GENERATIZED HOPF BTTURCATION
IN RN
AND
h-ASY}IPTOTIC STABIT.,ITY
by
S. R, Bennfeldrt and L.
TeehnicaL RePont
0ctobe::
r
Salvadoridsft
No.
l-2L
l-979
*Unlvensity of Texae at Ar:llngton, Reseanch pantlalJ-y supported by
C.N.R. (Nittonal Council of Resealrch ln ltal.y) and by U.S. Arny Gnant
DAAG-29-77-G0062.
*rtDlpar:tfinento
dl
Matematlcao unlvensita
di
Tnento, rtaly.
GINERALIZED HOPF BIFURCATION
AND
IN
RN
h-ASYHPTOTIC STABILITY
by
S. R. Bernfeldtt and L.
The concept and analysis
autonomous
gsfy6dq1ri:'ct'r
of stnuctural stability of two dfinensional
differentiai- systems was pnovided by
Anchronov and Pontnyagln
in a fundamenta,l paper [f] tn l-937. Wlthin the next fifteen
And:ronov and
his
gr3oup
developed a majon par.t
years
(incl"uding teontovlch, Chalklnr t{itt, et al.
)
of ttre pnesent theony of two dimensional stnuctun-
a1ly stabLe ancl unstabl-e systems of diffenential equations.
Hotivated hy probi-ems in cl-assicaL and celestial mechanLcs, mathematicians and physicists have been very concenned with pnoviding a qual.-
itatlve deecriptian of the trajectonies c,f pertunbed systems in a neighbonhood
of a structunal.ly unstable equillbnlum point in nn(n:Z).
Tc be
mc)l:e
specific let us considen a differential
i<
= fo(x)
systern
,
(1)
ftUnivensity of Texas at Ar,lingtono Reseaneh partlallv supponted hy
C.N.R. (National Council of F.esearcir in ltal"y) and by t.l.S. Anny Gnant
DAAG-29-7?-G006?
:t*Dipax'tirnento
.
di Matenatica, tlniversi.tb <li Tnenton ltaly.
whene fo 6 c*[Bn{r,o)rRn], fo(0) = 0, pn(oo} = {x € Rn: llxll . ,:o}.
Assume
that the
eLgenvalues tl
.Iacobian
matrix fot(O) has two punely imaginany
that the ::enaining eigenvaLues fijij:;z
satisfy
r, I mi (m=0ntlrt2r.,.), For- those f e c-[Bn(no)rRnl, f(0) = 0,
whlch ar"e close
turbed
and
to fo {in an appnopriate topol.ogy)
conslder" the pen-
systenr
i = f(x)
We
shall
norr addnese ourselves
(2)
to the following
1-oca1 pnoblem.
those f nean fo give .e complete deseniption of the
invaniant se'ts of (?) lying in gn(n) fon r smalL.
ProbLem.: Fon
compact
fn
a.pproacl'ring
this
pr,obLem we wL1l considen
integer k the faJ-lowi"ng two pnopenties:
(a) thene exi*ts a nei.ghborhood
of fo
ol"0r
^J
and dl>0
for any posit:i.ve
and two numbens
suchtha'bforeveny f 6il thene
are at moe't k nontr"j-vial periodic onbitc of (e) lying
in Bn(r..r)
ha'r-ing pcr.iodo
ir'r [?n-6r.r2r+6r1;
tb) fon each i.nteger j, 0 5 j : k, fon each n 6 (0rr"r1
an<l d e (0,61J and for each rrelghbonhoerd N of fo,
N <= lll, there exists an f G N sueh tirat (2) has exactly
j nc,ntrivial ireriodic onbits
per:iod in [2n-dn2r+6],
In R2,
Andnoncv
et aL. t2l
J.ying
pr,ovS.ded
in nn(r)
a partial
and having
answen
to the
Pr.oblem
by pnovi.ng
that prope::ties (a) and (b) are a consequence of the onigin
of (1) being (Zt+:.)*asynrptotically stable on (?k+,f)-conpletely
unstable. The or"igin of system (f) is said to be (Zk+l)-asymptotically
stable ((Zt+f)-completely unstabl-e) if
integer. such that the onigin of (2)
Zk+l is the smallest positive
is asymptatically stable (completely
is o(1"17'k1-2) (tirat
unstable)fonal-l- f fonwhich f(x)-fo(x)
is, the stabiJ.ity
pnoper.ty
is recognieable by terms up ta onden 2k+1 in
the Tayler. expansion of f,r)
asymtritotic
stability).
