The Focus-Cen e -Limi Cycle Bi u ca ion in
Discon inuous Plana Piecewise Linea Sys ems
wi hou Sliding
En ique Ponce, Ja ie Ros and El´ısabe Vela
Abs ac Plana discon inuous piecewise linea sys ems wi h wo linea i y zones,
one o hem being o ocus ype, a e conside ed. By using an adequa e canonical
o m unde ce ain hypo heses, he bi u ca ion o a limi cycle, when he ocus chan-
ges i s s abili y a e becoming a linea cen e , is comple ely cha ac e ized. Analy ic
exp essions o he ampli ude, pe iod and cha ac e is ic mul iplie o he bi u ca ing
limi cycle a e p o ided. The s udied bi u ca ion appea s in eal wo ld applica ions,
as shown wi h he analysis o an elec onic Wien b idge oscilla o wi hou symme-
y.
1 In oduc ion and main esul s
Nowadays, he analysis o discon inuous piecewise-linea sys ems is an ac i e ield
o esea ch since ce ain mode n de ices a e well modeled by his class o sys ems,
see [1]. Fo he simples si ua ion howe e , as is he case o he agg ega ion o wo
plana linea sys ems, he e a e bi u ca ions ha s ill equi e a ho ough analysis.
Recen ly, in [2] i has been p oposed a canonical o m o he case o plana
discon inuous sys ems wi h wo zones o linea i y, o be deno ed D2PWLS2 o
sho . In he quo ed pape , he e a e shown some bi u ca ion esul s o he case
when bo h linea dynamics a e o ocus ype wi hou isible angencies, ha is, he e
a e no eal equilib ium poin s in he in e io o each hal -plane. He e, by eso ing
o he canonical o m gi en in [2], we conside a di e en si ua ion when we ha e
an equilib ium poin o ocus ype in he in e io o a hal plane wi hou speci ying
he linea dynamics ype in he o he hal plane. Ou goal is o desc ibe quali a i ely
and quan i a i ely he possible bi u ca ion o limi cycles h ough he change o
En ique Ponce, e-mail: eponce[email p o ec ed] ·Ja ie Ros, e-mail: ja ie [email protected] ·El´ısabe Vela, e-
mail: eli [email p o ec ed]
Depa amen o Ma em´a ica Aplicada II, E.T.S. Ingenie ´ıa, 41092-Se illa (Spain)
1
2 En ique Ponce, Ja ie Ros and El´ısabe Vela
s abili y o such an equilib ium poin . Thus, his wo k is a ele an gene aliza ion o
discon inuous ec o ields o he bi u ca ion s udied in [3] o he con inuous case.
To begin wi h, we assume wi hou loss o gene ali y ha he linea i y egions in
he phase plane a e he le and igh hal -planes,
S−={(x,y):x<0},S+={(x,y):x>0},
sepa a ed by he s aigh line
Σ
={(x,y):x=0}. The sys ems o be s udied become
˙x =(F+
1(x),F+
2(x)T=A+x+b+,i x∈S+,
F−
1(x),F−
2(x)T=A−x+b−,i x∈S−,(1)
whe e x= (x,y)T∈R2,A+= (a+
ij)and A−= (a−
ij)a e 2×2 cons an ma ices,
b+= (b+
1,b+
2)T,b−= (b−
1,b−
2)Ta e cons an ec o s o R2, and he de ini ion o
he ec o ield in
Σ
is no ele an o ou pu poses.
We assume he gene ic condi ion a+
12a−
12 >0, which means ha o bi s can c oss
he discon inui y line in opposi e di ec ions, allowing he exis ence o pe iod o bi s
ha use he wo hal planes. By applying P oposi ion 3.1 o [2], sys em (1) can be
w i en in he canonical o m
˙x =T−−1
D−0x−0
a−i x∈S−,
˙x =T+−1
D+0x−−b
a+i x∈S+,
(2)
whe e
a−=a−
12b−
2−a−
22b−
1,b=a−
12
a+
12
b+
1−b−
1,a+=a−
12
a+
12
(a+
12b+
2−a+
22b+
1),
and T±= (A±),D±=de (A±)a e he linea in a ian s o each zone.
The canonical o m (2) has se en pa ame e s; apa om he men ioned linea
in a ian s,we ind he wo pa ame e sa+and a− ela ed o he posi ion o equilib ia,
and a pa ame e bwhich is esponsible o he exis ence o a sliding se , whe e bo h
ec o ields canno be conca ena ed in a na u al way. In ac , he e exis s a sliding
se which is a segmen joining he o igin and he poin (0,b), see [2] o mo e de ails.
These wo endpoin s a e angency poin s, he o igin o he le egion and he poin
(0,b) o he igh one. Fu he mo e, he sliding se becomes a ac i e o b<0 and
epulsi e o b>0, sh inking o he o igin when b=0.
By compu ing he sign o ¨xa he angency poin s, we ob ain ¨x|(x,y)=(0,0)=a−,
¨x|(x,y)=(0,b)=a+,so ha he le ( igh ) angency is called isible o a−<0
(a+>0), being in isible o a−>0 (a+<0), see [2]. Thus he a±pa ame e s
a e ela ed o he isibili y o he angencies, and when hey anish we ha e boun-
da y equilib ium poin s, see [4] and also [5].
