SIAM J. APPL. MATH.c
2005 Socie y o Indus ial and Applied Ma hema ics
Vol. 65, No. 6, pp. 1933–1951
THE FOCUS-CENTER-LIMIT CYCLE BIFURCATION IN
SYMMETRIC 3D PIECEWISE LINEAR SYSTEMS∗
EMILIO FREIRE†, ENRIQUE PONCE†,AND JAVIER ROS†
Abs ac . The bi h o limi cycles in 3D ( h ee-dimensional) piecewise linea sys ems o he
ele an case o symme ical oscilla o s is conside ed. A echnique al eady used by he au ho s in
plana sys ems is ex ended o cope wi h 3D sys ems, whe e a g ea e complexi y is in ol ed.
Unde some gi en nondegene acy condi ions, he co esponding heo em cha ac e izing he bi-
u ca ion is s a ed. In e ms o he de ia ion om he c i ical alue o he bi u ca ion pa ame e ,
exp essions in he o m o powe se ies o he pe iod, ampli ude, and he cha ac e is ic mul iplie s
o he bi u ca ing limi cycle a e also ob ained.
The esul s a e applied o accu a ely p edic he bi h o symme ical pe iodic oscilla ions in a
3D elec onic ci cui genealogically ela ed o he classical Van de Pol oscilla o .
Key wo ds. piecewise linea sys ems, bi u ca ion heo y, limi cycles
AMS subjec classifica ions. 37G15, 34C15
DOI. 10.1137/040606107
1. In oduc ion and main esul s. Piecewise linea modeling o nonlinea
dynamical sys ems is especially success ul in some enginee ing p oblems, such as he
analysis and design o elec onic oscilla o s o con ol sys ems (see, e.g., [CFPT02]).
Howe e , in he amewo k o piecewise linea sys ems, he e a e no gene al bi u ca ion
esul s explaining he appea ance o disappea ance o sel -sus ained oscilla ions, as is
he case o he Hop bi u ca ion heo em in he con ex o diffe en iable sys ems.
Thus, he au ho s ga e in [FPR99] a comple e cha ac e iza ion o he ocus-cen e -
limi cycle bi u ca ion o symme ic plana piecewise linea sys ems. Now we show
how he co esponding esul can be ex ended o he 3D case.
We conside a common si ua ion in applica ions, namely, dynamical sys ems de-
fined by piecewise con inuous ec o fields wi h h ee linea zones and wo pa allel
on ie s. Fu he mo e, i is assumed ha such sys ems show symme y wi h espec
o he o igin; ha is, i we pu hem in he o m dx/dτ = (x) wi h x∈R3, hey
sa is y (−x)=− (x). In pa icula , (0) = 0, and so he o igin is an equilib ium
poin o all alues o he pa ame e s. By means o a linea change o a iables, i is
always possible o suppose ha he on ie s a e he planes Σ1={x∈R3:x1=1}
and Σ−1={x∈R3:x1=−1}. We deno e by L(le ), C(cen al), and R( igh )
he egions o R3a which x1<−1, |x1|≤1, and x1>1, espec i ely, hold.
To be mo e p ecise, we conside sys ems exp essed as ollows:
˙x =⎧
⎨
⎩
ALx−bi x1<−1,
ACxi |x1|≤1,
ALx+bi x1>1,
(1.1)
∗Recei ed by he edi o s Ap il 1, 2004; accep ed o publica ion (in e ised o m) Feb ua y 2,
2005; published elec onically Augus 3, 2005. This esea ch was pa ially suppo ed by g an s
DPI2000-1218-C04-04, BFM2001-2668, and BFM2003-00336 o he Spanish Minis y o Science and
Technology. The au ho s we e also suppo ed by Jun a de Andaluc´ıa, as membe s o he esea ch
g oup TIC-130, in he budge co esponding o 2003.
h p://www.siam.o g/jou nals/siap/65-6/60610.h ml
†Depa amen o Ma em´a ica Aplicada II, Uni e sidad de Se illa, Escuela Supe io de Ingenie osm,
Camino de los Descub imien os s/n, 41092, SEVILLA, Spain ([email p o ec ed], [email p o ec ed],
[email p o ec ed]).
