= {(x ,y):
(HI) and (H2), thereexistsa neighborhood
By hypotheses
IXla< 5, IyI < J3} of (0,0) containedin U0 and 0 < eI < e0 such that
the local stableand unstablemanifoldsJs'Jce)and WJ'ujge)existin B(J1)
theyare
fore E [-eu'e], where5, > 0 is a smallconstant.Furthermore
C3manifoldsand are givenby
(1.2)
(1.3)
WIC(e)= {(x ,y) e B(51): y = hs(x ,e), IxIa < 3}1
Wc() = {(x,y) e B(t51):x = hu(y,g),IyI < J1},
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S.-N.CHOW AND BO DENG
542
satisfying
wherehs and hu are C3 in theirarguments
(1.4)
(1.5)
9 E[-e1,1],
hu(O,e)=0,
hs(O,e)=0,
Dxhs(O,0)=O,
Dyhu(O,O)=0.
(See Henry[8, pp. 112-116].)
For e E
[-1,
'e1] we definea diffeomorphismH(.
,* , e)
by
(X,y) = H(x,y,e) = (x - hu(ye),y)
(1.6)
map by (1.5). Its inversemap is
whichis neartheidentity
(x ,y) = H_ (x 5,y,e) = (x + hu(.y e)
(1.7)
8 E [-el 5 l].
yj)
Let (x(t) ,y(t)) be a classicalsolutionof (1.1) (see Henry[8, pp. 53-55]) and
U I(e) = H(B(c1) e) . Since y(t) is finitedimensional,H maps the solution
(x(t) ,y(t)) of (1.1) intothesolution(x(t) ,y(t)) = H(x(t) ,y(t) ,e) of
,
x = Ax +
(1.8)
f (.x,y
(X ,yV)E U I(9) 5 8
e),
Y = Ay+g(x,y ,8),
(1.8)
E
[-81 81]
E
U1(e), e[-el1],
(x,y)
where
f (x
y
8)
f
(1.9)
H-l (_ 5y 5e) e) + Ahu(y, c)
e)[Ay
-Dyhu(y
g(x ,y ,e)
=
+g(H-
(x
y
e)
)]
,y ,e) ,e).
g(H-(x
let (xc(t),y(t)) be a solutionof (1.8). Note that y(t) is finite
Conversely,
dimensional.By (1.7), (x(t) ,y(t)) = H'l(x(t) ,y(t) ,e) is a solutionof (1.1).
Thus,we have shownthatequation(1.1) is conjugateto equation(1.8) via the
changeof variables(1.6).
Since H(. 5, ,e) is near the identitymap for all E E [-e1 ,61], without
loss of generality
we assumethereis a neighborhoodUI of (O,0) suchthat
U1 c UI(e) foreverye E [-ci ,e]I Let 52 > 0 be so smallthat B(52) =
< 2, IYI< 62} C Ul
{(x, y): j&VIa
Note that(1.9) impliesthat f E C2[B(2) x [-elIe] ,X]. By thedefinition
of H, {(x ,y): x = 0, (x gy)E U1} is the local invariantunstablemanifold
of equation(1.8) forall e e [-e1 el] Hence f(O ,y ,e) = 0 for (O ,Y) E U1
we have
and e e[-e81 e1]. Therefore,
f (Xx
e) = P(x
9
,E) -x
where
P(X ,Y ,E) =
J Dff(sx ,Y ,e)ds:
B(52) x [-El ,el ] ) L(Xa ,X).
2
.
Notethat -PP iS C
C I1 SinCe
f iS C.
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BIFURCATIONOF A UNIQUE STABLE PERIODIC ORBIT
543
It follows
thatbyan appropriate
changeofvariables
wemayassumethatthe
localunstable
manifold
of equation(1.1) is givenbythey-axisin theneighborhoodU1 of (0,0)and
(1. 10)
f(x,y,C)
= P(x,y,)
*x.
However,
we willlose twoderivatives
in thisprocess.Thissaysthatif we
P is onlyC1.
assumef hastheform(1.10),thenthefunction
In thefollowing,
we assumef takestheform(1.10), P is C and U1=
B(32). By hypothesis
(H2) we choose 62 > 0, 0 < 62 < 61 and a constant
C1 > 0 so thatforevery(x,y ,) E B(62) X 1-62,621,wehave
(1.11)
IDXhs(x ,E)j < Cixlj,,Xj
(1.12)
IDyhu(y,)1j* < C1IyI,
(1.13)
(1.14)
IP(x ,y ,6)1' < Ci(IXIa+ IYI)
ID(XYP(x,y,)lIa < C1,
Ig(x ,y ,)j1 < C1(IxIc+ jyj)2,
|D(X,y)g(x,Y,E)| < Cj(IxI + IYI).
(1.15)
(1.16)
, there
Foreach (1 , 6) E B(32) X1-62 ,621
existsa uniqueclassicalsolution
(X(t; s,?,6),y(t;4, 1,6)) E B(c2), 0 < t < T < +oo , forsome T which
ofconstants
dependson (?, ,6) satisfying
thevariation
formula
x(-4 j'
)
e4
At
eAtj7+|
y(t4
e-,e)
'eX'-)((
4
eA('-S) g(x(s;
P1e)
,ys e)
,Y(
y(s;
?I,e)
e) ds,
e) ds.
Furthermore,
forfixedt, x(t; ., , ): B
1-EXA
X and y(t; -,*,):
(-2)
=
X
R
are
and
the
derivatives
C3
D(4,)x(t ; , , 6) and
p(t)
21
B(Q2) [-62
q(t) = D(4 I)y(t;?I, , E) are theuniquesolutionsof theintegralequations
-
(1.18)
p()= e-AtpO
e-A(t-s)[Dxf (x(s;,,e
I~~~~~~
+|
p
+Dyf(x(s;4,
q(t) = eA'q(?) +
t
eA(t-S)[Dxg(x(s.;4
ys,,r,),)p)
q,tE) ,y(s;4
, Q , e) ,y(s ;
+ Dy g(x(s; ;,
,q,e),c)q(s)]ds,
,'I , e) ,ep(s)
?I ,E) ,y(s; ; , I ,Ec),e)q(s)] ds
for0 < t < T. Notethat(p(t), q(t)) E L(Xa x R, Xax R) and (p(O), q(O))=
I: X' x R- Xa x R (seeHenry(8, pp. 62-65]).
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S.-N.CHOW AND BO DENG
544
Underthe assumptionon the operatorA and its spectruma(A) we may
assumethenorm I on X = X has been chosenso that
(1.19)
le-Atxl
(1.20)
le
t > 0,
< e-MtIXI
At
-xl,
<t
a e-t
>
whereCa is a constantdependingon a (see Henry[8, p. 31]).
To obtaingrowthestimatesforsolutionsin B(652)x [-62 '62Y we need the
following
versionof Gronwall'sinequalities:
Lemma1.1. Let u , a: [O , ox)
constant.If O< a < 1 and
[O , o) be locallyintegrable,and b be a positive
7, u(s) ds,
u(t) < a(t) + b |
then
u(t) < a(t) + [bF(1 - a)I1/1a
O < t < xo,
/
E/((t - s)[bF(l - a)]l/1a )a(s)
ds,
where
00
> 0
[bF( 1 -)l/la
E (t)
1 - a) + 1)
Etn(l)/v(n
-
n=O
and F is the usual Gamma function.
Proof. See Henry[8, pp. 188-189].
Remarks. (1) E'(t)
5
a) is boundedas t -- 0+. Both E'(t) (1 - a)e t
t areboundedas t -? +x,which impliesEa(t)(l - a)e t
tcT(l
-
and Ea(t) (1- a)e
is boundedon [0, + o) .
then
(2) If a(t) a, a constant,
u(t) < a(1 + E(t[b]F(1 - a)]
Lemma 1.2. Let T> 0 be fixed and e E
(x(t
;4
51, 5)
5Y(t
;4
)).
Suppose
[-62 592I
51, 5))
Ei B(J 2
for all O < t < T. Then
jx(t;4
, 1 58)10, <
1410,(l
+E E(0(j23t))e
Pt
O <
t <
T,
where 0(3) = [C C F(l - a)J/2]1/1'.
Proof.Fromthevariationof constantsformula(1.17),
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BIFURCATIONOF A UNIQUE STABLE PERIODIC ORBIT
545
Since f(x(s;E, ,c),y(s;g,r/,c),c) eX for (x(s;E,r,c),y(s;g,t,c))
B(J2), 0 < s < T, e e[-e2 ,e2j, by(1.20),
le A(t.S
A PX(xS;g,r1/,)
<
,8)
,y(s;g
'l7-x|f(X(S;,
g
e
1,)
1,8 ) Y(s; 4, 1, 8)
, 1)
By (1.12) and (1.13), we have
lf(X(S;
;,
, 8) ,Y(s;g,
1, 8) , 8)1 '
C1521X(S
;g
5,6
la.e)
Thus,
jx(t;g,j,g)1a
<e
Let u(t) = emtIx(t;g,
(t
1utjgla+,
t
--,e -j(t
S) c
Jl(sgCe)a
, 6)Ic,. Then
U + +J
1;Ca(t
u(t) < u(O)
1J2u(s) ds.
-S)a
By Lemma 1.1, thedesiredestimateis obtained. E5
growthof
Next,we willderivean estimatefortheexponential
(D(
,x (t;
, 1 ,6),D(e j)y (t;g
1 5))
whichis theuniquesolutionofthevariationalequation(1. 18) withinitialvalue
I, theidentity
map on Xa x R.
By the remarksafterLemma 1.1, [1 + E (t(O2))] *(1 - a)exp(-tO(J3)) is
boundedon (0, + o) . Hence,thereexist 0 < 33 < 2, a constantC2 > 2 and
a fixedy1> 0 suchthat
(1.21)
(1.22)
(1.23)
/= i - A)/4- 0(3) > y1 > 0(3) > 0 forevery0 <
- A)/4< ,u,
+ A)/2> A5
(1.24)
(1+E a(t6(3)))e
at<C2 ,
f+
(1.25)
0 <t<+Xo
0< 3 <?3,
<C
T e(-HhiA)sds
e(-A+A)sds < C2
(1.26)
f+00
(1.27)
J
e(-As
ds < C2.
In the following,A E (0, ,) is a givenconstant,and
constantsatisfying
( 1.28)
<33,
64<
34 >
minJ3,(A- A)/(2C,), 1/(2vC1
(A/2C+1C(8C2)
C~C2)
11[2C eC,C2(3
0 is a small
+
+ 2C2) + 8C, C2]1.
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546
S.-N.CHOW AND BO DENG
I L and 1
To simplify
thenotation,we willuse
as theoperatornormsin
II I
spaces L(Xa x R, Xa) and L(Xa x R, R), respectively.Also, we will use
as a normin Xa x R whichis definedby
(X,Y) E XAx R.
Il(X,Y)IIal= IXIa + IYI,
Lemma 1.3. Let A,34 be as above, and i,)
be as in (1.22) and (1.23). Let
T > O. Suppose (x(t; , , ,e), y(t; c, r , e)) E B(64) for O< t < T . Then
< t < T,
OO<t<
T, 9
E [ -8'6 92
ee[-e2,eJ]
x(t ;4, I , )1a< 2e-jt,',
ID(<D(;^Zx(tE,r/,1a2e
ID(Q7y (t; t ~,)1 < 2e
Proof. Note that (D(, Z)x(t;
of (1.18) such that (D(;, )x(0;
it
O<
Z)Y(t;
,),D(,
,
,,
t<
e E
T,
,))
,
, e) D(;, )y(0;
,t,
[ -82'6
is theuniquesolution
,e)) = I, the identityon
X' x R (see Henry[8, pp. 64-65]).
Definea metricspace
V
=
{(p ,q):
and q: [O,T]
x R,Xa)
. L(Xa
p: [O,T]
L(Xa
-.
x R,R)
are continuousin theoperatornormsI *la, I *I, respectively}
with-metric
d((p1 ql)
(p24,2q)
=
max
-P2(t)lI
O<t<T (IpI(t)
ql(t) - q2(t)l),
+
I and I also denote the norms in L(Xa x R, Xa) and L(Xa x R, R),
respectively.
Let V be thesubsetin V definedby
where
-
V ={(p,
-
q):
q) e V, (p(O),
(p,
q(O))
= I,
x
the identity map on Xa
R,
and Ip(t)Ia< 2eAt, q(t)l < 2eit, O?<t
Clearly,V is a closed subsetof V. Let 1: V
definedby
(1.29)
'p(t) = e-Atp(O)
+
V,
(p,q)
(p,q),
=
be
t
, r
(x(s;
e-A(t-s)[Dxf
+ Dyf(x(s;,
q(t) = eAtq(O)+ |
-
T}.
eA(t'
e()-,s[Dxg(x(s
;,
+ Dyg(x(s;
. I ,8)
rl, e , y(s; ;,
I)
4
,
e), y(s;
, e)
,1, 8), 8)q(s)]ds,
,y(s;4,
r,
~,6 ) , y(s;4
e)p(s)
,
e)p(s)
,,1 ,6)
,6)q(s)]
ds.
Firstwe showthat ID maps V intoitself.Becauseof (1.12), (1.14), (1.16),
as in the proofof
(1.19) and (p(O),q(O)) = I, by usingsimilararguments
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547
BIFURCATIONOF A UNIQUE STABLE PERIODIC ORBIT
Lemma 1.2,we have
e <
a
e -u' +
a~~JO
14
Ct 2Ce-#(t-5)lx( s;qg
+t
=1I
sIP(s)I ds
-)'3C34e
(t -S)a
+I2
)l lq(s)lds
for(p,q) E V
+I3
Hence,we have Il = e -' and
3C 35e (Ip(s)Iads
'2f 0o(t.s)Q
f) _s)a41
=2 6CC_4~
a ( 16410(t
a
I )e(-"+A)('-s)
=6C C164e 8,A t
ds.
By (1.25) we have
I2 <
6CaC1C264eMt.
By Lemma 1.2 and (1.24), we have
t,
IX(t;
(1.30)
Hence,
I3=
|
e)1a < KU'(1+ Ej(t6(34)))e
ft Ca
2Ce_Y(t s)jxs~j81j()d
Ix(s;4,t,e)IaIq(s)Ids
S)1
(t _)a
(t )ae
< 4CaC1C234
< 4CaCIC234eMt
By (1.25) we have
Therefore,
'ut< C264e
f
I3 < 4Ca
M(ts) e(-+A)s es
ds.