to'tic stability
(,.:r.:e
Thene
[10] for frrrther. infr:r.mation on (2k+l-]-
i.s;rn equivalence between the (Zk+1)-;rsynrp-
(Zt<:tt)-conr1:letr,:
-i.nst;rbjIity)
c.F tltr,:
r:ri.gin r:f (1)
the exi.stence of a computable l,i;rpunov function V(x)
;rlong sal-utions
of (1)n derrcteql by r?(*), is
and
whose der-ivative
rregat-ive
clefinite
and
satisfi.es ti(*i = Gzkozlx[2k+z + 0(l*lzk*u) whene Gzkoz is a negat'ive
(J:ositive) cr:rnputable constant (see Ij-0]). 'I]ris algebnaic pnocedr:re nf
conrputing V(x) is originalJ.y due 'bo Poincard lLzf.
and Sal-vadori rel,ated
the
(.2k+:L)-aswr4:to'tic
In [10]
stabif.ity
Negnini
({?k+t-compl.ete
instabil-j.ty) of: tlLe origirr of (l ) with the existence r:f aeymptotically
stabl.e (completeJ"y unstat,l.e) l:i.funcating pe:riod or:bits of t2) when the
c.lass
of per.tunbati-ons f is
y'estr:i.r:tec1 "to
a one parameter family
0 < p < l- (the usual Hopf'bifur.cation) such that the oni-girr c'f
{f..i,
u
(?) i,s compJ.etely unstable (ersymptct-ica1.Ly ..;tahl,e) for u > 0 and this
stabi.).,ity pnoperty isi recogni"zabl-e by the linei-lr pant of f u. Ttrey showed
fon each U > 0 (U small.) ther.e exists exactly one Der,i,:dic r:r'bit of
i = f..(x) with peri-ocl nean 2n and lying close to the onigin,
fn recent wor:ks l-3-l , [4.], the authors extend the::esults of
II
And:ronov
et aI. by p:rovlng pnopenties (a) and (b) f,on n=2
ane eguLvalent
to
the
stablltty ((zk+f)-eomplete lnstablllty) of the onlgln
of (1). We also pr.ove fon n=2 that the (zk+1)-asymptotlc etablllty
of the onlgln of (1) ((2k+1)-complete lnstablllty) implleg
(2k+1)-asymptotlc
(e)
frJ=
fr s.
r
j=o
and
on
,
each Uj has a nonempty lntenlor wlth fo lylng
its
boundar-y
for each j,
whet'e
Sr = {f € Al such that (2) has exactJ-y J nontrivlal
J
periodic or:bits lylng
period
ln en(nr) with
in Izn-6,,zn+611] i
(d) as f+ fo the set of peniodic orbits lying in Bn(nr) with period
in [2n-6r,2n+6rJ tend to t0] in the flausdor:ff topoJ.ogy.
In the. genenic case, k=lr necessary and sufflclent conditions
given fon those
f
contal.ned
in SO and in St in
Poincar6 constant G+ and the neal pant
tenms
of the
of the elgenvalues of the
ilacobian ft(0) (whlch we refer: to as c). Namely f e 31 lf
only if
oG*
Flnally
. 0 (f e S0 if
we eonsiden
and only
if
were
oGU
and
:0).
the tnanscendental case uhen Gl = 0 for
a1l i; that ls the origin of (1) is not (Zk+1)-asymptotically stabj.e
nor (2k+1)-completely unstabLe for any posltlve intege:r k. lfe provided
a chanactenl.zation of the tnaJectot'ies of (Z) in a sultable neighbonhood
of the origin.