1933
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1934 EMILIO FREIRE, ENRIQUE PONCE, AND JAVIER ROS
T>TT<T
cT=T
cc
Fig. 1.The ocus-cen e -limi cycle bi u ca ion in he case D>0,γ>0. The ocal plane and
he complemen a y one-dimensional in a ian mani old a he o igin a e shown, along wi h he wo
pa allel planes which sepa a e he h ee linea egions. In he si ua ion ske ched, as deduced om
Theo em 1.1, he bi u ca ing limi cycle is o saddle ype.
whe e we ha e aken ad an age o he con inui y and symme y o he ec o field
in ol ed; in pa icula , he ma ices ALand ACdiffe only in hei fi s columns.
F om P oposi ion 16 o [CFPT02], unde he gene ic condi ion o obse abili y,
e e y sys em (1.1) can be w i en in he gene alized Li´ena d o m
d
dτ ⎡
⎣
x1
x2
x3⎤
⎦=⎡
⎣
−10
m0−1
d00
⎤
⎦⎡
⎣
x1
x2
x3⎤
⎦+⎡
⎣
T−
M−m
D−d⎤
⎦sa (x1),(1.2)
whe e sa (x1)is heno malized sa u a ion
sa (x1)=⎧
⎨
⎩
−1,x
1≤−1,
x1,|x1|<1,
1,x
1≥1,
so ha , ega ding sys em (1.1), we ha e
AL=⎡
⎣
−10
m0−1
d00
⎤
⎦,A
C=⎡
⎣
T−10
M0−1
D00
⎤
⎦,b=⎡
⎣
T−
M−m
D−d⎤
⎦.
No e ha sys em (1.2) is a pa icula ins ance o he mo e gene al Lu ’e o m
dx
dτ =Ax+bsa (cTx)
o he case A=ALand c=e1, whe e e1s ands o he fi s ec o o he canonical
basis.
Clea ly, he pa ame e s ,m,dand T,M,Ds and o he ace, he sum o p in-
cipal mino s o o de wo, and he de e minan o each ma ix, and hey comple ely
de e mine he dynamics o he sys em.
Choosing Tas he bi u ca ion pa ame e , o he c i ical alue Tc=D/M wi h
M>0, sys em (1.2) has a linea cen e in he zone C(see Figu e 1); ha is, he
ma ix AChas a pai o pu e imagina y eigen alues. We wan o analyze whe he a
limi cycle bi u ca es om his configu a ion as he bi u ca ion pa ame e T a ies.
No e he simila i ies wi h he classical Hop bi u ca ion scena io.
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3D FOCUS-CENTER-LIMIT CYCLE BIFURCATION 1935
I will be use ul, in o de o know he s abili y o such a limi cycle, o es ima e he
cha ac e is ic mul iplie s o he limi cycle, ha is, he eigen alues o he de i a i e
o a Poinca ´e e u n map defined in an adequa e sec ion o he phase space. We will
deno e he loga i hms o hese cha ac e is ic mul iplie s by μ and μa, om adial
and axial, espec i ely. Ou main esul is he ollowing.
Theo em 1.1. Le us conside sys em (1.2) wi h M>0,Tc=D/M, and
γ=DM −Dm +dM − M2=0.Fo T=Tc he sys em unde goes a ocus-cen e -
limi cycle bi u ca ion; ha is, om he lineal cen e configu a ion in he cen al zone,
which exis s o T=Tc, one limi cycle appea s o γ(T−Tc)>0and T−Tc
sufficien ly small.
The ampli ude “a” (measu ed as he maximum o |x1|), he pe iod Pe , and he
loga i hms o cha ac e is ic mul iplie s μ and μao he pe iodic o bi a e analy ic
unc ions a 0, in he a iable (T−Tc)1/3; namely,
a=1+(6π)2/3M4/3
8γ2/3(T−Tc)2/3+(6π4)1/3a4
960M1/3γ7/3(T−Tc)4/3+O(T−Tc)5/3,
Pe =2π
√M+π(M−m)√M
γ(T−Tc)−62/3π5/3M5/6P5
20γ8/3(T−Tc)5/3+O(T−Tc)2,
μ =−(48π)1/3M7/6γ2/3
D2+M3(T−Tc)1/3+O(T−Tc)2/3,
μa=2πD
M3/2+(48π)1/3
M5/6M −D
γ1/3+M2γ2/3
D2+M3(T−Tc)1/3+O(T−Tc)2/3,
whe e
a4=−120 M5+120D+2 3+21m +72dM4
+−93m+27 2D+27m−2 2dM3+2 2m+25d −27m2DM2
+25D3+23(m −d)D2M−25mD3,
P5=[M(M−m)2+(M −d)2](M −D).