(t - S)ae(-"+)(t-s)
C2Cd4e
ds
t.
? (1 + CaCI C2(6 + 4C2)94)e
IIY(t)Ia
< 2e At
Mt
since 34 < 1/[CaC
CC2(6 + 4C2)] by (1.28). Similarly,
Iq(t)l < e+ + feA(tS)2C5
e
(()2CI4+q(s)l
ds
= II + I2 + I3
whereIl = eAl and
I2=
Joe
?4C,34
ds
2C
)2C54Ip(s)Ia
JleA(ts)e
ds es
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S.-N.CHOW AND BO DENG
548
Becauseof (1.23), (1.27) and -(A+,)
4C1 C234e
I2 <
I3
=|eA(
52C,641q(s)l
f
< 4C134
=
+
=
e (ts)eAs
4C1(4e
f
ds
ds
e( +A)(ts)ds.
By (1.26) we have I3 < 4C,C2.4eAf. Hence,
Iq(t)I< (1 + 8C1C254)e < 2e
(1.31)
since <? 1/[8CIC2]by (1.28). It followsthat V: V -- V. Next,we willequip
V withanothermetricd suchthat(V, d) is completeand themap VP:V , V
mapping.
underthisnewtopology(V, d) is a contraction
Let (p1, ql) (P2 q2) E V and define
d((pl, q ),
(P2 q 2)) = Omtax (e
.4
IPI(t) - P2 (t)1,} + e Atlql (t) - q2(t)D.
It is easy to verifythat (V,d) is complete.To show 4): (V,d) -- (V,d) is
contractive,let (pi ,qi) = F(pi ,qi), (pi),qi) E V for i = 1,2. From (1.29),
we have
IIPI(t) - P2(t) la <
+
( tt
C
saPI
Jt
)
(s) - P2(S)lads
2C1e
-(
S)Ix(s;
(s)- q (s)Ids
I,r1,e)IaIq
=IIX+ I2'
where
I = 3C1Cp14
=
(t S)a
(tCS)e
3CC14j
a 105
(t
< 3COC,64e-8/
IP((s)
-p2(s)Iads
-A S)e es(efsIlp
S 1
(-2()
(t1),(
S)a
X8
(s)-p2(s)I)ds
a d
(p,Q),p,2)).
By (1.25) we have
I < 1-3Ca CC264e
124
d((p,
1 ,ql), (p2,Pq2)q
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BIFURCATIONOF A UNIQUE STABLE PERIODIC ORBIT
549
Because of (1.27) and -,u + 2/+)A = 0 we have
e
)l2C,
fto(
=
I2
)u
4q 4c ) la lq, (s)
1t-S lx(s;(-)(-ij+)s-
2CaC1C254 (I
<
?2C
JoCa->|(+2-+A)s
(-(t+-)(ts-)
(t
2
aC C24e'1
/
=2C,CC26~4e
(e
e
s)le
t
S)'e(
++)ds
S) e
+iL(ds
~
e
q2 (S) I dS
q(s)-q2(s)D)ds
(-
d((pq)(p(pqq)P2))
d((p, ,ql),~(P2,~q2).
By (1.25) we have
I
<
2CC1C254e td((p, ql) ,
P2 ( q
Hence,
(1.32)
e 'lpg (t) -gp2(t) la< CC, C2(3 + 2C2)94d((p
Moreover, from (1.26) we have
Iq2(t)
I2(t)I <
-
f
e (s)2CI541pI
j
+
(s) - p2(s) la ds
(s) - q2(s)I ds
2C(tS)2j>41q
0
, ql) ,(p1 q2)).
I+12
where
I=
jeA(ts-)2Ci54Ipi(s)-p2(s)Ids
1 (eusIp (s) - p2(s))
=2CIf4 fei(t)e
2C 64 At|e(-A+A)(t-S)
e
(1+A)s ds
ds
d((p, ql,
(P2
q2))
By (1.26) we have
d((p, ql)) ,(P2
IIl < 2C, C264e'
fj
'2=
(54-lq (s)-q2(s)I
0
=2Ci54
< 2C54e
j(e
e (e
ft
j
e(
, q2))
ds
Iq,(s)-q2(s)I)ds
+A)(tS)dsd((pi
q1 ),(p2
q9)P
By (1.26) we have
I2 < 2C C254ei
d((p1
ql) , (P2 , q2)).
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550
S.-N.CHOW AND BO DENG
Hence,
e
Atlql(t) -
,ql), (P2 , q2))
q2(t)l< 4C1C264d((p1,
Therefore,
from(1.32) and theestimateabovewe have
d((1
1q1) 1 (g2
42)) = max(e" IP1(t) - p2(t)Iar+ e 1t1(t) - 42(t)1)
< [C CIC2(3 + 2C2) + 4C1C2]54d((p ,ql) '(P2 q2))
2 d((p, ) ql) ) (P2 ) q2))
since ?4 < 1/[CaCiC2(6+4C2)+8C1C2] by (1.28). By the contractionmapping
theorem,thereexistsa unique ( E,q) e V, such that (p, q) = b(f, q). By the
uniquenessof thesolutionof (1. 18), we have
p (t) = D( ^q)x(t;4, I1, 8) ,
(t) = Dg ^Z)y(t;
41^
8
twoestimatesof thetheoremhold:
Thus,thefirst
JD(<:)x(t; I, ,e)
I
< 2et,
0 < t < T, e E
[-62 ,'62J,
Next,we show
C)
' Iei,
ID, Y(t;( 181
O < t < T.
By (1.18), we have
(1.33)
D17y(t; ,18)
= e + |e
(
[Dxg(x(s;4, r18) ,y(s;4,q^,8),
E)
*D,,x(s;4,r/ ,e) +Dyg(x(s;4,r1,,e),y(s;4,C
*Dty(s;
,q,8)=
DIy(O,
,l), E)
,j,e)]ds,
1.
Let
T, = sup{t: O < t < T , Aly(t;,
, E) >
0}.
Since D y(O0,,I,e) = 1, T1 > 0. We will show T1 = T. Suppose the
contrary,T, < T, D17y(T,, ,6) = 0. By differentiating
(1.33) in t, we
have
tl
dt
=
[R+ Dxg(x(t; ,I ,e) ,y(t; ,I ,e) ,e)]JD,Iy(t;4,t1,e)
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BIFURCATIONOF A UNIQUE STABLE PERIODIC ORBIT
551
Thus,by (1.15), (1.16), (1.28) and (1.30), we havefor 0 < t < T1
dD?y(t;~
I,t ,)
6 dt
> DA
C.4Dy(;
e
t
t4]Dly(t)- 22ClCC5e(eI+Iit
2C,
I,,) - 2CIC264
> ADIy (t;
2
e(+)'
It followsfromtheabove inequalityand (1.28) that
D, y(t;' , 1, 8) > I eAt,
Thus, D y(T1;I,
0 <
t < T1.
e) > Ie" T, # 0. This is a contradiction. Hence, T1 =
t,
and thedesiredestimateholds. o
T
By (1.21) and (1.24), we have
(1.34)
< C2 fort > 0, 0 < ?<(4.
1[1+Ea(tO(6))]e'"tj
Let
(1.35)
Q(64, p,e) = {(x,y):
(1.36)
pe,e) = {(x,y):
Q+(34
4/2C2, Iy- hs(x, e)l < p},
Ixa<
(x,y) Ei (64,p P8)
p
( 1.37) Q (64, p, e) = {(x, y): (x, y) Ei 0(64, P),
,< y
-hs(xe,e)
<
p},
<
y - hs(x,
e)
<
- p
O},
where e e [-62, 2]. Note thatby (1.11) and (1.28), 41/462 > 0 and
IhS(x,,e)j< C1(5Afor IXIa? (4 and e E [-62 ,62J. Thus,forevery(x,y) E
< lY- hs(x, 8)1 + Ihs(x 8)1 < 6414
(34, p ,e) with 0 < p < 64/4 - CA52 IYI
Hence, Q(54, p ,e) c B((4/2) for every e E [-62 ),62J
Lemma 1.4. Let (x(t;
0
<
<
p
exists
4/4 -C1
e)
T(,
>
,e) ,y(t;
. Then for every (,
I,
9); ( , I , 9) - Y(T(4,
if (4, J)
_
J412
-
4roof
1fis
>
Proof.If (~')is
if (4
412 ,
E Q+(44,
p, e) uQ
),I
hs(4
E
t)jE
4 ,p ,e) U Y(%4 ,p ,e)
Furthermore, T:
5e)
)
0 such that
Y(T(4,
T(4
bea solution of(1.1) in B(54) and
,))
,
1
l_In
__
4
__
_
_
_
Q
(64 ,P,
e)
Q
(64),P
, e)
-*
T(4,t
_
-A4
in
); (,
((4, p, e), there
, 8), e)
R is C2 and satisfies
1
ie) < - In -
?4
?5
8C1 4
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552
S.-N.CHOW AND BO DENG
then (x(t ;g, I,E) ,y(t ;g, e,E)) is not in W!jOj(e). Hence, the solution cannot
stay in B(?54) for all t > 0. If 1g1< ?4/2C1, then by Lemma 1.2
forO<t<
Ix(t;4, ?,e)I, <C264/2C2=?54/2
T,
where T> 0 is such that the solution (x(t ; g , , e) , y(t ; ,q, e)) is in B(64)
for all 0 < t < T. Hence, it has to leave BQ(4) througheither hyperplane
y = ?54 or hyperplaney = -(4. Therefore,
T = inf{t > 0:
Myt;4 ,21 IE
= 64}
is well defined.
Let
,I 1,E
A(t ;
, ,E-y(t
c)
y(t ;g
2
Note that A is C2. Since (g,)e
JA(?
Since (x(t;
1,E
0 < t < T.
;g, hs(g I,E) It) I
Q(4,p,e),
I = I c1h(
) I < p < 6414 - C 64 < 64/2.
,hS(4,E),E)) isin Wjsc(e),by (1.11) wehave
,hS(g,E),E),y(t;
l^(T;4,^1,E)|~~~~~~~~~~~
|yT4?,)-yTgh
gE,)
>
b-14
4/2.
Hence, by the intermediatevalue theorem,
T(4,,E
(1.38)
) =inf{t:<t<
=4/2}
t<I(A(t;g,,,e)I
is well definedforevery (g I1) E Q+(?4, p, E) U Q? (64I p E) .
Next, we will prove that
A(T(g,I, e);~ In
(1.39)
We only prove A(T(g,I
-4/2
e)=
if(
othercase followsthe same argument.
Suppose the contrary,
Since (x(t;,hs ( ),e),E),y(t;
(1.28) we have
(1.40)
y(T(g,t6,E);4,t6,E)
,)
,)
,hs(g,e),e))
+
=
E
W(4
for (,)E
for
(0t)e~?4Q+J,p
I,E);
g,
E) = 2/
~~~~~~~~~~~~~/
4
)
)
the
ic
h
Ip I )) since
= -64/2.
isin WOc (e)Ithus,by(l.ll)and
,E) <-?414
Define
Since (,)
E Q+(4, p, E), we have
d
(1.;11 nE)( =
But, by ( 1.40),
- ha v(e
> 0.
( 1.1 1 ) and ( 1.28) we have
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BIFURCATIONOF A UNIQUE STABLE PERIODIC ORBIT
553
Hence,bytheintermediate
valuetheorem
thereexistsTo, 0< ToT
< T( ,1iE),
suchthat
(,
d(TO;
0.
=
1,e)
This means (x(TO; , n,e),y(TO; ,J,8)) E Wj'(e) and,therefore,
(xt; (, C, 8), Y(t; I,^1 8)) C wic(g).
This contradicts(4, C) E n(64 ,P , ) - Woc(e). So (1.39) holds.
Since A(t;4,t1,e) is C2 and
,0i(t ;, ?1,e) = A/\(t;,qe)
?1, + g(x(t ;, 11,e) ,y(t;, 11,e) ,e)
aA
by (1.15) and (1.28) we have
(1.41) ~
- (t(;4''
~
> 2 -4- C,6 >O
forevery(, t) E Q+(4, p, E). By applyingtheimplicitfunctiontheoremto
theequation
A(T;,
,)
we havethat T iS C
6412,
2
.
) E Q (64 'P'8)
(4
'
9 E [8 82 582]'
Moreover,byLemma 1.3 we have
54/2= Iy(r(4, ,E);X, ,E) < 2eAT( ' I'6)l-hs(
, j)I < 2e 'A
Thus
T(4,
>
,)
1
_
_
Iln4
'6) p.
_
_
5(
1
Similarly,
by usingLemma 1.3 we can show
< - In
T(4,ti,)
I~hs(~ , e)1V
Finally,let ( ), E) E
(, p, e) U QY (d5p, e). Using the chain rule and
differentiating
(1.39) withrespectto (4, ), we obtain
0
=
at
(t;
4X1
e)
DD(4'
t=l(4,Q,6)
rT(4, 15
) + D
)A(t;
45
5, e)|
,6 )
~~~~~~~~~~~~t=t(Q,
Thus,by (1.41) and Lemma 1.3 we obtain
ZJ4 -8 Cl 24
The proofis complete. o
The following
maybe foundin Henry[8, p. 71].
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554
S.-N.CHOWANDBO DENG
Lemma1.5. Let (E ,,r)E Q(N4 Ip ,e), CE [-E2 ,C2I, T > 0 . Forequation
( 1. )
thereexistsa constantC3 independent
of I,r, e2, T and 54 (see (1.28)) such
thatifthesolution(x(t; , , e), y(t; g,rI,e)) is in B(54) for0 < t < T, then
forevery0 < < 1I, d X(t;,i,e)
E L(R,X4) and
I
d
O<t< T, ee[-82e21
0
x(t;,I,e) Ifl? C3tal e",
Remark.In particular,
choose ,B= a. Then
4dX(t;* n e) <f3Le A
Notethatx(.; ',ii,e) E C [(O,T),X] but x(*; ,,e)
in generalif a #0. However,+x(t;
boundedby C3eTP/t for 0 < t < T.