We
omit the detalls hene but point out that if the
et aL. [ZJ) ttren
i is equivalent to the orlgin of (1) being a centen.
function fo ls analytic (as assumed by
Gl = 0 for: all
Andnonov
Thle is not tnue in the C* case and a rather- extensive tneatment of
this
case can also be f<rund in [5J,
In
L9?80
prion to oun papers [3] ana [4], Chafee [0] considened in
Rn the same aspect of the
Pr.ob1em
as Andronov et al.
Using the
alter-
native metliod [?-1, he obtained a deter:mining equation S(E'f) = 0,
of the amplitude of the bifut.cating onbit of (2)
and f repr'esents the right-hand side of (2). He found that {,(0rfoi =
and by assuming that tire multiplicity of the zero root of 0('rfo) - 0
whene E is.a
is equal to
(e1)
mea$ur?e
some numben
k he pnovet! that
c,enclusiorrs
hold. Since the cletermination of 0('r')
is
the determination of k in Chafeets r.esults is a
Thus for. the
case n=2
r+e
h;rve shown [4]
(a), (b), (c)
otrS-y known
and
impl-icity
new problem
in itsel.f.
t]rat the k of Chafee [6] is
pr.ecisely equal to the k associ.ated r+lth the (:tk+l)-asymptotic sta-
bility or:
'rfe
(Zt<+t)-complete instal:i.J.i'by
are now attempting to characterize the k of Chafee ln
n : 3. Fir:st
ti
of ttre origin of (I).
su6rpose
tha't for(0) l:as two punely
lmaginany eigenvalue.s
and the renainirrg eigenvalues have nonzero r"eal
Center M.anifold Theorem
[3],
we then recognize
Rn,
parts.
By using the
that ther:e exists a
2-
say llg, tangent at the origin 0 of Rn is the
which is locally inccrr,raesponding to the eigenvalues !i
dimensiorral, manifoldo
eigenspace
vaniant wittr nespect to the unper:tunbed equatJ.on (1)'
have
In this
the following characterizatiorr of pnope::ties (a) and (b).
ca$e
we
Ihgoran 1., A necessany and sufflcient conditlon fon pnopentles (a)
and
le .that 0 is fon equatlon (1) elther (2k+1)-asymptotlcally
stable on (2k+1)-completely unstabLe on HO (1,e, with
respect to ini_
tiat points J_ying on Hn).
(b) to
occul^
Under solne rnore panticulan hypotheses on
the
stabllity pnopenties in
?heonem
the
spectnurn
of for(0),
1 can be expnessed d1-ectJ-y in
tenms
of the unpe'tunbed equation (t), r.rithout any explrcit invor-vement
of
Ho. Pnecisely the following theonem hol"ds,
Theorsn
2,
Suppose
that for(0)
has two
punely inraginary eigenvalueg
ti
and the rernaining eigenvalues have negative neal pants.
Then a
necessa::y and su:Fficient condition for" pnoperties (a)
and (b) to occun
is that 0 is fon equation (L) either
(2k+1)-unstable (in the whole).
(
Zt+f ) -asympto.tically stable on
(Notice that we ar:e using the concept of (Zt+t)_instabiLity
whose
definition is analogous to that r:f (2k+t)_cciniplete instability). A
similar theonem can be stated when for(0) has two puner.y imaginany
eLgenvalues tt
oun proof
of
anrl the nemaining eigenvalues have
Theorem
posltive neal pants.
2 involve the extension of the poincand
proced,une
given by Liapounorr in his treatise on stabii.ity of motion
[g], in onde'
to pnove tlre equivalence betvreen the stabi).lty pr"operties of 0 fon
equation (1) occuring in Theor:em l an<I in Theoren z, respectively.