In pa icula , i γ>0and D<0, hen he limi cycle bi u ca es o T>T
cand is
o bi ally asymp o ically s able.
This heo em desc ibes a codimension-one bi u ca ion, simila o he Hop bi u -
ca ion o diffe en iable dynamics (see [CH82]), bu some diffe ences should be no ed.
In pa icula , he exp essions cha ac e izing he bi u ca ion a e in e ms o he pa-
ame e o he one hi d powe ins ead o he one hal powe , and, mo e impo an ,
he limi cycle ampli ude’s leading o de is O(1). Thus, he s abili y change o he
o igin is accompanied by he ab up appea ance o a limi cycle o significan size.
This commen also applies o he plana case, as appea ed in [K 87] and [FPR99].
When he coefficien γis no equal o ze o, i allows a comple e cha ac e iza ion
o he bi u ca ion c i icali y. I s ole is analogous o he coefficien o he cubic e m in
he Poinca ´e–And ono –Hop no mal o m. When γ= 0, he bi u ca ion is o highe
codimension, equi ing a specific ea men ha will appea elsewhe e.
We wan o ema k ha i is possible, wi h he same echniques, o ob ain sim-
ila bi u ca ion esul s o he asymme ic case o single-sided sa u a ion. Thus, he
p oposed me hodology is able o cope wi h a wide class o piecewise linea sys ems.
The es o he pape is s uc u ed as ollows. In sec ion 2, we show how he
abo e esul can be use ul o accu a ely p edic ing he bi h o symme ical pe iodic
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1936 EMILIO FREIRE, ENRIQUE PONCE, AND JAVIER ROS
oscilla ions in a idimensional elec onic ci cui , which can be buil by aking a Van
de Pol oscilla o as s a ing poin . The p oo o Theo em 1.1 is gi en in sec ion 3.
2. P edic ing he onse o symme ical pe iodic oscilla ions in a 3D
elec onic ci cui . In his sec ion, we conside he elec onic ci cui o Figu e 2(a),
genealogically ela ed wi h he classical Van de Pol oscilla o , in o de o show he
applicabili y o ou esul s. Rega ding his ci cui , he nonlinea conduc ance NL is i s
ac i e elemen , implemen ed by means o an ope a ional amplifie wi h he eedback
s uc u e o Figu e 2(b), and he cu en - ol age cha ac e is ic is shown in Figu e
2(c). No e ha we a e dealing wi h a nonlinea i y cha ac e is ic quali a i ely simila
o he cubic one appea ing in he classical Rayleigh–Van de Pol oscilla o . In ac , i
we elimina e he capaci o C2and make R=R0= 0, hen he esul ing plana ci cui
could be hough o as a mode n elec onic ealiza ion o such classical oscilla o s; see
[K 87] and [FPR99].
Thus, he 3D ci cui o Figu e 2 can be buil by adding he capaci ance C2 o
a bidimensional oscilla o ci cui . In he con ex o chao ic ci cui s, such opology
was o iginally p oposed in [SYM81], and was s udied a e wa ds in [FGA84] and
[FRGP93] in he case R0= 0 and assuming a nonlinea posi i e conduc ance o he
esis o R. Wi h sligh modifica ions, his ci cui has been ex ensi ely s udied in he
las wo decades; see [GK92] o [HBCJM91]. Taking R0= 0 and subs i u ing he
nonlinea elemen by he so-called Chua diode, many pape s ha e also been w i en;
see [CWHZ93], [Ma93], and e e ences he ein. Anyway, he onse o symme ical
pe iodic oscilla ions was ne e accu a ely p edic ed, since in mos cases he ci cui
was analyzed by aking polynomial app oxima ions. Thus, he apid bi u ca ion o
he limi cycle obse ed in p ac ice was ne e jus ified.