,
t ,e) is an element in
Cl [(0, T), Xa]
L(R , Xa) and
Lemma1.6. Let
where(,11) E Q+(4I4,pI), p < 64/4- C1(2 and E [-e3, E331.Then,there
existsa constant
C4((4) independent
of,, I, e and &3, suchthatfor(4j I ni) E
Q+ (64, p,I
i1,2,
IXI (4I ' nl I 8) - XI (42 ' n2 I 8) la + lyY1
(4I I 11 I 8) - y1(42 '12
< C4( 4)l(g I I'1, I2
21a + II
?12)a(Il-
8) I
21)
-1
where a = min{j/PI, (u - A))/).} and
l(4I I l '42''12)=max{'1i-hS(gje):
Proof.For (4j I 17)E W (64 I p I E) I i = I I 2, let
(1.42)
g(s,e)=(l-S)l+S4,2
6(s, IE) = 71 - hs(gj I E)
Then
0 < S< I I
+
( I(l ,e) I ?(l IC)) = (42
) - hs(g(s
01(s
i= 1 ,2}.
hs(g(s, IE) I ?) I
- hs(Qj
I1
I)
0
< S
<
.
+ hs (2 IE)) 5
C) I ) = n - hs(gj I,E)I
0 < S< 1
Thus, (g(s, ), I (s,I)) E Q (34 I p IE) for 0 < s < 1. Let
( 1.43)4(S,
lE) =
(S I )
214
(
)1SAO( IE) + S?72-
Then
(s
,e)
-
h
(s ,e))
(4
?z(s,I)
-hs(g(s I c),
= (
s(
h))=(1
I)
)
1))
E) 5 ?z(l IC()) = (42 I ?12)I
E) =(
-s)(1I,
- h(S(g I e))
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BIFURCATIONOF A UNIQUE STABLE PERIODIC ORBIT
555
Thus,
0 < 6(,)-h((,)8
0 < s < 1
,
and
(64, P, 9 ),
(4(S, 9), ri(S,9)) E Q
0< S <
Next,
I 8)0
XI (42 ' 12
IXI(4,
=x1X(4I'
11,
a)1
) - Xl (42,q(
1, 8,
0
+ Xl (42,q(
1, 8,
- Xl (42 '72
= IJ +-xT(,(s, e), q(s, e), e) ds + dX (,(s, e), q(s, e), e) ds
< ||D(
5XI (4(s,
+
D(
e), q(s, e), e) d-(4(s, e), q(s, e)) ds|
(s,e) e) d-((,
x ( ,(Se)
'
1
a
a
e), (s, e)) ds
Il +12.
By usingthe chainrule,Lemmas 1.3, 1.4 and 1.5 and p < 64/4 - C154, we
have
x
JDtM
(4,1,8)la
_d
=
-X(T(4,
-dt
,;8)
,,e)D(eT468
M
+
|
C3
e!'r(~'
e
A54
D(
,)X(t;
e
C5+12
-
,17,)=
, ,)|
)| + l2e-jk(4sq ?,)
-ln(l 18CI4
-4CI4)
+ 2 (4)4
h
Furthermore,
by (1.11), (1.42) and (1.43) we have
(')
,
i(s e))
,
= Il
-
421a + IDxh (4(s,
a~~~~~~~~~~~~~
e), e)l |-4(s,
e)
< (1 + C 54)I11 - 42Ia
and
|T;E(4(S,
a), 6(
j e) + hs172
1 )(I < 12 -III Ihs(17
t = 12 +
<
(1
+
C
{1640
_
21
11
II
_721)'
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a
S.-N.CHOW AND BO DENG
556
if we let
Therefore,
4
2 1-4
)C(1+~+
( 44
-Iln(l -4CI 64) *(AJ4- 8CI j)+2
142'12) = max{i -h (j ,e): i= 1,2}
l(,,1
.
C4(4)
a = min{(,u-
,t/I
A0 ,
thenwe have
II <5 C4 (64) lo16(s,
-
e)
hs(4(s,e 8, .),a
ds
C4(J4)I1 -hsQ(1 e
.)Ia
-
2 1a + I'll
11 ' 2 ' 112) (4l-
<C4(054)1(4,l
721)s
Similarly,
'2
< C(4(34)
j
i6(s 8) - hs(4(s , e) , I)a ds
,
< C4 (04) 1 (4 ,I1
42 '12)
6l
1a + 17
(1-2
121)'
Hence,
'7I
IXI(4Il
-
(42
8)XI
12 8) la
s
9 11 Q42 9 112) (I4l - 2Xla +1 I'll
2C4(054)1(4l
121)
Next,by ( 1.11) and ( 1.39) we have
IYl(4l , 111,E)= IY(T(4l
Y1 (42 412
,
I
,e );41
-
hs(X(T
,
4/2 +Y (T(42
il I ,;e)1
e)
-
,1 e )-Y(T(42 ,'12 ,e);
'11,,E); 41 , hs(l
= 164/2+ y(T(1
=
4C)
'
,E) ,e)
1126) 42 ,hs(42
,hs(1 ,e)
hs(X(T(2
2,112 ,e)
))
e) e)
n2 ,E); '2
hs(42 e)E), ,E)|
<CI(41X(T(l( , n, e) ;l ,hs(4 se) se)
- X (T (2 9 q2 ' );2 , h (42 E)E).
we can show
Similarly,
IY,(4l,n 9, E-
YI (42 9n2 9E) I
< 2CI&4'C4((54)1(4l,
9
' 2 '9 2)a (1,
-21a
+
?ln -2)
Finally, if we let C4(54) = 2(1 + C154)C4(54), then we have
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557
BIFURCATIONOF A UNIQUE STABLE PERIODIC ORBIT
we let (4 and p < (4/4 - C1(42 be fixedand let
In thefollowing,
po < min{[2a+2(1 +
(1.44)
B(po) = {(,
S(64 E) = {(g 1):
(1.45)
/ p/2},
CI64)C4(64)1a
IJLj< p0o 1QI< po}l
= h (g, E) + 412, lxle < 4121.
t):
6
Since Wisoj(e)and Q(54 , p ,e) vary continuouslyin E E[-63
small 0 < E4 < 63 suchthatfor po as in (1.44)
(E) 0
B(po) n WiOc
(1.47)
B(po) C Q(4, 'PpE),
We definea map
3], thereexists
6 E [-E4 ,64],1
(1.46)
7C B(po)
,
6 E [-64 ,64J.
X [-64 ,641 -+ X x R
by
(1.48)
Ir'
g
if (g,j)
,y1(g, ,E)),
if (g,I)E{fi2(4,p,E)-Q
(x1(g,6)
)
(0,64/2),
EfQ
(4,p,E)nB(po),
+(,p,e)}nB(po),
where (x1(g ,1,6 ),y(g ,6 ,E)) is as in Lemma 1.6. Note thatforeach 6 E
of
,* ,6) maps B(po) into S(64 ,6) as in (1.45). This definition
[-64,64],
ft'(4
ft' is based on Sil'nikov'sideas in [9 and 10].
Let Y beaBanach space, f: Y - Y'beamapand 1: YxY -R beareallemmaand sections,we say f is Lipschitz
valued function.In thefollowing
continuouswithLipschitzconstantl(y , ,Y2) if
lf(YI)
- PYA)
< l(Y, ,YA)YI
- Y21
forevery(yI, Y2) E Y x Y. We notethatthe"constant"1 is dependenton y
willletus havemorepreciseestimateslater.
and Y2. This definition
in (g ,11,6) E B(po) x [-64 ,64] and is
Lemma1.7. ft1
(g , ,e) is continuous
in (g, t) E B(po) withLipschitzconstant
Lipschitzcontinuous
C5 (64)41(
of 6 E
independent
[-64,
64],
)a
42 ' 2
41
l
where
C5((4) = 2(1 + Cl (4)C4((4),
l(l
'l
'2
'12)
=
max{I
i - hs (j ,e)I: i=
1 ,2},
and C4((4) and a are as in Lemma 1.6.
of ft followsfromLemmas1.3 and 1.4 and Definition
Proof.The continuity
to showthat
of ftI and Lemma 1.6,it suffices
(1.48). By (1.44), thedefinition
llft(gl Il
,6)
-
fk'(42 112 ,6)IIa < C5((54)1(4, Il
' 2 ' 12)a(,l, -
21a + Ill -
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21)
558
S.-N.CHOW AND BO DENG
- hs(4j ,) > 0 and
holdsforthose ( ,i1)
t and (42 ?12) satisfying
AI =
J.
A2= 112-hS(42,) ?<0, where 11f is thenorm1a+l I in X' xR.
Let
12(S) = s(hs(42 ,8) + A0) + (1 - S)n2,
lift1
( (~ 1'61)
-
(4'2
(
-
e)j
112 )1<11
a
jf (4
If
1(~'1
Because
(4'21')1
,12(l)
- ft(~2
'61)
+ llt (422
=I+
O < s < 1.
"6)Ila
8) - :f'(42 ' 2S8)
212
Il
I2.
ByLemma 1.6,
h2(1)-hS(22,6)=A1 >0, (42 2M) E+ (64 pE)
1 < C4(t54)Q1(1
s?
1 '4
2 2) (1 - 421a+ I1l -12(1D)1)
= lhSQ(4
Since 1117 112(1)l
,) - h(4he)I
II
by (1.11), so
< C13411 -42la
< ( 1 + C134)C4(34)1(1
' 11
' 42' 12)a
_-
21a-
Note that 2( )- hS(4, e) = A > 0, but 42(0) - hs(42 . e)= 12 hs(4,2 e)
A2 < 0. So thereexists 0 < s < 1 such that 42(S) = hs(42 , ) and i2(s)hs(42, ) > O fors < s < I1,implying(42 62(S)) E Q+ (64 ,p
P, ) fors < s <1.
Hence,by Lemma 1.6,
r
f 72(1)
7(442
(1.49)
f8)
< C4(64)l(41
- f
(4~21f72(S)
' 8) - 't
12(1)
(R2
,112(S)
1 ,4 }212)a1/2(1)
=
1i2 '
'
-
172(S)l
(, q,e) in (?7)
in e E
uniformly
,e)jj
? C4(54)4'1
C4(54)l1Q
8)1Ia
' 11 ' 2 , 12)a, 12(1)
of
for s < s < 1. By the continuity
[-84 e84] and (1.46), we have
IIkl('2
6
-2MI
1)a,1(,l-)[(h
< (1 + C1.54)C4(34)(1
( II'4b 212)a (
-
s
h(1 e)) + (ti
- 121).
421a+ 1171
s(42 e)
-
-
112)11
By the definitionof f' in (1.45), it1(4,2 2(?), E) = fr1(42' 2, 8) = (0 . 54/2).
Hence, we have
'42' 112)(I,
I2 < (1 + C1c54)C4(54)l(1 I1'
-21a
+ 117- 1121).
Finally,
7r (
11l(
?1,)
8 - 7 (42 'M2 3,E)la
< 2(1 + Cl(54)C4(54)l(41
) n14294 2)(Kag- 421a+ 1ll- 1121)This proves the lemma.
O
2.
HoMOCLINIC
BIFURCATIONS
Consider the evolution equation in Xa x R with a parameter 6 E R,
(2.1)
*=-Ax+f(x,y,e),
jI
=).y+g(x,y,e).
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BIFURCATIONOF A UNIQUE STABLE PERIODIC ORBIT
559
In thissection,wewillconsiderthebehaviorofsolutionsneara homoclinic
orbit
of (2.1) whichis asymptotic
to (x ,y) = (0,0) at e = 0. We notethatthe C3
local changeof variables(1.6) is valid locallyat theoriginforequation(2.1).
Thissaysthattheestimatesobtainedin ?2 aretrueforthenewvariablesand the
map ir is welldefinedin thenewvariables.In termsof originalvariables,it is
notclearthatsimilarestimatesin Lemma 1.3 willhold. However,themap 7I
is welldefinedin theoriginalvariablesand is Lipschitzwitha smallLipschitz
constantsince 7rl is just a map and the local changeof variablesis C3 and
is neartheidentity
map. We also notethatthedomainof themap xl is also
givenby theoriginalvariables.
Considerthefollowing
hypothesis:
=
to the
(H3) At E 0, equation(2.1) has a homoclinicorbit FO asymptotic
equilibrium(0,0).
Remark.In fact,hypothesis
(H3) impliesthat Fo belongsto Xa x R.
In thissection,we considerthebehaviorof solutionsof equation(2.1) near
Fo for IE? 1. To do this,we considercertainreturnmapsnear FO.
(i) Construction
ofthemap 7r1. Recall from?1 thatunderhypotheses
(H1)
and (H2), thelocal stableand unstablemanifoldsof theorigin(0, 0) existand
theyare givenby ( 1.2), (1.3) in ?1:
< 1I}
y = hs(x ,E) l ItxIa
Wju(e) = {(x ,y): x = hu(y, ?),IyI < a}'
W.oc(E) = {(x,y):
where sl > 0 is as in ?1 and hs and hu are C3 and satisfy(1.4), (1.5).
Note thatthechangeof variablesH(. , E) of (1.6) is neartheidentity
map.
In the new coordinatesystemit has been shownin Lemma 1.7 thatthemap
Zl: B(po) x [-64 ,E4] ) Xa x R is welldefinedby (1.46) wherepO ,64 are as
in Lemma 1.7. Define
(2.2)
(2.3)
e) = H
1(54,e) = H
G(po
? E [-84
(B(po) ,),
(S(54 ,e) 5E),
,84],
e E [-e4
,641,
whereS(34, e) is as in (1.45). Since G(po , e) is open,thereexistsp1 satisfying
(2.4)
(2.5)
p1 < po/(2+ C1c54),
B(p,) c B(p,) c G(po00)
where B(pl) is the closureof B(pl) in Xa x R. Since both ''oc(6)and
in 6 E [-E64,64], thereexistsa small 0 < E5 < 64
G(po , 6) varycontinuously
suchthatthecorresponding
properties
(1.46) and (1.47) in ?1 hold for B(p1),
i.e.,
(2.6)
(2.7)
B(p1) n c(e) 5) 0
B(fp,) c G(po, E),
EE [-E5
,E5],
EE [-E5,E5].
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560
S.-N.CHOW AND BO DENG
Note thatby ( 1.6) and ( 1.11),
(2.8) IID(41)H(
< 2+ C134,
,68)lla
( ,11
(2.9) IID(4,)H
a
(, ?e)
< 2 + C1(54,
Thus, H(., -, e) maps B(pl) into B((2 + C,54)pl)
we have
H(B(p)
X [-65 ,E5],
E B(pl)
(1,8,E)
E B(p)
for 6 E
, E) C B((2 + C,54)pl) c B(po)
X[-5,eS]
[-6 5, 5].