?he rnuch more
dlfficrrl-t problem in Rn
concerns the case
in
whlch
fot(0) has 2[t purely imaginary eigenvalues (+ Zp : n). In this
:
situation the center Hanifold i-s zrs dimensionaf and the poincan€-
.#Baddp o1 .uoTlpDdnJTq JdoH pup .;ua+sds ilfqF+s
[11ea1-]c+dtu{se-q Jo sar+*ado"rd ar;aual *r"repEnTRS "l prp pTiuu.rafi .g
tq]
'(.readde o1; 'V'!'i'I ' 'TpuV " Irroq ;io 'I' 'd+TT Tqc'+s t;+o1<iurr{se-q
pue uoTl.l?stiniTq Sciog pazgTl!'.Iauag 5tur.:penle$ "T pup FTaJu.Iall .S tt]
' l,ruaddu o1;
dTB+I ca;sar.r;, (stiotlenbg TI?T+ua,JsJJl{j ilf sa)upn.pv lirara'd {.1.) ;Jcria
-.raJuo3 ;uo;ctrel,.rnjTq jdoH pazrTpJarr.ap {1,.ropen1eg .T pue p;iiJurrd{ .S te
_l
.ua1Hsn,ial .Tl.BT 5 {ju{tT+BIsuF,-{jr cr.5-rtrui:rcf,
3o ue.rHo.rd TaEJSJ {,,ane'19 ar{} rIJ uullris:.dg ;elrureufg {o suoT}lr},fnJI{
.V pfiB rUop.toS .1 sq;:-r.ro+uoal .[ 6A.0r1o,I1..,r:y 'v
3;O r{.:OaqJ,,,
(!IUTp1a1
''tqt,^
.)inEN'pp)tii'T{og Ts.larsscd3 sauqrlslig,ur8a'L,l:"uc4
t,ht'
.11
Id]
(tu6TJfiT .us$s
pup .trouo.:puv
.V iT]
$:1JNSd3"J3u
.uaTgodd aq+ ?luTu.ra::uoc suorlsanb
1ue1
-.rodu1 pue 8u1+saJo+uT fueru uloniad rrT:|s a.ror{l- +pij+ a}ou d11eu1g ag
'(1u;.rBog 'd tl;.Tl'r ,{T}r:tot auup ?urilq sT Z pup
r
sua"ro;rr{tr,8u1pn1cur )tJorr{ rjTri+)
([a]
aounodpr't aas) sanTs.nuaSra oq]. uo
uoT+Tpuoc asueuosa{-r-uou aTqe}Tns E ,Japun s+rq"Io crpol.rod JoJ sp TTaA sp
sl"rg,ro alpol*adSsenb ;io acualsrxa ari+ .IoJ panrE'}cio uaag aApq $+rnsalr
auos 'tTT] u3 uaal3 o.rnpaco.rd gsTEaurod
e Eulsn adp
or't asps
aq+
JO uU
uT uoT+pzr-geaauaB
srq+ uralqea11dde,ra3uo1 ou sr d,:oaq1 uosxrpuag
t6l l'i.
Chafeeu Gr:r.reralize<i Hopf
neighbonhood
(rsze),
t7l J.
bifuncation and pentunbation in a full
of a gJ-''ren vecton field, lndi.ana Univ. Math. J.r
27
17:t--tgt+.
Ha.ler trOr.clinai.y Diffenentj-al. llquationsfi., Ilrtenscience, l{ew York,
_1,q69.
tS] A. M. Liapormov, trProbl6ne general de l.a stabil.itd
Atrn.
of Math.
Studies
,
1-7,-
du mouvementtr,
Princeton Univer'sity Pr"ess, Princeton'
lle'ni Jensey, 1{147.
tgl G. Mar.sden end fi.
llcC:.acken, rrThe tlopf.
bifurcation and its applica'-
tionsr', l{otes irr Ap1:lied Math. $ci. 190 Spninger', New Yonkr 1976-
l-fpl P,
Negr:'ini anci
L.
Sal-v;r<1ori,
Attnactivity and Hopf bifurcation,
r TMAx 3(1979) o 87- 100.
sulla s:tab,ilLt.i delltequilibrio nei casl cniticl,
1'krnlj.nea:: Ana.i-.
[],rl L.
Salvado:ni,,
Ann, I'l;:t . Pur:a Appi-.
[12i G.
Sans;one
{
'r
)LXIX( i965 }
;ind R. Contl,
'
"Non-1.inea'r"
f4aemill;:rrro i'lev Yor,k, 1961r'
l-33
"
DifJ'eyential llcluationsrro
J"