I should be ema ked ha he cha ac e is ic o Chua’s diode is quali a i ely sim-
ila o he one p esen ed in Figu e 2(c) bu he zone o nega i e slope is made up by
h ee pieces wi h wo diffe en slopes. Fo ha , a leas wo subci cui s wi h ope a-
ional amplifie s like hose shown in Figu e 2(b) a e needed. Thus, he Chua ci cui
cha ac e is ic has fi e linea segmen s ins ead o only h ee, as in ou case. Howe e ,
in modeling Chua’s ci cui , usually only he h ee inne mos pieces a e ep esen ed,
since he wo ou e mos pieces o posi i e slope a e no physically used; see [Ke93].
As s a ed in [K 87] and [FPR99], he e exis s an excellen ag eemen be ween he
ac ual esponse o he nonlinea de ice NL in he ci cui and i s symme ic piecewise
C2
R
L
R0
NL
R3
R1
R2
i
(b) (c)
C1
(a)
iLi
Fig. 2. (a) The 3D elec onic ci cui . (b) Implemen a ion o he nonlinea conduc ance NL.
(c) Piecewise linea cu en - ol age cha ac e is ic o NL.
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3D FOCUS-CENTER-LIMIT CYCLE BIFURCATION 1937
linea ma hema ical model. The e o e, we a e led o conside he piecewise linea
dynamical sys em
C1
d 1
dτ = 2− 1
R−i( 1),
C2
d 2
dτ = 1− 2
R−iL,(2.1)
LdiL
dτ = 2−R0iL,
whe e 1and 2a e he ol ages ac oss he capaci o s C1and C2, espec i ely, while
iLis he cu en h ough he induc ance. The nonlinea cu en - ol age cha ac e is ic
is
i( 1)= 1− ( 1)
R1
wi h ( 1)=Esign ( 1),| 1|> E/σ,
σ 1,| 1|≤E/σ,
whe e
σ=1+R2
R3
is he gain o he ope a ional amplifie configu ed (using eedback) as a nonin e ing
amplifie and Eis i s sa u a ion ol age.
Wi h he ollowing linea change o a iables and ime escaling,
1=E
σy1,
2=E
σy2,i
L=E
σC2
Ly3,τ=RC1¯τ,(2.2)
and defining he ollowing fi e nonnega i e dimensionless pa ame e s,
=R
R1
,c=C1
C2
,μ=(σ−1) R
R1
=RR2
R1R3
,ρ=R2C2
1
LC2
,κ=RR0C1
L,(2.3)
we can exp ess sys em (2.1) as ollows:
d
d¯τ⎡
⎣
y1
y2
y3⎤
⎦=⎡
⎣− −11 0
c−c−√ρ
0√ρ−κ⎤
⎦⎡
⎣
y1
y2
y3⎤
⎦+⎡
⎣
μ+
0
0⎤
⎦sa (y1).(2.4)
Fo he subsequen analysis, we will choose μand ρas he main bi u ca ion pa ame-
e s. In p ac ice, o de ec he bi u ca ion in a expe imen al way, i is be e o une
he pa ame e μby means o a a iable esis o R2, which is equi alen o a ying he
gain σ.
The obse abili y ma ix o sys em (2.1) is
O=⎡
⎣
eT
1
eT
1A
eT
1A2⎤
⎦=⎡
⎣
100
− −110
( +1)
2+c−c− −1−√ρ⎤
⎦,
which has ull ank o all he alues o componen s o he ci cui . F om P oposi ion
16 o [CFPT02], sys em (2.1) can be exp essed in Li´ena d’s gene alized o m (1.2)
wi h he ollowing alues:
T=μ−c−κ−1, =− −c−κ−1<0,
M=(c+1)κ−(c+κ)μ+ρ, m =(c+1)κ+(c+κ) +ρ>0,
D=(cκ +ρ)μ−ρ, d =−(cκ +ρ) −ρ<0.