By (2.4)
6E [-E5 ,65].
DefineHxI: B(p)x[_5_,E5]
Xa xR by (HxI)( , 1,?E) = (H(, ,6,E),
__
and ft'x I: B(po) x [-E65, E5] Xa xR by (ft x I)(, 1, E) = ( (4
'g ) 6E)
Then & : B(pl) X [-65 ,56]-1
x R is welldefined,
where
(2.10)
7r =H
((ft1 xI)o(HxI),.).
By Lemma 1.7, (2.8) and (2.9), it is clearthat 7t is continuousand Lipschitz
continuousin ( , ) witha Lipschitzconstant
(2.11)
(2+C134) 2C5(4)1(1
,15 ,$2,?12)
a
whereC5(34), 1 and a are as in Lemma 1.7, and (4j 5qi) = H(j 5qi,'E) and
(4j ' Ji) E B(pl )
i = 1 52,
E E [-E65,
E5]
.6
Next,let p =( 0? 10) E B(pl) n rO be fixed.For every
0< p< dist(po ,OB(pj)),
thedistancefromp0 to theboundaryof B(pj), thereexists0 < 66(p) < 65
suchthatfor B(p?0 P) = {(4,1): 1-~ 0 la< P,1 - 01< p} it satisfies
(2.12)
B(po ,p)
Since H(.,,6)
6
n Wi.c(6) $ 0,
6
maps B(po ,p) into B(H(
(41 ' 61
4 2 ' q2):
(4i ' qi)
forevery
, q,o),(2+C134)p)
E [-E66(p), E66(p)],
sup{
E [-E6(p) ,66(p)].
=H(i
51i,
)
(4i
' qi
, E)
E B(p ,p) x [-E6(p), E6(p)], i = 1 ,2}
< 2V2(2 + C 14) p ,
wherefunction1 is definedin Lemma 1.7.
of it' to B(p?0,p) X [-E6(p) , E6(p)]
Finally,we define7r' to be therestriction
where7r' is as in (2. 10). In otherwords, r' is definedby
(2.14)
(2.13)
1
-
1
)
1
E
5 5E)
E B(p 0(P),
6(P)]
Define
(2.15)
p' (e) = ( '(e),6 '(E)) = H
((0,54/2) , E),
6 E
[-E6(p)
,e6(p)]-
It is clearthatp' (e) is theonlyintersection
6E).
pointof Wj,(6) with (64/22,
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BIFURCATIONOF A UNIQUE STABLE PERIODIC ORBIT
561
Wemaysummarize
theproperties
of r1inthefollowing:
Lemma2.1. Let r'I: B(po,p) x [-E6(p),E6(p)] Xa x R bedefined
by(2.14),
where0 < p < dist(p?,aB(p )) and B(po p) ,E6(p) satisfy
(2.12). Then70
thefollowing:
satisfies
(a) Let ( ,tJ) E B(p0,p), E E [-E6(P),E6(p)J* If q > hs(@,E), then
70I(4,? ,E) is theintersection
pointof L(54/2,E) withtheorbitof thesolutionof(2.1) withinitialvalue( , 1) andparameter
E. On theotherhand,if
rI< hS(, E), then 1'(E,i,E) = ( ((E),eq(E)), where( '(e), tjI(e)) is defined
by(2.13),
1, ll,E) is Lipschitzian
constant
in (,,r) e B(po,p) withLipschitz
(b) nr(4
in E e [-E6(p) ,E6(p)], where
C76(j4)pauniformly
C6 (94)=
23 /(2 + Cl 4)
C5(a4)
C5(34) and a areas inLemma1.7.
Proof.(a) follows
fromthedefinitions
of ft1and nr.
from(2.11) and (2.13). o
(b) follows
2
01
ofthemap a . Let p and p (e) be as in Lemma2.1
(ii) Construction
and(2.15)respectively.
Sincep0 ,pI(0) areon J70and p1(0) E Wluo,(0),
there
existsT> 0 suchthat
(x(T;4'(O),
(0) ,0) ,y(T;l'(O)
,&(0)
0))= (4? ?,)
?p.
on theinitialvaluesandparameter
Bythecontinuous
dependence
property
forsolutions
ofequation(2.1) (seeHenry[8,pp. 62-70]),thereexistconstants
r, > 0 and E7 > 0 suchthatthemapping
(x(T; -~, ),) ,y(T;- ), ),)): B(pI(O) . r,) x [-E7 ,E87]- B(Pl)
is C3, whereB(pl) is as in (2.6) and(2.7). Wedefine
,r1) X [-E7, E17]. B(pl )
by ir2=(x(T; ,,.),y(T;,-, -)). Since 2 iS C3 at (p1(0),0),thereexist
0 < r2< r1 and 0 < E8<E7 suchthat
7t2:
IID
forevery (
(2.16)
B((p?()
7r2(1'I1
E)
<
ID(,
qD
)
7r2(pI(0),
0)11a + 1
) EB(pB(0),, r2) X [-e8, e8]. Let
C7 = IID(
e)72(PI(P(0), ,0)Il + 1.
2 maps
Notethatforevery
0<r<r2 and 0<E<E8, B(pr(0),r)x[-c,e]
intoB(p , C7r) and 72 is Lipschitzian
withLipschitz
constant
C7.
=
7.
Construction
the
Let
(iii)
of
map
p' (E) (4I(e) ,6
?I(E)) be as in (2.15).
in 6 E [-E6(p) ,E6(p)] forevery
Since p1(e) is continuous
0 < p < dist(po OB(p1)),
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562
S.-N.CHOW AND BO DENG
thereexists0 < e9(P) < e6(P) suchthat
lXI(e)
(2.17)
(2.1)
1
(O)l < C6(.54)pap,
-
(e)-1
(O)I < C6( 4) PaP,
e [-e9(p) ,89(p)]
9 E [-eg(p)e 9(p)]
where86(P) , C6(54) and a are as in Lemma2.1. Let
(2.18)
p2 =mi
10=
{ [2C7C6(64)l
min{68
-
[2C (4)l /]
,dist(p? ,OBGoi))}
e
,e9(P2)}
wherer2, C7 and 68 are as in (2.16). By Lemma2.1 and (2.17), I maps
B(p?,P2 x [-en
] into B(p'(0) ,2C6(54)pap9. By (1.19), 2C6(c54)pap2<
r2 and e1O ?< 8 . Hence, by the constructionof r2, rc2maps
B(p1 (0)
,2C6(54)p2P2)
X [-e10
'610I
into B(p0,2C7C6(34)ppa . By (2.18) we have 2C7C6(c54)pa< 1/2, thus,the
composition
(2.19)
7r(.
- 8) =
n2(7n((
) ,e)
it is clearfrom
maps B(p0 , p2)X [-61o , 60] into B(pO,p2/2). Furthermore,
Lemma 2.1 and (ii) that in is continuousin (4 ,7, ) and ir(. ,., e) is Lipschitzcontinuousfor everygiven 6 E [-e8 , 6I].
The Lipschitzconstant
is C6C6(34)pa whichis less than 1/2 by (2.18) and independentof 6 E
Therefore 7(, *,e) is a contractionmapping for every
[-el,6
06]0.
6 E [-e,10 e 90] -
We maysummarizetheproperties
of in in thefollowing:
Theorem 2.2. (a) in is continuous
and in(., *,6) is a contraction
mapping
from
in
constantlessthan 1/2 uniformly
B(p , P2) into B(p , P2) withcontraction
8 E [-610 ' 610].
(b) If (4, ) E B(pO,p2) and r > hS( , e), then r(, r,e) is on theorbit
of equation(2.1) containing( , q) . If (E ,t) satisfies1 - hS( ,6 ) < 0, then
7c( ,,1 ,) isaconstant map with rq, 6, e) = 72(p (6) ,e),whereir and p1(6)
are as in (ii) and (2.15).
(c)
B(pO P2) n
B(pO, P2) n j(
,r:
n)
q - hs(4 ,8) > ?} 7&0 ,
): q - hs(4 ,6) < O} 740 ,
E [-810
9
10]
E [-,610 910].
of in as in (2.19).
Proof. (a) followsfromtheconstruction
(b) followsfromLemma2.1.
(c) followsfromcondition(2.12). o
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563
BIFURCATIONOF A UNIQUE STABLEPERIODIC ORBIT
(iv) Homoclinicbifurcation
theorem.Let pl(e)
(2.15), and
E
(2-21)
E [-E10,E10]. Let
P2(e)
(2.20)
Wu(e)
(e)) be as in
=
=
=(2
(,,):
(E) I 72(e))= ir2(p(e) ,e),
(47,)
E E [-E10 I,10]
,y(t;4OIt0
= (X (t ;4O?o~o10,)
forsome (40,I170)E Wu-,,(g)
I qo - hs(40 E >
))
?
I t _>?}
E E [-E10,
E610]*
is partof the globalunstablemanifoldof (0,0). Note that p2(e) E
thatFo = HU(0) .
B(po Ip2)n Wu(e) . We willassumewithoutlossofgenerality
In thefollowing
by a neighborhoodN(ro) of Fo we meanan open subset
in X' x R whichcontainsFo u {(0, 0)} .
W:(e)
Theorem2.3. Considerthefollowing
semilinearevolutionequationwitha real
parameter E E [-EO egO]eE > 0:
(2.22)
(
(x,y,e)eX
ix=-Ax+f(x,y,e),
=Ay+ g(x,y,E),
xRx[-eo0eo],
E X X R x [-e ,E0].
(X,y,E)
Suppose(2.22) satisfies:
(HI) A is a sectorialoperatorwithRe (A) > ,u> A > 0;
(H2) For some fixed 0 < a < I, f E C3[Xa x R x [-eo,Eo],X],
3a
C [X x R x [-EO,O], R] and
f(0,0,e)=0,
g(0,0,e)
D(X
=
0,
Y)f(00I0)=0I
D(X y)g(050I0)
?
=
0,
Ee[-%o
g E
I80 I
E E [-eo ,60I;
to the
orbitrO asymptotic
(H3) At E = 0, equation(2.22) has a homoclinic
equilibrium(0, 0).
< go suchthat
Thenthereexistsa neighborhood
N(I0) of rO and 0 <
=
exists
a periodic
Wij(e)c N(FO) and W+(e)n WjsOj(e) 0 ifand onlyifthere
orbitin N(Io) , whereW+(e) is theorbitof (2.22) through(E,, io) satisfying
thisperiodic
(4o I7) E Wu,U(e)and rO = W-uj(0)(see (2.21)). Furthermore,
stable.
orbitis uniquein N(Fo) and is exponentially
orbitally
asymptotically
lemmas.
To proveour mainresultabove,we needthefollowing
Let
Lemma2.4. Let p' (e) and T > 0 be as in (2.15) and (2.16), respectively.
ro = {(x ,y): x = x(t;pI(0) 0) ,y = y(t;pI(0) ,0) ,0 < t < T}. Thenthere
N1 and N2 of (0,0) E X" x R and FO,
exist e81 > 0 and neighborhoods
suchthat
respectively,
orbitro;
(a) N(ro) = N, u N2 is a neighborhood
ofthehomoclinic
(b) if y isan orbitofequation(2.22) at E E [-el11 1] satisfying
ynN1$ 0,
y C N(FO), then
l- hs(E,e) >O
forevery(4,I) E yfnN1,
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564
S.-N.CHOW AND BO DENG
wherehs is themapwhosegraphis thelocalstablemanifold
(see (1.2));
(c) if (e ,) EN2 and e E [-,el,
then(x(t;E,j,,e), y(t;E,J,e)) E
B(p , P2) forsomet > O, whereB(p , P2) is as in Theorem
2.2.
Proof.Let 34 , p and Q(34 , p, e) be as in Lemma 1.4,and 64 and S(34, e) as
in Lemma 1.7. Define
~N(8) =
( )23
(2.23)
for8 E
[-E4 , 4] .
j,t):
{(
=
,
141, < 34,
054/1 < Q
=
<
i) IIa ?54
64/4
Then
D
N(E
(2.24)
(4P8,
eE[
By Lemma 1.7, forevery(,
that (x(t;4, I,6),y(t;4,J,9))
(1.6) and
N, =H
)
84,8]
e E [-64
N(e) D S(34, e),
(2.25)
hs(EIXe)< 34}
- hs( ,e)}
E N(e) n
Ea N7(6)
n
84].
((4 , p , e)
for 8
thereexistst > 0 such
[-e4,84]. Let H be as in
E
aN7 = H' (N0 (), 0).
(N(O), 0),
Since H is continuousin e, thereexists0 < I< le (see Theorem2.2) such
that
(2.26)
f~~~~~~~~~~~~~~~~~~~~
N, D H_ ()(64, p
P,E) ,8) D H (B(po),8) = G(po,8) D B(p ,P2),
jN,
E [-ell
I 911
e~~~~~~~~~~~~~~~~
<
D
H ((564
,6) ,e) = 1(34 ,e),
,
8 E [-el ,ell
6
[
whereB(p0), G(p0, 6), Z(34 ,6) and B(p0, P2) are as in (2.2), (2.3) and Theorem2.2. This impliesthefollowing
properties:
(2.27)
if ( ,1t) EN, nHM'(-
then (x(t;
and e [-e611, 1],
willleave N2 throughN7 ;
( p4,
p, e),e)- Wlc(e)
,t6,6))
,t6,e),y(t;
fN,
ON, is closedwithTo n
(2.28)
= 0.
By continuousdependenceon initialvalues and parametersand property
(2.28), foreveryf = (x(t ;pl (0), 0),y(t ;p' (0), 0)) E ro thereexistr = r(t) >
O and E = E(t) > 0 suchthat B(p-,r) n aN, = 0 and if (4, )E B(-, F) and
6
E [-t-,iJ, then
(x(T -
t;4
I6 ,E)
,y(T -
t;4 ,6I,E))
E B(po ,p2).
Note that UO<t<T B(pf F) is an open coverof ro. Since ro is compact,there
existsa finiteopen subcoverN2 = Ul7 B(fi , P,),where
,
fi = (X(ti;p2(0)
0) ,Y(ti;p2(0),
0))
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BIFURCATIONOF A UNIQUE STABLE PERIODIC ORBIT
r
565
P
and 0 < to < t1 <
< t, < T. Without loss of generalitywe
r(ti)
may assume to = 0 and fp(0) = p I(0). Define eI = min0<i<n{e(t1)} and
N(I0) = N1 U N2. We need to showthat N(ro) and eI satisfytherequired
=
properties
(a), (b) and (c).
It is obviousthat N(F0) is an open neighborhood
of Fo U {0} . This yields
(a). Since N2 nON1= 0 and ONJ c ON1c ON(FO),theboundaryof N(FO) .
This and property
(2.27) imply(b). (c) followsfromthedefinition
of B(p-,r) .
0
Lemma2.5. Let p' (E), N2 and T be as in Lemma 2.4. Thereexist 0 < fo <
and F > 0 suchthatthefollowing
hold:
properties
6le
(a) if (4,rl,E) E B(p (0) ,)x[-Uo
N2 for all 0 < t < T;
fo], then(x(t;
,?I,e),y(t;
,i ,E)) E
(b) if 6 E [-10, %], thenIpI(E) -p (0)ll,< F/4;
(c) ifp*(E) = (e*)(, q* (E)) is theuniquefixed
pointofthemap ir in Theorem
2.2 and E E
[-0Eo, 50],
IpI(e) -
then
(E()
,hs(* (E) ,e))Ila < min{C7r/4 '.P2}
wheretheconstantsC7 and P2 are givenin (2.17) and (2.19), respectively.
Proof.Let r(t) be as in theproofof Lemma2.4. By Lemma2.4, if ( , i?) E
B(pl (O) r(O)) and E E [-11,1 1J],then (x(t;,
I, E), y(t; , , E)) is defined
for all 0 < t < T. We claim that thereexist 0 < 612 < I and 0 < F < r(0)
suchthatforevery(, 7,t,E) E B(p (0) , F) x [-e12 ,612],
(x(t ; X, r1 E ) ,y(t ; X, q,
E))
E N2
forevery0 < t < T. Suppose the contrary.Then,thereexistsa sequence
1(4k 5 nk 5ek,
k = 1,525 .. *
. such that
t
WI
(X (tk ; 4k 5 ?1k 5ek) 5Y (tk ; Xk 5 1k 5Ek))
EON2 , (4k ,1k ,ek) (4(0), 52(0) ,0) as k- +oo and O< tk < T forevery
k = 1,2, .... Since [0, T] is compact,we mayassume tk - + to E [0, T] as
k +oo. By passingto thelimits,we have
r03(x(to;4
-
(0) ,q (0) ,O),yt0;4
lim (x(tk;k
k--*oo
,k
5
(0)
6ek), y(tk;
51 (0) 50)
k 5r,k 5Ek))
E aN2.
This contradictsro c N2. This yields(a). Next,since p*(E) and p1(6) are
continuousin 6, thereexists 0 < e13 < 610 such thatproperties(b) and (c)
hold true for E E [-e13 ,E13]. Define
0o=
min{610,ell
12 '6 13}.
This yieldsthedesiredresults. 0
Lemma 2.6. Let to and p*(E) be as in Lemma 2.5 and N(Fo) be the neighborhoodin Lemma 2.4. If y is a periodicor homoclinicorbitof equation (2.22)
at E E [-E-,fo%]and y C N(I0), then p*(6) E y.
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566
S.-N.CHOW AND BO DENG
Proof. Let y be eithera homoclinic,or periodic orbitof equation (2.22) at e E
[-,o ,o] . We will show thatthe inclusion y c N(FO) implies p* (e) E y, where
p* (e) is the unique fixedpoint of the map r(. , *,e) definedin Theorem 2.2. If
y is a homoclinic orbit,then p I(e) E y and i(e) = 7l2(,P
8 O)E B(po,
where P2 is as in (2.18) and e e [-fo, 0 Since y c N(FO), then y f N ln
H-1(IQ (54, p , E) , E) = 0 by Lemma 2.4(b). By Theorem 2.2(b), (7)k (p (e) , E)
is on y forevery k = O, 1, 2,... ,where (7r)k(fi(e), C)= 7((7t)kl1(p(e), E), E)
is the kth iterate of 7r(. ,. , e). Since y n Wj3S(e) #&0, there exists K > 0
such that (7K) (p(E) , ) E Wlso(e) . By Theorem 2.2, i((7f) K(p(E),)C)
=
(
Thus, 15(e) is a fixedpoint
8) = (8) ie p(7) K+1 (p() e)=
7r2(p(e)
of (7)K+1
(K
*, e) . Since every fixed point of any given iteration of a contraction
mappingis a fixedpoint of itself,we have p5(e) = p*(e) . Similarly,if y is a periodic orbitin N(FO), by Lemma 2.4(b), ynH1 (Q7 (454,p , e) , c) = 0. By the
hyperbolicity
of (O , 0) E N1, y ? N1 , i.e., ynN2 $ 0. By Lemma 2.4(c), there
Since ynH 1(Q (64,p,e),e)=0,by
mustbeapoint q0 EynB(p0,po).
Theorem 2.2(b), qk = (7r)k+1
(q? e) is on y for every k =0, 1, 2, .... Since
there
K
is
exists > 0 with qK = q , i.e. q = (t)K(qI,
y
periodic,
E) . This
implies that ql = q*(e) by the uniqueness of the fixedpoint for (r)k( , , )
k = 1, ~2, ... .a
Proof of Theorem 2.3. We show that the condition in the theorem is necessary. Suppose W<V
(e) c N(I70) for some e [-to, 90], then by Lemma 2.4(c)
WCu(e)n H l(Q ((54, p,8), E) =0 . We claim that
p* ( )
H-'(n-(64
p5e)
).
Suppose the contrary.By Theorem 2.2(b), p (e) = t(p*(e) e) = 72(p1(e), E) E
Wu(e). This is a contradiction.Since p*(e) 0 H 1(Q- (4 ,p
,p) ,), by Theorem 2.2(b), 7t(p*(e) ,) = p*(e) is on an orbit y of a solution of equation
(2.22). It followsthat y is a periodic orbit. Next, we need to show y c N(IFO).
Let the real valued function z be as in Lemma 1.4 and
T*()
=
T(H(p* (e) ,e) ,e).
Then, by Lemma 1.4, (x(t;p*(e) ,e),y(t;p*(e),8)) E N, for 0
By (2.18), Lemma 2.1 and Lemma 2.5(b), (c) we have
(2.29)
< t < T*(8).
l1(p*(e) ,e) -P1(O)II< 7IIr(P(*(8)
,e) --P (C)11
+ Ilp1(e)
-P'(O)jIl
< C6(64)p min{C7r/4, P2} + t/4
< r/4 + r/4 = F/2,
where C6(64) a, P2 and C7 are as in Lemma 2.1 and Theorem 2.2; r- is as
in Lemma 2.5. Thus, by Lemma 2.4(c),
(X(t + T* ()
;p* (e) ,)
,y(t + T* ()
;p* (e) ,))
E N2
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567
BIFURCATIONOF A UNIQUE STABLE PERIODIC ORBIT
for 0 < t < T. Thereforey c N(70). We notethattheperiod wo*(e)of y is
o ()
+ T.
= T*(
Let y
We now show thatthe conditionin the theoremis also sufficient.
be a periodicorbit. By Lemma 2.6, p*(C) E y. Let f(C) = it2(p (C) ,) E
Wu(e)nB(po P2) be as in theproofof Lemma2.6 and T be as in Lemma1.4.
Define
zk
p (C)
z
for k= 1, 2 .
(2.30)
(C)
~k
=
(ir)
=
T(ft(C)
E [-o
)
,)
,C)
E]'
CE
'C),
- ]
[_Ce
We willshowthat
pf() E H1(Q+ (J45P2 ,8) '8),
ee [-o ,Jo], k = 1,2,...
and
U
O<t<T
(2.31)
{(X,y):X=X(t;p(C)C8)),y=y(t;p
U
(C) ,)}
c N(FO),
{(x ,y): x = x(t;fk (e) Ce)y=y(t;fk (C) ,)}cN(F),
O<t<TA(e)+T
8E
-oeoi]k=
152....
Note thatif (2.30) and (2.31) hold true,then,givena p E W(C), p is in
either
8)),y=y(t;p (e) ,C)} c N(ro)
U {(X,y):X=X(t;p1(C
O<t<T
or
U
{(X
O<t<tk (e)+T
,y):
x = x(t;fk
()
,C)
,y
=
)}
y(t;fk(C)
-theconclusion W:(C) c N(FO) holds
forsome k = 1 ,,2..
Therefore,
true. Thus it suffices
to show (2.30) and (2.31). Suppose (2.30) is not true,
-K
thenthereexists K such that f (e) E H-1(Q7 (34, p2 ,E) ,E ) U WI'c(E) and
t (C) H (Q 4,4 P2, ), E) U Wis(C(), k = 1, 2,.. ,K-1 . By Theorem
-r(iK((C) 5)=
(7 K(e
6
,C) = p(C).
,C) = 7((7)
It followsthatp(E) is the
2.2(b), r(f
(p(C) ,C))58
K+ i
fixedpointof (7r) . Thus,it is thefixedpointof 7r and p(C) = p*(C) E y.
-kc
and proves(2.30). By Theorem2.2(b), ft(e) is on
This is a contradiction
Wu(e) foreveryk = 1, 2, .... By Lemma2.5, we have
U {(X, y): x = x(t;p
O<t<T
(C), C),y = y(t;p
(E), E)} c N(ro).
By Theorem2.2(a) and Lemma2.5(b),(c) we have
(2.32)
jjtl () -P*() jj_ < (1) IIjf(E)
*
()jjl
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568
S.-N.CHOW AND BO DENG
whereft*(e)= (5*(e) ,hs(s*(e),e)) E Wjsc(e).ByLemma2.1, (2.18) and (2.30)
we have
(2.33)
1171(-k(8) 8) e)I
(f e)e
(p*(g
?
2
C6(64)P211P
()-k
(ek)1
*81c
< C6 (34) C7 P2(1/2)ky/4<
F/8
By (2.29) and (2.33) we have
lit (fk (e) e)- p'(0)ll < t/2+r/8 < F.
It followsthat,by Lemma2.5(a),
U {(x ,y): x =x(t;k (e) e) y = y(t;pk(e),)}
-k2
T
(e)<-t<jk(e)+T
By thedefinition
of N, and Lemma 1.4 we have
U
{(x,y):x=x(t;pz
0<t<T (6)
(e) e),y=y(t;f k(e) ,)}CNJ.
Therefore
(2.31) is proved.Finally,theexponentially
of y
asymptotic
stability
followsfromTheorem2.2(a). o
Corollary2.7. Supposethesame hypotheses
of Theorem2.3 holdforequation
(2.22Y.Thenthereexistsa neighborhood
N(I7o) of 1Tou {(o , 0)} in Xa x R and
0 < tf < e0 suchthatifthereis a homoclinic
orbitin N(I7o) forequation(2.22)
<
at 1e1 u., thenthereexistno periodicorbitsin N(I0).
3. HoMOCLINIC BIFURCATION FOR RETARDED
FUNCTIONAL DIFFERENTIAL EQUATIONS
In thissection,we willderiveanalogousresultsof theprevioussectionsfor
retardedfunctional
differential
ofretardedfuncequations.Forthebasistheory
tionaldifferential
of Hale [7].
equations,we followtheusual treatment
Let Rn be the n-dimensional
Euclideanrealspace,let r be a fixedpositive
constantand let C be thespace of continuous
from[-r , 0] into Rn
functions
withtheusual sup-norm1/1= sup-r<0<O 10(0) |
Considerthelinearautonomousdelayequation
x(t) = L(xt),
0
E C
-r
where I is an n x n matrixfunctionof bounded variation and x, e C with
autonomousequation
xt() = x(t + 0) for -r < 0 < 0 . Considertheperturbed
witha realparametere E [-e0 e,0]
(3.1)
(3.2)
L(0) = f[d?1(0)]$(0),
x(t) = L(xt) + f(xt 8)
,
where f: Cx[-e0, e0] Rn is continuousand takesbounded setsintobounded
sets, eo > 0 is a fixedconstant.
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BIFURCATIONOF A UNIQUE STABLE PERIODIC ORBIT
569
Let x (q) and x (q , e) be thereal-valued
solutionsof (3. 1) and (3.2) respectivelywithinitialvalue 0 E C at t = 0. Let T(t) be thesemigroup
generated
by the solutionxt(q) and let A be its infinitesimal
withdomain
generator
9(A) = { : 0 E C1 ,qS(O)= L(0q)}. The spectruma(A) of A containsonly
pointspectrum
and A E a(A) if and onlyif A satisfies
thecharacteristic
equation detA(A) = 0, where
A(A)
AI - |ei
dtl(0).
Our discussionwillbe based on thefollowing
hypotheses:
(H4) The characteristic
equationdetA(A)= 0 has a uniquerootA > 0 which
is simpleand the real partsof all othersolutionsof detA(A) = 0 are strictly
less than -A;
(H5) f(q, e) is C3 in
f(O,e)=O,
k and e
and satisfies
6E[-e60,e01,
D?f(O,e8)=O,
e8E [-e0 ,60].
By hypothesis
(H4), C is decomposedby the eigenvalueA of A as C =
S E U, whereU is theone-dimensional
eigenspaceof {A}A.Let qp = (p;(0)e
to theeigenbe theeigenvector
of A and v/' be a roweigenvector
corresponding
A*
with
thebilinear
value A of the formaladjointoperator of A associated
form
(3.)
(,
q$)=
(O)q$(O)-
ffoc4
r?r
di(()q$(c4)dc
-)
on [0, r] and 0 E C. It is known
wherea is a 1 x n continuousrowfunction
that
U = {0: q e C,
S=
Forevery
$E C,
=
0$
=
{ 0: q EC,(yi,(A$)
+
ai, forsomea e R},
=0}.