(2.5)
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1938 EMILIO FREIRE, ENRIQUE PONCE, AND JAVIER ROS
M = 0
D = 0
MT = D
–0.05
0
0.05
0.1
0.15
0.2
0.25
ρ
0.2 0.4 0.6 0.8 1
μ
Fig. 3.The pa abolic a c ( hick line) in he plane (μ, ρ)co esponding o he bi u ca ion locus
o P oposi ion 2.1 o c=0.2and κ=0.05. The ho izon al line indica es he pa h ollowed as μ
a ies o a fixed alue o ρ. The dashed line ep esen s poin s wi h D=0, so ha abo e i we ha e
D<0. A he do ed s aigh line we ha e M=0, and abo e his line we ha e M>0. The e ical
line co esponds o μ=μ∗.
No e ha hese coefficien s a e he linea in a ian s o he wo ma ices in ol ed, so
ha hei compu a ion is s aigh o wa d, and ha i is no necessa y o explici ly
compu e he linea change o a iables equi ed o ge he Li´ena d o m o applying
Theo em 1.1.
The equa ion MT −D= 0 leads o
(c+κ)μ2−(c+κ)2+c+2κμ+ρ(c+κ)+κ(c+1)(c+κ+1)=0,(2.6)
which can be ew i en as (μ−μ∗)2+ρ−ρ∗=0,whe e
μ∗=1+
(c+κ)2−c
2(c+κ),
ρ∗=μ∗2−κ(c+1)1+ 1
c+κ
(2.7)
ep esen he coo dina es in he (μ, ρ)-plane o he e ex o he quad a ic (2.6); see
Figu e 3. Now he applica ion o Theo em 1.1 allows us o s a e he ollowing esul .
P oposi ion 2.1. Le us conside sys em (2.4) and assume ha c>0and he
pa ame e κsa isfies
0<κ<κ
max(c)=√c2+c−c
2.(2.8)
Then he sys em unde goes he ocus-cen e -limi cycle bi u ca ion desc ibed in Theo-
em 1.1 a he poin s o he (μ, ρ)-plane belonging o he pa abolic a c defined by he
quad a ic equa ion
(μ−μ∗)2+ρ−ρ∗=0(2.9)
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3D FOCUS-CENTER-LIMIT CYCLE BIFURCATION 1939
1
κ
κ
MAX
κ
=0.25
κ
0
0.1
0.2
0.3
0.5 1 1.5 2
c
Fig. 4.The g aphs o he unc ions κmax(c)and κ1(c), which de e mine diffe en egions in he
plane (c, κ)as desc ibed in s a emen s (a) and (b) o P oposi ion 2.1. No e he ho izon al asymp o e
a κ=1/4.
and sa is ying
ρ>(c+κ)μ−(c+1)κ.(2.10)
The endpoin s o he abo e pa abolic a c a e
(μ1,ρ
1)=1−c+c
2(c+κ),c(1 −2κ)−c
2,(μ2,ρ
2)=1−c−c
2(c+κ),c(1 −2κ)+c
2,
whe e
c=c2(1 −2κ)2−4c(c+1)κ2.
In he poin s o he abo e pa abolic a c, he inequali y D<0holds, and he ollowing
cases a ise:
(a) I 0<c<1and 0<κ<κ
1, whe e κ1=κ1(c)is he only posi i e oo o he
qua ic
(c+κ)4+4cκ(c+κ)−c2=0,(2.11)
hen μ1<μ
∗<μ
2and wo subcases appea ; see Figu e 4.
(a.1) I μ1<μ<μ
∗, hen a he bi u ca ion poin s o he pa abolic a c gi en by
(2.9)–(2.10) one has γ>0. Consequen ly, when ρ a ies, he bi u ca ion is
supe c i ical and he limi cycle is o bi ally asymp o ically s able.
(a.2) I μ∗<μ<μ
2, hen γ<0a he bi u ca ion poin s o he pa abolic a c. He e,
when ρ a ies, he bi u ca ion is subc i ical and he limi cycle is uns able.
(b) I 0<c<1and κ≥κ1,o c≥1, hen all he bi u ca ion poin s o he pa abolic
a c (2.9)–(2.10) sa is y γ>0. The e o e, he bi u ca ion is supe c i ical and
he bi u ca ing limi cycle is o bi ally asymp o ically s able.