U with qU=(y,
Note that U and S are closedsubspacesof C.
definedby
Let X0(0) be thematrixfunction
0,
I
)
E U and 0
$= 0 -u
E S
- r < 0 < 0,
60=0.
of T(t) to includeitsactionon X0 (see [7]). X0
We mayextendthedefinition
has thefollowing
decomposition:
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570
S.-N. CHOW AND BO DENG
Note thatif v E Rn, then T(t)XYOuv
E U forall t E R and T(t)Xsv e S for
t > r. Furthermore,
thereexistconstantsK1 and u > A> O suchthat
' Kle JUt
losl
< K e 5t
IT(t)X81l
t >O O,5E S,
IT(t)osl
t >
T(t)0(0) = T(t + 0)q$(0),
T(t)0(0) = O(t + 0),
T(t)qU =qUeAt,
-x <
qu(6) =qu(O)eAO
Let the solutionx(q ,)
O,
t + 0 > O,
r <
t+ 0 <
t < +x,
E [-r O],qu
0
O,
u
E U
E U
of (3.2) be withinitialvalue q. Then x t(q5 ,)
c), with xs(q, e) E S and
can be decomposed as xt(q, e) = xts(o, e) + xu(q,
xtU(0,) E U. Then we have thevariationof constantsformula(see [7, pp.
143-147, 185-188]):
xt (q, e)
(3.6)
=
f
T(t - a)X8 f(x,,(k, e), e) daa,
T(t)/s +
t
xt(?> e) = e t/Y+I e(tfa)X
f(xa(O
,e)
,
e) da.
S and Pu: C -- U be the projection operators, i.e. if 0 =
,then Ps5q=qOs and Puq=$$u Since f is C3, xt(q$,e) is C3
Os+quC
in (4 , e) forall t > 0 in the maximalintervalof existence.Let ys4(q,) =
D?,Xts(+e) and ytu(q e) = D x u(q , e) . It followsfrom(3.6) that
Let Ps: C
-*
,
(3.7)
yts(q,e) = T(t)Ps +
Yu(q, e) = T(t)Pu +
rt
j
T(t - a)X8sD,f(XQ
(:,
e),
e)
.
Ya(q
,e)
ea,
t
T(t - a)XOuDqf(xa(k , e) , e) *ya(q, e) dea.
Notethatif q E C, then
T(t)Pu$=
eAtqU
te R.
By hypotheses
(H4) and (H5), thereexist 6, > 0 and 0 <el < e0 such
thatthe local stableand unstablemanifoldsWlsoc(e)
, W"uc(e)existin B(31)
for e E [-el ,el], whereB(61) = {q E C: kk51
< 15 1 < 61} (see [7, pp.
210-214]). Moreover,
Wloc(e)= {O = Os+ u
(3.8)
)Wluoc()
= {q = Os5 +?U
u= hs(zs5 e)
$s
= h
(,qu
e)
1
sI
<
,I,uI ?1}
whereh5 and hu are C3 . hU is givenby
(3.9)
n
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571
BIFURCATIONOF A UNIQUE STABLE PERIODIC ORBIT
e), t < 0, is theuniqueboundedsolutionof (3.2) with
(q
wherext4
t < 0,
lx*(qU, ,e)I < K21buIe.t,
for some constant K2 independentof (ou, e), and hu(OU e) = (xOu(/, ))s
qu = (xoQu,u))u.
Because of hypothesis(H5), we may assume for constant
< k1
?< 6 and
K2 > 0 that the followingestimates also hold for kfOsI
,I
e E [-el1 '61]:
Ihs(s
K210s 1
~,P)l'
Dphs(s
,e)-
Is <K2kksIlIvsI,
ID shs(OS,e) *(,s
Y/seS
E S,
'
) < K2IIIKI
i=
1,2,
y
K21,u12U
Ihu(hU 5e) '
lDuhu(o , 8) . /ul < K210ul ly/ul
/u E U
2
v 4 EU,
ID h
i=1,2,
operatorwithrespectto 0.
whereD denotesthe kthpartialdifferentiation
We willuse D1 for D 1.
We havethefollowing:
Proposition3.1. For every lIul
0 E [-r ,0].
<
61 and e E [-e1 ,e1], hu(q a,e)(0)
is C4 in
Since qU +
Proof. Since oU(O) = aq#(O)eA for some a E R, qu is C'.
forevery
=
q
in
manifold
unstable
local
the
element
is
an
e)
,
WluJ'(e),
hu(u
=
by the
This
implies
q.
suchthat xt(vI,e)
t > 0 thereexists Y/E Wluoc(e)
of thesolutionoperatorthat q(O) is C4 in 0E [-r,O].
smoothing
property
o
Hence, hu(ou,8)(0)= 0(09(o) is C4 in 0E[-r,0].
Let lIUl< s and lei<
1.
thenotation,we let x* = x (qU,e)
To simplify
(3.9) withrespect
and hu = hu(qU , e) . Note that x*, hu e C. By differentiating
to 0 E [-r, 0], we have
(3.11)
hu(G) = XO(O)f(x*) + L
-
J| d
(I
T(-a)Xof(x4+0
(O - a)v(0)f(x
,
,e) da)
e) da.
Since dhu(0) is a functionin (ou ,e) withvalues in Rn, D4u dhu(0) is a
continuouslinearmap fromU to Rn. On theotherhand, d h is a function
in (u ,e) with values inC and
D.,
d
h is a continuous map from U into
the orderof
C. The followingpropositionshowsthatwe may interchange
differentiations.
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S.-N.CHOW AND BO DENG
572
Proposition3.2. Let qu E U, IQu < 6
(a)
d
E [-r,O] and
d
/ dO
u)(o)
(D'b,hu
,
u
l,
(o) )
Then
<
< e?e1
-e
E U;
(b)
d(D2
h2(A
u
(DOqh
,dU\(6)
ON'
(0)
D;u
(DTod
u)
hu(6))5 (du
EU;
/,qu
(c)
* 5)
(0) =
dO (Dfu d69hu
Proof.
Let fi (t) = Deu [f(x
(Du
h
(qu
* Xu=O/fT(-a)XosfL
$u
E U.
,e)]. Then fi (t): U-)
f2(t) = D,u [f(x
and bilinear.
f2(t): U x U -- Rn is continuous
0
2
01
and
,e)]
and linearand
Rn is continuous
By (3.9) we have
DDuhh )(
*,
do2 hu(6))
(a) *
(q
=X|O-a)Xof2(a)
u
f
+uE
daX
U
u
u
,q2)da
u
E U.
-00
By direct computation, we have
(3.12)
H(Duhu u*
and
~
~
(0) =Xo
~
(f9)
(Jo
*+ L
T(-a)Xosf2
(a + 0)
u
da)
~
-
t)oi(6 )y()f(a)* -lda~,q5)d
10fA0-
h
H
D== (dQ
uE8) (X
0
q$h"())',q$
(DQ
(6) =a (D0) f, (a)
hu(D,/,,,2U)d(}*u XOu(0) =(D$u 7h()2uu
(Dou
dO
2
U
X,+
-u
*
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eU
BIFURCATIONOF A UNIQUE STABLE PERIODIC ORBIT
573
to showby
To showthat (D,I,,de *XU)(0) is C1 in 0 E [-r, 0], it suffices
<
t
Recall
in
0.
(3.12) that f1(t) is differentiable
.
f1(t) = Doff(x* , e) *Doux.
Since f is C3, we onlyneed to show x* and D x * are at least C1 in t.
Since x*(qu ,e) is a solutionof (3.2) and D ,ux*(ku,8) is thesolutionof the
variationalequationalong x*,
L(yt)+D, f(xt ,e) *ytS
=
whichis definedfor t < 0, we have
ax
at *(ru/ ,)
= (X*
k
,8))t
- D x**Que) = (D0JC*(0U,8))t'
where (x*(Ou,e))t(O) = S*(qU,e)(t
+ 6). Hence f(t) is C' in t < 0. By
(3.12)
differentiating
(3.11) withrespectto 0 and qu and thendifferentiating
withrespectto 0, we obtain(c). ol
of Dciu,uh(6),D,uh(6) and D5 dh(0) in
Remark.The smoothness
properties
6 E [-r, O] are notoptimalin theabove proposition.
3.2
3.3. Underthehypotheses
ofProposition
Corollary
d[(Do2uhu) (u 1.
u)] and d[(D;u 5
hu) 0u]
.
K3
there
existsa constant
areelements
in C forall 9u,qu E U. Furthermore,
on ,, l, K1 and K2 suchthat
depending
d [(D2 hu)*(ouq02u)]< K3jql
TO
i
qu u,qE U;
u)*I1]|31OU11011,
(DQdO
1,E
Proof.This followsfrom(b) and (c) of Proposition3.2, (3.5) and (3.10). o
u
above.
Notethat I I appearson theright-hand
sideofthesecondinequality
This is becausetheunstablemanifoldis tangentto thelinearspace U at the
origin.
Note thatby (3.9) we have
-Idh (O)
=
u
L(hu) + X0(O)f(hu + q$
~)
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S.-N.CHOW AND BO DENG
574
if qu $ 0 . However,we have
This saysthat hu(qU, e) 0 -(A)
3.4. Let qu E U, I uI < 3, 0 E [-r ,0] and JeI< e . Thenthere
Proposition
on b,, e and K1- K3 suchthat
existsa constantK4 depending
d
(a) T(t)hUIt=+= hu- Xof(hu+ u 8);
(b) IT(t)dT(T)h 2o+I < K4Iq| e-A,tt > 0
d is taken
in L?.
where
thederivative
Proof. (a) By (3.5) and (3.9) we have
(3.13)
T(t)h^(6)
=
f [xoo+
t+6
L(T(f6)Xo)dfl6f (x
f
[XO(O)+
f
,
e) da
,t
+<
O
fo (P(t0-)NJ()(O,)a
-|)
T(t)hu(0) =
rt+Oa_
t > 0.
L(T(fl)Xo)d6] f (xo, e) da,
(3.13) we have
If 0 < 0, thenby (3.11) and bydifferentiating
d
= d hu(0) = d hu(0) - Xo()f (hu +
Od
T(t)hU(0)
dt
~~t=O+
u
e).
If 0 = 0, thenby (3.19), (3.11) and (3.13) we have
d
T(t)hU(0)
= rlin [T(t)hu(0) - hu(o)]
t-O~~~~
t-
= lim - |
t+0+ ta
=
L
(J
[
f(xJ
L(T(J)Xo)dflJ
,)da
T(-a)XOf(xa e) d)
dOhu(O) - XO(O)f(hu+
q$U ,e).
This proves(a).
(b) By Proposition3.1 and (a) above,
T(t) dr T(T)hu __
is well defined. Moreover,since T(t)hUE S is continuousfor t
for t > r and S is a closedsubspace,
differentiable
T(t)dr h
dt (t)h
eS,
t>r.
It followsfrom(a) of Proposition3.2 that
T(t),d T(T)h
,
t > 0,
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>
0 and
BIFURCATIONOF A UNIQUE STABLEPERIODIC ORBIT
575
is C3 in (qu , c). Moreover,since
D -dhU(0,e)=0?,
dO
E [-C1,C811'
we have
DollT(t)
d
t > 0, C E [-C1 ,C1].
= 0,
T(T)hU( ,C)
Therefore,by (3.10) we have
Kand
sup
K depends on
l, elC K1,
and K3 . Thus, by (3.5) we have
K2
< K1Ke
T(t) d+ T(T)hu
< +00,
T(T)hu
e2lr D?UT(2r)d
lu
t > 2r.
2
Again by (3.10) we have
K=
sup e28r DoT(t)TT(T)hu
< +00
le1<6,
0<t<2r
and K depends on
T(t)d
1,
el, K1,
K2
<
T(T)hu
and K3. Thus,
Ke'
Define K4 = max{KK I, K}. This proves (b).
Define a change of variables 0 = H(0 ,
s
+ =
(3.14)
0
tlq$uj2
<
t < 2r.
o
) , 0 E B(31) , II <
- hu(ou ,C),
1 by
?, =0 ?,
It is clear from (3.10) that H is near the identityand in the new variable
1
WiU(C) = {0:
E H(B(.)
, qs = 0} . The inverse H
of H is givenby
S
q U
=qU
=
s+hUW
C)
J
Note that since xu(q,e)
is C3 in (t, ,C), t > 0, 1?k1
< 3,e
< e , the
composition map h (xt(q C) , ) is C3 in t and ( ,C). By this change of
variables, (3.6) becomes
tsW e) = T(t)
(3.15)
+
T(t - a)Xsf(xFa, e) da
+ T(t)hu(0u ,C)
xu($
)=eAtu
+j
-
eA(t-a)Xu
T(O)hu(Fu ,C),
(x
)d&
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S.-N.CHOW AND BO DENG
576
where
X = XI(,E) =H(x,(q,
e), e) and f is defined by
f (Xa, 8) = f (H I (xaz ,e), 8).
(3.16)
Note thatin the new variables, xt(5, e) may not satisfya functionaldifferential
discuss the behavior of
equation. However,we can make such a transformation,
xt(0,e) as a solution of the integralequation and then returnto xt to obtain
informationabout the originalequation.
By using Propositions 3.2-3.4, (3.15) can be writtenas
rt
-ts(+,
T(t)+s
+
IJT(t
e) == T()
+1a
It(k3,8)
(3.17)
-u
xt (,
e) =
At u
+
etU
0
ft (t-a)
JeA
-
a)g(xa
e) da,
,
,)a
u~,
XJxf(x, c) da
where g is given by
g(+, e)
=
(3.18)
~d
S
Xof(
-
-
,Pe)- -tT(t)hu(0U,e)
*
+ X4f (
Diuh"(QU 8) [)Au
=X""(0
e)
+ X0[f (
dhu (o
o-
e)D,O,,hu
C) + f(hu(Ou , C) +
, 8)]
(o
+ xof(
,)[Ao
,u , 8)].
Note that X0 appears in (3.18). This says that g(q , e) has a discontinuityat
= L??([-r ,0] , Rn). Since H is near the identity
0 = 0, hence g(q ) E
map, by (3.10) thereexist J2> 0 and 0 < C2 < C1 such that
,E
B(62) C H(B(3,),)a
(3.19)
ICI< C2
Therefore, g: B(62) X f-C2,621-* L? is C2. By (H5), (3.10) and (3.11),
?2 and IEI< C2. ByProposition3.2, g(q, C)(0) and
D(E))g(O,C) = 0 for VkI<
in 0 E -r , 0) and have a discontinuity
differentiable
are
actually
Dtg(q, c)(0)
at 0 = 0 in general. However, the followinglemma says that if q E WIU0C(E),
< 62 and 1C1<? C2 theng(k, e) = 0, thatis, the discontinuitywill disappear
101
on the unstable manifold.
0 < 33 < 32 and 0 <
3.5. Thereexistconstants
Proposition
F: B(33)
x [-83 ,
g(,c)
-*
L(S
, L?)
C3 <C62
suchthat
= F(q5,C) .4,
E B(53),
I
anda map
< E3,
we have
whereOs = Ps5. Furthermore,
(i) F is C';
s
(ii) if E S, then(F(Q,e) q s)(O) is C1 in OE [-r,0);
on 33, C3 and K,- K4 suchthat
Ks depending
(iii) there
existsa constant
estimates
hold:
foreverykE B(3), ICl < C3 and @ E S, thefollowing
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577
OF A UNIQUESTABLEPERIODICORBIT
BIFURCATION
IF(Q,e) S IL?
l,
KI
L55IsII
d
< KI
sup |j(F(,6).
l)(6)?0Ks)
1
thatthereexist 0 < J3< J2 and 0 < 63 < 62 such
Proof. We willshowfirst
thatif I0I< 3,qE U and 1I1<?3, then
(3.20)
Recall that q
0 E [-r,0].
)(0) = 0,
9,(
6) is definedby
= H(q,
s
- hU(oU
U=
eg),
Assumethat II < J2 and 1el < 62. Let
to= to(o ,) = sup{t > 0: Xt(O 8) E H (B(J2) g)}.
thatif 32> 0 is sufficiently
It is nothardto see fromthesaddlepointproperty
small,then to > 2r forall 1q$< 32 and 1el<612. Since H is neartheidentity,
thereexist 0 < 83 < 62 and 0 < J3< J2 suchthat
(3.21)
B(33) C H(B(32)
I1I ? 3.
,6)
e U and 161< 63.
Now, we will show (3.20) holds true for all 101< 33,
Suppose the contrary;thenthereexist q$ E U, J$oj< d33 I6oI? 63 and
00e [-r , 0] such that
g(q$ ,o)(0o)
(3.22)
#
0.
IXo
Then 00 E Wju0c(6o)and
Let 00 E H1'(qug,o).
< 32. By the choice
of 32' to = to(00,0o) > 2r. Hence, [-00,to - 00] is nonempty. Let Zt =
H(xt(q$0,go) ,6o), - 0 < t < to- 0h.Since WOC(60)c { : 0S = Ps = 0} and
0 for -0 < t < to-6 0, x0 = H(q$ ,60) =qu. By (3.17), we have
xFC
0=
100 T(t
Xt+00
-
,o)
a)g(xca+60
da.
Let t = -00 + a and 0 < a < to with a < -00 if 00 $ 0. Since the above
integralis an elementin C, by (3.5) we have
0
(3.23)
-00+a
= 1
=
[T(- 0 +
a-
-I00+a
0
f
g
g(Fca+o
T(a-
d)
o0)](00)
ca)g(xa+o0,
I e0)
(a-
a) da
,
a)[g(xF ,6o)(0)]daQ,
-
00 > a>
O,
00 = 0.
Dividing (3.23) by a and lettinga -) 0+, we have
0=
lim
I
-00+a
[T(-0o +
a - a)g(xFc0 ,60)](60) da
= g(q$g ,6o)(0o)
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S.-N.CHOW AND BO DENG
578
forany -r
(3.22) and proves(3.20). Therefore,
00 < 0. This contradicts
<
e)
= g(q$S +q,
g(0qe)
-
(I
- g(0U,
e) =
j
+ouq
Lg9(aqS
e)da
C
e) da).
~D<sg(aO4 +
Define
F(b, e) =f
(3.24)
D,sg(aqs
+ u , 8) da.
Since
D
-
(DOhu) . (Xo ft) + X011
whereh=hu (ku, e) and fi
e , a) = Ddsf (a)s + 4u , c) , F($, e): B((3) x
L[S ,L??] is C1 and [F(f,e) *y-S](Q() is C1 in 0 E [-r ,0) for
[-83 ,83]
(3.25)
+
(as
=Xu
,u)
-
it is not hardto see from
all 5 E S. This proves(i) and (ii). Furthermore,
(3.25) that F(O ,e) = 0 forall IeI < e3. Thus,thereexistsK5 dependingon
62' e3 and K,-K4 suchthat
JF(>, 8) - sl
K-5
1?>1< 63,
1> I.s,
181 <
Moreover,by Corollary3.3 thereexistsK5 dependingon
suchthat
- r<
yo(F(+ ,e) *Vs)(0)| < K51sl ,
(53, E3
1eI <e3
and K,- K4
< 0,
0
forall q$I< (3,
proves(iii). o
SEES.
'635
and Vs E S. Choose K5 = max{K5,K5}.
This
By (3.18), we have
T(t)g(q ,e) = T(t)Xof- T(t)d T(T)hU
(3.26)
- T(t)D
hu * ())WU + Xou ),
(qu ,e). Since S is a closed subspace, (Dp hu)
and hu = hu
+ Xouf) e S . We also have that T(t)Xosf e S for t > r. Therefore,
by
where f=f(,e)
(,?u
Proposition3.4(b), (3.5) and (3.26) thereexistsa constantK6 dependingon
e and K1- K suchthat
(3,
for all q
t> 0
jT(t)g( ,e8)j< K6e'HtkkjjlSj
(3.27)
E
B((3)
and 1Ie<I?3
Now,we rewrite(3.17) as
S(
)
.
rt
= T(t)qs +
(3.28)
T(t - a)F(x(qc
, e), e)
t
Xu+
E) =eiU
+
e (t-a)Xuj(Xa(
,
e), e)
*
,
e) da ,
da.
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BIFURCATIONOF A UNIQUE STABLE PERIODIC ORBIT
579
It followsby differentiating
(3.28) with respectto q in L?? thatthe derivatives DgxY(t,e) and D,x1't(q, e) satisfy the variational equations along
U=
f
rt
T(t - a)[D F(x(,ce),e)
T(t)Ps +
(ua,xS
,e))
F(x(c ,) 5c) - us ]da,
~~~~~~~~~~~~+
(3.29)
0<
At
=u e P +]e
u
u
D)XJ f (Xa (Pe),)
(ta
)ua da,
t < to
0 < t<
to
where ut = D,?xt(q, e) and [O , to] is a definitionintervalfor x(q+, ) .
Note that even though
xt,
X,u and Xs are C3, we cannot differentiate(3.28)
more than once because F is only Cl. This says that by using the change
of variables (3.13) we have lost some differentiabilityproperties. However, the
special forms in (3.28) and (3.29) will be sufficientto derive similar local results
as in ?1.
By using (3.28), (3.29), (3.5), (3.10), (3.27), and Propositions 3.1-3.5, the
proofs of the following lemmas can be adapted from those of Lemmas 1.21.6. We will therefore omit the proofs of Lemmas 3.6-3.8 and 3.10. We will
present the ptoof of Lemma 3.9 since it is differentfrom the case of semilinear
parabolic equations in ? 1.
on (3, e3
Lemma3.6. For equation(3.2) thereexistsa constantK7 depending
and K1-K6 such thatif X E(b,e) E B(63) forall 0 < t < to whereto > 0 is
then
anyconstant,
lt (o
5 )l < K710 le
A,
0 <
t < to.
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Lemma3.7. Forequation(3.2) thereexist0 < 34 < 33 andconstantsj A, , A > O
then
suchthatif
WXt(b,E) E B(64) for 0< t < to whereto is anyconstant,
ID xt + ,E)| < 2e
ID+Xu,E)1
(
ID X
O0<
,
< 2eA
[-E3,E3]
0 < t < to, E E [-E
,
E)<!e
t < to, E E
< t < to, E E [-E3,E3].
0,
Note that by the change of variables H in (3.13), the local stable and unstable
manifolds Wloc(e) and Wju(Ce) are given by
JIoc(e)
= {1
WJ1j(e) = {
4
:
qs
hs(W0,E), WI < 64},
=0?I a I <34}
=
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S.-N.CHOW AND BO DENG
580
whereh3 is C3,
h (0,C)=
0, e - [-C3,e3]
wedefine
forevery ee[-E3,83]
(3.30)
and Dhs(O,0)
= 0.
As in ?1,
ISi < 64/K7,I I(i I u_
o-:
hs(0S,))I < p} C B(64)
Q(64I P Ie) = {q
Q+04
,P I 8) ={:
,e)) < PI
IXu- hs(0s5
Q(6S4,P I C) I? < (vA1
E
Q2(04,P, )={
'20S4,P I8) ,-P < (,V,Xu- hs(0s,C))< 01,
0E
has
where (*,*) is the bilinearformgivenby (3.3). Note thatsince WlsJc(e)
u
U U Wjsoc(e) where Q =
codimensionone and Q+n Q- = 0, Q
2(64,p,e) and &f = Q (S4 , e)
(3.28) in B(34) and 0 < p < 4/4.
Lemma3.8. Let (XYs(q,C)xu(f,Ce))satisfy
If E Q (44, p, e) u Q- (34, p, e), thenthereexistsT = T(,C) > 0 suchthat
Xu(?
,e) C)
+ hs(S
E) - Xu (s
6412, if
-b4/2,
if+ E Q2 (54p,g,) I
,In 4l
(5~4
-A u _
?z
S
1l
In
< T <
#5e)
I1A
C3
181< 83'
and II ?<3, then
E)U Q-(34,p,I)
if Q+$(34,p,
E
Furthermore,
1l 4I
181<
,P ,IC) I
E fl+(64
354
1t?u-h(/5e)
32
34 -8K 24
whereK2 is givenin (3.10). o
Lemma39. Let 'b E (34 ,p,I), ICI<?e3 and to> 2r be as in (3.21). There
on 33, 83 and K1-K7 suchthatifa solution
existsa constantK8 depending
is in B(34) for 0 < t < t , thenxts(q,e)
(3.28)
equation
e))
of
(,s (4IC) I,X1u(q$
I
is differentiable
in t E (r , to) and satisfies
d
xS(k,e)
< K81bje yt,
dt is takenin L?' .
wherethederivativedt
and x =
Proof. Let t > r, X1=x (,)
T(t +
t~~~~~~~~
xts(0) =
+/
rt
Q)q$s (0) +
f
t+O
t < to,
. By (3.28) we have
x (,e)
T(t - a + 0)[F(xca, e) *xs](0) da
[F(.x,e) *xS](0+t-a)da,
Since [F( C,e) *@s](0) is discontinuous only at 6
and (3.25) we have
tt
r<
d
tf(
0 e [-r, 0].
=
0 for all
ViS
E S,
+_d
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by(3.24)
BIFURCATIONOF A UNIQUE STABLE PERIODIC ORBIT
581
where
F(a, 6)
*
6
[F(xYc,e)
V-s=
@
s]()
-Xo0(6)j
f1(x
e ))
d,B
*
s
and f (q 5,e, fl) is as in (3.25). Note thatby (3.24) and (3.25), F(a , 6) is C
in 0 E [-r , 0] and by Proposition 3.5(d) we have
(3.31)
<K
TOF(a,6)*@S
?Ks
IF(a,60).* I 'KsIsII
f9E[-r
<a?<to,
I,
O<a<to,
,0],
6E[-r,0],
for all Vs E S. Hence
d-Y (6) = L(T(t + 6)qs) + [F(Xt+05) *
ot+0~~~~~~t
+
L[T(t+6, 0).
- F(t
J(0)
a + 6)F(xc, 6) *xs] da + F(t, 6)*xs
t+O
--
/
t+0 00
F(a
,
6 + t - a) Xs
da.
a
Therefore,the desired estimate follows simplyfrom (3.5), (3.31), Proposition
3.2 and Lemma 3.6. 0
Define 1: B(54) x [-c3, 3]3
(3.32)
1(
,
2)
,
R by
=
max{(q' ,1u - hs(i ,))} ,
i= 1 ,2
where ( ,.) is as in (3.3). The followinglemma is analogous to Lemma 1.5.
Lemma 3.10. Let
and 6 E [-83 ,83].
) = X,(?> )
X 1 (?
Q+(54,p, ), p < 54/2
58), where qE
ThereexistconstantsKg and a > 0 depending
on64, 63
and K1-K8 suchthatif E1' K2e Q+(54 ,p,6 ) and ]e [-63,e3], then
lX1 ($1 ,6)
-X
(k2 6)I < K9[l(?1
5 ?>2 ,6)]a1j,
- >21 '
where1 is givenby(3.32). o
Define
(3.33)
Wu(e)
where
=
{q: thereexist t > 0 and V E Wuoc(6)
u
with (V. ,
hs(V,SE)) > 0 and k =
nX;(54/2
1(8) = WuJ(6)
X( 5,e)={q
(YA
$ - hs(q
,6)},
161 ?6<
3S
<
,e),
,e))
xt(q,
11?6
= ,5$
= H($, 6)}.
Let 0 < pO < 64/2 be fixed. Since H is near the identitymap, there exist
0 < pi < pO and 0 < 64 < 63 such that
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582
S.-N.CHOW AND BO DENG
Let q0 E Wloc
(O) withk01I
of Wlsc()
property
I < p1 be fixed.Bythecontinuity
in , for every 0 < p < dist(OB(p)I, 00), thereexists 0 < e5(p) < 64 such that
n W 6c(g)7$0,
1:1_
B(Oo,p)
<5(p).
By usingtheabove lemmas,it is nothardto see thattheproofsof thefollowingresultsare exactlythesame as Lemma2.1, Theorem2.3, and Corollary
2.4, respectively.
Lemma3.11 willbe neededin thenextsection.
Lemma 3.11. Thereexistsa continuousmap
satisfying
r1: B(0,
p)X[-65(P)
65(p)]
5,
-*
C
0 - Xo < p and (V.A, ou - hs(Os, e)) > 0, then ,1 (0, E)
18I< e5(P)
is the intersection
point of X((54/2, 6) and the orbitof thesolutionof (3.2) with
initial value q and parameter 6 . On theotherhand, if (YA, Os - hs(Os ,6)) < 0,
then 7rl (q)
=l(e);
(b) 7r ( ,6) is continuousin ( ,6) and is Lipschitzian in X for each fixed
6. Furthermore,thereexist constants a and Klo dependingon 654, 64 and
K1-Kg such that
(a)
If
17I (ob1 5,)
-
7r1(02 ,68)1 <
K,opalq$
- 021
E]
In thefollowing
theorem,
we mayassumewithoutloss of generality
thatthe
homoclinicorbitIF = WJu'(O).