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1940 EMILIO FREIRE, ENRIQUE PONCE, AND JAVIER ROS
Table 2.1
Lis o componen s o he ci cui .
C=C1=C2100 nF
L220 mH
R110 kΩ
R32200 Ω
R1kΩ
R0220 Ω
P oo . Condi ions T=Tcand M>0 o Theo em 1.1 lead o MT −D= 0, which
is equi alen o (2.9), and o (2.10). A e some manipula ions, we ge he inequali y
(c+κ)μ2−(c+2κ)μ+(c+1)κ<0,
whose disc iminan , namely c2−4c2κ−4cκ2, is posi i e due o (2.8). In ac , his
exp ession coincides wi h c2. The endpoin s o he pa abolic a c can be ob ained by
sol ing he equa ion M= 0 and (2.9).
To show ha D<0 a he bi u ca ion alues, as we a e wo king a poin s whe e
MT −D= 0 along wi h M>0, i suffices o show ha T<0, which is a i ial ask.
To p o e s a emen s (a) and (b), i is enough o s udy he sign o he coefficien
γin Theo em 1. Using he condi ion MT −D=0,weha e
γ=MT(M−m)+M(d− M)=M[T(M−m)+d− M],
and wi h M>0 we ge sign (γ) = sign [T(M−m)+d− M]. Thus, using (2.5),
(2.6), and canceling a ac o +μ>0, we conclude ha
sign (γ) = sign (c+κ)2+c+2κ−2(c+κ)μ= sign (μ∗−μ).(2.12)
Assume now ha 0 <c<1 and 0 <κ<κ
1. Thus, he le -hand side o (2.11) is
nega i e, which implies (c+κ)2<c.Then μ1<μ
∗<μ
2,and s a emen (a) ollows.
When c≤1 and κ≥κ1, we ha e (c+κ)2≥c. I c>1, hen we ha e (c+κ)2>
c>c
.In bo h cases, we conclude ha μ∗≥μ2,and s a emen (b) ollows.
Fo he sake o comple eness, i we define, o 0 <c<1, he cons an s
q1=3
27c+26+3
81c2+ 156c+75>0,q
2=q1+1
q1−4>0,
we ob ain
κ1(c)=c
6q2+c6c
q2−cq2
6−2c−c,
which is ep esen ed o 0 <c<1 in Figu e 4.
The abo e p oposi ion enables us o design he elec onic oscilla o by choosing
adequa ely he componen alues o he ci cui . In pa icula , in o de o minimize
he signal dis o ion om he sinusoidal wa e o m, one mus selec pa ame e s no a
om he bi u ca ion cu e whe e he onse o pe iodic oscilla ions has been p edic ed.
To assess he accu acy o piecewise linea modeling o his ci cui , a SPICE
implemen a ion o he ci cui was made; see [QNPS93]. The alues chosen o he
componen s a e in Table 2.1, while he ope a ional amplifie used was an LM324,
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3D FOCUS-CENTER-LIMIT CYCLE BIFURCATION 1941
M = 0
MT-D = 0
D = 0
0
0.2
0.4
0.6
0.8
1
ρ
0.2 0.4 0.6 0.8 1
μ
Fig. 5.The pa abolic a c ( hick line) in plane (μ, ρ)p edic ed by P oposi ion 2.1 o κ=0.1.
The ho izon al line indica es he pa h ollowed as μ a ies o he fixed alue o ρused in he
simula ions. The dashed line ep esen s poin s wi h D=0, so ha abo e i we ha e D<0. The
do ed s aigh line indica es poin s wi h M=0, and abo e i we ha e M>0.
wi h a supply ol age o 9V and a measu ed sa u a ion ol age o 8.5V (wi h sligh
a ia ions a ound).
Fo hese alues, we ha e
c=C1
C2
=1,κ=RR0C
L=0.1<√2−1
2≈0.2071,
so ha we can apply P oposi ion 2.1 and, in pa icula , i s s a emen (b). No e ha
μ=RR2
R1R3≈R2
22000,ρ=R2C
L≈0.4545,
so ha by a ying R2we mo e μ, desc ibing a ho izon al pa h ha c osses he cu e
co esponding o he locus o bi u ca ion poin s, as shown in Figu e 5. Fo he abo e
alue o ρ, he bi u ca ion akes place o he alue ¯μ≈0.4924, in acco dance wi h
(2.6), ha co esponds wi h he alue R2≈10833Ω,and oscilla ions will appea by
inc easing R2abo e his c i ical alue.