Theorem 3.12. Consider the nonlinearautonomous delay equation witha real
parameter
(3.34)
6
E
[-o0
Xgo]
60 >
x*(t)
0:
= L(xt)
+ f(xt
,
8),
where L: C -* Rn is a bounded linear operatorand f: C x [-60o go1
Rn in
C3. Suppose (3.34) satisfiesthefollowing:
(H4) thecharacteristic
equation detA(A)= 0, whereA(A) = AI-JfreAO[d(6)],
has a unique positivesolution A > 0 whichis simple and the real parts of the
othersolutionsof A(i) = 0 are smaller than -i;
(H5) f (q, 6) satisfies f (0, 6) = 0 and D, f (0, 6) = 0, where 6 E [-6o ,g]
(H6) at 6 = 0, equation (3.34) has a homoclinicorbit FO asymptoticto the
equilibrium 0.
Then thereexist a neighborhoodN(Io) of FO U {0} in C and 0 < 6f < 6o
such that WuZ(6)c N(FO) and WuJ(6)n Wlsc() = 0 ifand onlyif thereexistsa
periodicorbitin N(Io), where WuJ(6)is theorbitof(3.34) through00 satisfying
0 E Wlu4c'()and (v. , Ou - hs(Os0,6)) > 0 . Furthermore,thisperiodic orbitis
stable. o
unique and exponentiallyasymptotically
We willonlyoutlinetheproofofthistheorem
becausetheproofis exactlythe
same as thatof Theorem 2.3 afterwe have establishedLemma 3.1 1. Let to > 0
be thetimesuchthat xto(I(0) 5,0)= 00 and fo = {q0:0 = xt(qI(0) ,0)
, 0 <
t < to} C
FO. We definea familyof maps
7r2(.,)
from B((0)
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5,p) into
BIFURCATIONOF A UNIQUE STABLE PERIODIC ORBIT
583
small e
a neighborhood
of 00 by 7t2(q$,e) = xt0(q, e). Then forsufficiently
witha
and po, thecompositionmaps 7r(.,,e)= 72 (7r1(,e) ,e) are contractive
a neighborhood
uniform
contractive
constantlessthan 1/2. Then,we construct
thefollowing
< co satisfying
properties
N(17) c C of FOU {O} and 0 <
(see Lemmas2.4 and 2.5):
of {0} containing
(1) N(FO) = N1 U N2, where N1 is a neighborhood
of FO;
B(q0 ,po) and B(q$I(0) ,p1), and N2 is a neighborhood
(2) if y is an orbitin C of equation (3.34) at e E [-U ,Uo] satisfying
yn N1: 0 and y c N(FO), then (WA,
ou - h5(ou,e)) > O forevery0 E yn N ;
(3) if 0 E N2 and Iel < U0, thenthereexists t > 0 such that xt(q$,e) E
B(0o Po);
(4) if 0 E B((0) ,p1) and eI?< U0, thenxt( ,e) E N2 forall 0 < t < to;
closeto 0)
(5) 01(e) and thefixedpoint 0*(e) of 7r(-,e) are sufficiently
and 00, respectively(see Lemma 2.5(b), (c)).
AfterhavingdefinedN(FO) and %0,we can showthatthe conditionthat
,e)) >
(e) = 0 is equivalentto (V. , *u-h
W+u(e)c N(FO) and W+u(e)nWloc
0, where =
(e) is thefixedpointof 7r(-,e). However,thelastcondition
is equivalentto theexistenceof a uniqueperiodicorbitof equation(3.34) in
property
oftheperiodicorbitfollowsfromthecontraction
N(Jo) . The stability
property
of themap 7t.
ofTheorem3.12 holdforequation
Corollary
3.13. Supposethesame hypotheses
(3.34). Thenthereexistsa neighborhood
N(FO) of FO U {0} in C and 0 <
orbitin N(Fo) forequation(3.34) at
fo<,e0 suchthatifthereis a homoclinic
IeI< o, thenthereexistnoperiodicorbitsin N(FO) .
4. APPLICATIONS
As an application,we considerthe followingfunctionaldifferential
delay
equation(see [6 and 11]):
(4.1)
x*(t)= af(x(t - 1)),
x E R,
where a is a real parameter.We assumethat f: R -* R is periodicwith
minimalperiod -A + B, A < 0 < B, f (A) = f(B) = 0 and f'(A) = 1. Note
thatif x is a solutionof (4.1), then x + n(-A + B) is a solutionof (4.1) for
everyintegern. x _ A and x _ B are equilibriaof (4.1). A solutionx of
solutionfromA to B if
(4.1) is calleda heteroclinic
lim x(t) =A
t- -oo
and
lim x(t) = B.
t-*E+oo
A solutionx of (4.1) is called a periodicsolutionof thesecondkindif there
existsp > 0 suchthat
foralltER,
x(t)=x(t+p)-(-A+B)
forall0<q<pandsometER.
x(t)7x(t+q)-(-A+B)
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S.-N.CHOW AND BO DENG
584
Note thata periodicsolutionof the second kind correspondsto a periodic
solutionfrom
rotationof the statevariableon the circleand a heteroclinic
to a homocliniccurveof thestatevariableon the circle
A to B corresponds
r(t) ). We are
(bya homocliniccurve17(t)we mean limt-00 1(t) = limt-++00
ofperiodicsolutionsofthesecondkind
interested
in theproblemofbifurcation
solutionfromA to B. It is obviousthatour mainresult
froma heteroclinic
will
to thisproblem.However,a slightmodification
cannotbe applieddirectly
be sufficient.
delayequationwitha realpadifferential
Theorem4.1. Considerthefunctional
rametera e R:
x(t) = af(x(t - 1)).
(4.2)
Suppose(4.2) satisfies:
(H7) f is C3, periodicwithminimalperiod -A + B, A < 0 < B,
solutionxao fromA to
(H8) at a = aO > 0 equation(4.1) has a heteroclinic
B;
equation0 = A- a e-A has onlyonepositivesolution
(H9) thecharacteristic
Re A < -AO
solutions
and
all
other
A. satisfying
AO
Then thereexists a neighborhoodN(Xao) in C of
U
{xao}
U{A} U{B}
-oo<t<oo
and a+ > a0 such thatif equation(4.1) has a periodicsolutionof thesecond
kind x at a E (aO, a+) satisfying
(4.3)
XtE
U{N(xao) + n(-A + B)},
nEZ
t E R,
stable.
asymptotically
thenit is uniqueand exponentially
Proof.Since solutionsof 0 = A- a e i are theeigenvaluesof the infinitesicontinuoussemigroupdefinedby the linearized
of the strongly
mal generator
equation xc(t) = aox(t - 1), by (H8) and (H9), local resultsderivedin ?3
are applicableto equation(4.2) nearthe equilibriaA and B. Let B(3) be
a sufficiently
of A suchthatthelocal stableand unstable
smallneighborhood
and Wuc(a) of A aredefinedfora > ao closeto ao. Since
manifoldsWlsoc(a)
has
codimension
one,we mayassume B(6) is dividedby Wlsoi(a)into
(a)
WlsJ''
two disjointparts Q+(a) and Q (a) (see (3.30)). Withoutloss of generality,
a
we assume x c Q+ as t -* -oo. Since
lim X ao _(_A +B) = A,
t-*+oo
we can choose 00 E Xax- (-A + B) with40 E B(5). By (3.33) we can define
q (a) E Wluoj(A)
for a > aO closeto ao suchthat - 1(a)E Q+(a) is continuous
in a with q1(ao) = xta forsome t < 0. By Lemma 3.11, foreverysmall
p > 0, thereexist a(p) > 0 and a continuousmap 7r: B($0 , p) x [ao ,a(p)]
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585
BIFURCATIONOF A UNIQUE STABLE PERIODIC ORBIT
10 -qoI
the orbitof the solutionx(q , a)
satisfying(a) if a E [ao,a(p)],
is on
(- (a), then 7r1(q , a)
=
q 1(a);
> p and q E Q +(a),
then 7r (,a)
of (4.2). On the otherhand,if q E
(b) 7r1(q, a) is Lipschitzian in q for each
fixeda E[ao, a(p)] witha LipschitzconstantKp6, whereconstantsK and
p are independentof a E [ao ,a(p)]. Let to > 0 be the time such that
x (0, (ao), ao) - (-A + B) = 00 . Then, we definea map '2: B(0q (ao), pl) x
small number,by
B(3), where p1 and a1 are some sufficiently
[ao ,a]
aI
small,thenthe
if p0 is sufficiently
7r2(q , a) = x,( , a) - (-A + B). Therefore,
composition7r = 7r(Xr
(2 ,), ,)
B(q0, po) x [ao , a(po)]
B((0 , po) satisfies
( , a) is a contractionmap with a unithat for each fixed a E [aO , a(po)],
a neighborhood
constantless than 1/2. Next,we construct
formcontractive
{A
B}
satisfying
and
a+
N(x,a) c C of {xao: t E R} U ,
of A containing
(1) N(xa) = N1U N1U N2, whereN1 is a neighborhood
- (-A + B) EN1}
B(00, p0) and B(1 (ao), p), N1 = N1+ (-A + B) ={:
of {x,(1 I(ao) ,ao):
of B and N2 is a neighborhood
whichis a neighborhood
0 < t < to);
(2) if y is an orbitin C satisfying
y c U {N(ro) + n(-A - B)}
uEZ
and yn {N, + n(-A + B)} forsome n, then 0 - n(-A + B) E V+(a) UWjo (a)
foreveryq e yn 1NI+ n(-A + B)};
(3) if 0 E N2 and a E [ao , a+], thenthereexistst > 0 suchthatx,(q, a) (-A + B) E B(qo Ipo);
(4) if 0 E B(01(0) ,pl) and a E [ao,a+], thenx,(q5,a) E N2 for0 < t < to;
closeto 0q1
(5) ,1(a) and thefixedpointq5 (a) of 7r(,a) aresufficiently
(ao)
(see Lemma2.5(b),(c)).
and 00, respectively
anyorbitin C of a periodic
By (a) and (2) and (3) of theabove properties,
(4.2) mustcontainthe fixedpoint
solutionof the second kind x satisfying
q5 (a) of 7r(.,a) and q5 (a) E Q+ (a). This provestheuniqueness.By (b) and
theperiodicsolutionof thesecondkindis
(4) and (5) of theabove properties,
stable. 0
asymptotically
exponentially
is takenfromWalther[ 1].
The following
Suppose r > 0 and 1 I2 E R satisfy
A + 2r < 4j < 42 < -r < r < B - r.
Let g: R -R be C' andsatisfy
(i) g is periodicwithminimalperiod -A + B,
(ii) g(A)=0, 0<g in (A,0), g(0)=0, g<0 in (0,B),
(iii) IgI < r/2 in (A - r, A + r) U (-r,B - r),
(iv) thereexistsq E (O, 1) with Ig(4) I < q - AI forall 4 E (A - r ,A + r),
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586
S.-N.CHOW AND BO DENG
(v) a+ = g'(A) is positive with loga+ < -A(a+), where A(a+) satisfies
A(a+)= a+e-A(a+)
(vi) for qE C, A ?<
< 1 and q(O) = 41 imply c1 -+ f01 g q(s) ds <?2
and for y
E C,
Consider
41
<
<
V ?2
and
,(O) =
2 imply B
+ r<
*
42 +f1
gy(s)
*
ds.
E R.
(4.4)
fP() = g(4)/g (A),
Theorem4.2. Let f: R -* R be defined
conditions
by(4.4). Supposef satisfies
(i)-(vi). Then
(a) thereexists a nonemptyinterval [ao
,a)
with a,
> 0 such that if
> ao
a = ao, thenequation(4.1) has a heteroclinic
solutionxao fromA to B, and if
a E (ao , a1) thenequation(4.1) has a periodicsolutionofthesecondkind Xa.
foreverya E [aO, a1 thereexistsOa E ya (theorbitin C of xa)
Furthermore,
suchthatOa `(ao
as a- a0;
(b) thecharacteristic
equation0 = A- aoe
has exactlyonepositivesolution
AOand all othersolutionsA satisfytheinequalityRe)A< o
By Theorems 4.1 and 4.2 above, we have the following:
4.3. Suppose f: R -- R is C3 and satisfies
Corollary
in Lemma
theconditions
4.2. Thenthereexists0 < a < aO+ < a and a neighborhood
N(xao) c C of
{xao: t E R} U {A} U {B} such thatif a E (ao a+), thenequation(4.1) has
a uniqueand exponentially
asymptotically
stableperiodicsolutionofthesecond
kindin
G= U {N(xao) + n(-A + B)}.
neZ
Proof. Let N(xa,) and a+ be defined as in Theorem 4.1. Without loss of
generality,we may assume Oa - (-A + B) = +*(a), the unique fixedpoint of
the map 7r(. , a) definedin Theorem 4.1 for every a E [ao a+). Therefore,
the orbit ya of a periodic solution of the second kind xa must be in G for
all a E (ao , a+). Hence, by Theorem 4.1 Xa is unique and exponentially
asymptoticallystable. o
,
Example. Let p E (7r/2 7r) be given. Let g.(4) = p(sinw - sin(4 + w)) for
4 E R and w E (0, 7r/2). Then thereexists wo E (O , 7r/2) such that for every
w E (wo 7r/2) the real numbers A = -7r - 2w, B = 7r- 2w, r = B/3,
Ej = A + w, E2= -w and g.(4) satisfyhypotheses(i)-(vi) (see [6]). The
correspondingequation xk(t)= af(x(t - 1)) with a > 0 and f = gWlg(A)
models phase-lockedloops for the controlof high frequencygenerators(see [6
and 11]).
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BIFURCATIONOF A UNIQUE STABLE PERIODIC ORBIT
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DEPARTMENT
OF MATHEMATICS,
MICHIGAN
STATE
UNIVERSITY,
EAST
LANSING,
MICHIGAN
48824
Currentaddress (Shui-Nee Chow): Center for Dynamical Systems,Georgia Instituteof Technology,Atlanta,Georgia 30332
Currentaddress (Bo Deng): Departmentof Mathematicsand Statistics,Universityof NebraskaLincoln, Lincoln, Nebraska 68588
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