In Figu es 6 and 7, we show he compa ison be ween some expe imen al esul s
aken om a SPICE simula ion, once pu in o dimensionless o m, and he p edic ions
o Theo em 1.1 o he ampli ude and he pe iod o he bi u ca ing limi cycle. The
excellen ag eemen achie ed alida es he piecewise linea model assumed o he
ope a ional amplifie nonlinea cha ac e is ic.
3. P oo o Theo em 1.1. In his sec ion we p o ide he esul s necessa y o
p o e Theo em 1.1.
Fo he c i ical alue o he bi u ca ion pa ame e Tc=D/M, he ma ix AChas
a pai o imagina y eigen alues, so ha o Tin a neighbo hood o Tc he eigen alues
o ACwill be α±iβ and δ∈R. The cha ac e is ic polynomial o ACis
p(λ) = de (AC−λI)=−λ3+Tλ2−Mλ+D,
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1948 EMILIO FREIRE, ENRIQUE PONCE, AND JAVIER ROS
Due o he symme y o he o bi , i s pe iod is equal o 2(τC+τL). Subs i u ing
expansion (3.18) in o (3.9), and compu ing he abo e exp ession o he pe iod, we
ge he expansion gi en o Pe .
We will now de e mine he ampli ude o he pe iodic o bi . By using he a ia ion
o pa ame e s o mula, he solu ion o sys em (1.2) in zone Ris
x(τ)=eALτx3(τL)+τ
0
eAL(τ−s)b(τL)ds,(3.19)
so ha i s fi s componen is
x1(τ)=eT
1⎧
⎨
⎩eALτ⎡
⎣
1
−x1
2(τL)
−x1
3(τL)⎤
⎦+ ∞
!
i=0
Ai
L
τi+1
(i+ 1)!"b(τL)⎫
⎬
⎭.(3.20)
Le τ∗be he ime when |x1|a ains i s maximum alue in zone R. Taking
de i a i es wi h espec o τin (3.20), and imposing ha i mus anish a τ∗, we ge
G(τL,τ∗)= dx1(τ)
dτ τ=τ∗
=eT
1eALτ∗⎡
⎣
x1
2(τL)+T(τL)
x1
3(τL)+M
D⎤
⎦=0.(3.21)
Now using exp essions (3.11) and (3.13) and compu ing he powe se ies o Gin
(τL,τ∗)a (0,0), we ob ain
G(τL,τ∗)=M
2τL−Mτ∗+D−M
12 τ2
L+M −D
2τLτ∗+D−M
2τ∗2+O(τL,τ∗)3.
Hence, (3.21) defines implici ly in a neighbo hood o (0,0) a unc ion τ∗=ψ(τL) such
ha G(τL,ψ(τL)) = 0, namely,
τ∗=1
2τL+M −D
24Mτ2
L+O(τ4
L).
Subs i u ing he abo e expansion oge he wi h (3.11), (3.13), and (3.17) in o he
exp ession (3.20), we ge
a=x1(τ∗)=1+M
8τ2
L+1
1152M(13D2−11DM +15M2m−2M2 2)τ4
L+O(τ5
L).
Using exp ession (3.18) o τL, we ob ain he final exp ession o he ampli ude a.
Le us now compu e he cha ac e is ic mul iplie s o he bi u ca ing limi cycle.
Due o he simila i y ela ionship es ablished in P oposi ion 3.2, we conclude ha he
p oduc exp(ALτL)·exp(ACτC) co esponding o a solu ion o (3.5) has an eigen alue
equal o −1. We will deno e by λ and λa he o he wo eigen alues ha co espond
o he eigen alues o he de i a i e DpπLC (p0) o he ansi ion map associa ed wi h
he semio bi . The p oduc o he h ee eigen alues is hen equal o
−λ λa= de eALτLde eACτC.
Using ha de (eAτ ) = exp(τ ace(A)),we ge
−λ λa=eτL +τCT.(3.22)
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3D FOCUS-CENTER-LIMIT CYCLE BIFURCATION 1949
The expansion o he p oduc o exponen ials in (3.16) leads o an exp ession o
he o m
eALτLeAC(τL)τC(τL)=H0+τLH1+τ2
LH2+···.(3.23)
To compu e he abo e ma ices Hi, we w i e
I+ALτL+A2
L
τ2
L
2! +···
× eAC(0)τC(0) +τL
d
dτL
eAC(τL)τC(τL)τL=0
+τ2
L
2!
d2
dτ2
L
eAC(τL)τC(τL)τL=0
+···".
F om expansions (3.6)–(3.13), we ob ain τC(0) = π/M1/2,τ
C(0) = −1, τ
C(0)=0,
A
C(0) = A
C(0) = 0,and using hese alues in he abo e exp ession, we finally ge
H0=eAC(0)τC(0) =⎡
⎣
D2K−1−DMK M2K
0−10
D2MK −DM2KM
3K−1⎤
⎦,
H1=⎡
⎣
−D/M
m−M
d−D⎤
⎦D2K−1−DMK M2K
and
H2=M −D
2MH1,
whe e Khas been defined in (3.15), and i is emphasized ha H1and H2a e ank-one
ma ices.
The ma ix H0has eigen alues −1 (double) and λ0= exp(πD/M3/2). Fo he
single eigen alue λ0, we selec a igh eigen ec o 0=[1,0,M]Tand a le eigen ec o
wT
0=[D2/M 2,−D/M, 1]. We will deno e by λa he eigen alue o DπLC (p0) ha
o τL= 0 is equal o λ0. Since he eigen alue λ0o H0is simple, we can apply
pe u ba ion heo y (see sec ion 2.8 o [Wi65]) o assu e ha he equali y
H0+τLH1+τ2
LH2+··· 0+τL 1+τ2
L 2+···
=(λ0+τLλ1+τ2
Lλ2+···) 0+τL 1+τ2
L 2+···
holds o ce ain ec o s 1, 2... . As a consequence o P oposi ion 3.2 and (3.23),
we ge
λa=λ0+τLλ1+τ2
Lλ2+···.
A e some compu a ions, we a i e a
λ1=wT
0H1 0
wT
0 0
=M −D
M+γM
D2+M3λ0,
λ2=wT
0(H2 0+(H1−λ1I) 1)
wT
0 0
=(M −D)D2+M3+γ
2M2(D2+M3)2(λ0+1)
×(M −D)D2+M3λ0+D2−M3+2M2(dM −Dm)λ0.
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1950 EMILIO FREIRE, ENRIQUE PONCE, AND JAVIER ROS
The loga i hms μ and μao cha ac e is ic mul iplie s o he comple e pe iodic
o bi mus sa is y
eμ =λ2
,e
μa=λ2
a,(3.24)
while om (3.22) we ge he ela ionship
μ +μa=2 τL+2TτC.(3.25)
F om (3.24) and using he compu ed simple eigen alue λa, we ob ain
μa= 2 log λ0+λ1τL+λ2τ2
L+O(τ3
L)= 2 log λ0+ 2 log 1+λ1
λ0
τL+λ2
λ0
τ2
L+O(τ3
L)
=2λ0+2λ1
λ0
τL+2λ2
λ0−λ2
1
λ2
0τ2
L+O(τ3
L).
Subs i u ing he e λ1and λ2, and using expansion (3.18) o τL, we finally ge he
exp ession o μa ha appea s in Theo em 1.1. Using in (3.25) he expansions (3.9)
o τCand (3.18) o τL, we compu e μ .
Since he las asse ion o Theo em 1.1 is a di ec consequence o p e ious s a e-
men s, i s p oo is now comple ed.
Acknowledgmen s. The au ho s since ely app ecia e he ca e ul eading o he
anonymous e e ees and hei in e es ing sugges ions ha ha e no ably imp o ed he
pape . They also wan o acknowledge Jo ge Gal´an o his in aluable sugges ions
and commen s on a p elimina y e sion o he manusc ip , Manuel Rom´an o his
help wi h SPICE simula ion o he ci cui , and Fe nando Fe n´andez o helping wi h
figu es.
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