The Focus-Cen e -Limi Cycle Bi u ca ion in D2PWLS2wi hou Sliding 3
The possible equilib ia ( eal o i ual) a e loca ed a (a−/D−,a−T−/D−)and
(a+/D+,b+a+T+/D+)whe e i is assumed D+D−6=0. Wi hou loss o gene ali y,
we assume ha he e exis s an equilib ium o ocus ype in he le zone, ha is,
T2
−−4D−<0 (wha implies D−>0), and a−<0.
Ou in e es is o s udy wha happens when he ace T−passes h ough he c i-
ical alue ze o, ha is when he ocus passes om s able o uns able o ice e sa.
No e ha o T−=0 in he le hal plane we ha e a cen e con igu a ion which e -
mina es in a isible angency a he o igin. To a oid o he non-local phenomena i
is hen na u al o impose ha in he igh zone we ha e also a angency a he o igin
o in isible cha ac e , wha amoun s o equi e b=0. In ac , i we allow o mo e
his pa ame e bin a neighbo hood o ze o, hen we should ha e he possibili y o
new bi u ca ions, namely he collision o angencies, which has been epo ed in
[4]. Thus, ou s udy can be seen as a i s s ep in he analysis o he codimension-
wo bi u ca ion ha appea s when he pa ame e bis allowed o be mo ed. Such
codimension- wo bi u ca ion will be he aim o a u u e wo k, since i u ns ou o
be a non-gene ic case, no included in he ex ensi e analysis done in [6].
Thus, he e a e no p ope sliding se no jumps in he igh dynamics wi h espec
o he c i ical cen e . Unde hese assump ions ou i s esul is he ollowing.
P oposi ion 1. Unde he hypo heses T2
−−4D−<0, a−<0(le ocus dynamics
wi h isible angency a he o igin) and assuming ha in he igh zone we ha e an
in isible angency a he o igin, ha is, b =0and a+<0, sys em (2) is opologically
equi alen o he sys em
˙x=Tx−y,
˙y=Dx+a,i x >0,˙x=2
γ
x−y,
˙y= (1+
γ
2)(x+1),i x <0,(3)
whe e T =T+, D=D+,
γ
=
α
/
ω
, wi h 2
α
=T−,
ω
>0issuch ha 4D−−T2
−=4
ω
2,
and a =D−a+/(
ω
a−)>0.
The p oo o P oposi ion 1 appea s in Sec . 3. Wi h his esul , we manage o
desc ibe he le dynamics wi h only one pa ame e , needing o he h ee pa ame e s
o deal wi h he igh egion. The ollowing ema k should be aken in o accoun .
Rema k 1. Sys em (3) ep esen s in gene al a discon inuous ec o ield since o
x=0 we will ha e gene ically a6=1+
γ
2. The o iginal sys em (2) is con inuousonly
in he case a+=a−and b=0, bu e en in such non-gene ic case, he new sys em
p o ided by P oposi ion 1 will be discon inuous. Thus, he analysis o sys em (3) is
ele an o some con inuous cases ha could be also s udied by al e na i eme hods
wi hin a con inuous ec o ield con ex . This ac will be illus a ed la e in Sec . 2.
Rega ding equilib ium poin s, in he zone x<0, he e exis s a ocus a (x,y) =
(−1,−2
γ
), o be s able o
γ
<0 and uns able o
γ
>0. When
γ
=0, we ha e a
linea cen e . In he zone x>0, since a>0, he e will be no equilib ium poin s i
D=0; o he wise, hepossible equilib iumpoin will be loca ed a (−a/D,−aT/D).
We will ake
γ
in (3) as he bi u ca ion pa ame e , ha ing i s c i ical alue a
γ
=0, whe e he cen e con igu a ion akes place, see Figu e 1. Ou i s main esul
is he ollowing.
4 En ique Ponce, Ja ie Ros and El´ısabe Vela
01
Y
0011
Y
bx0
bx1
Fig. 1 (Le ) The c i ical si ua ion o
γ
=0. (Righ ) The bi u ca ing limi cycle o
γ
T<0 and |
γ
|
small.
Theo em 1. Conside sys em (3) wi h a >0and unde he assump ion T 6=0. The
linea cen e con igu a ion es ic ed o he zone x ≤0, ha exis s o
γ
=0gi es
place o a unique pe iodic oscilla ion o
γ
T<0and |
γ
|su icien ly small.
Mo e p ecisely, o T <0 he limi cycle bi u ca es o
γ
>0and i is s able, while
o T >0 he limi cycle bi u ca es o
γ
<0and i is uns able. I we deno e wi h
bx0= (0,by0)T he lowe c ossing poin o he bi u ca ing limi cycle, hen he peak- o-
peak ampli ude App in x, he pe iod P o he pe iodic oscilla ion, he cha ac e is ic
mul iplie
ρ
o he pe iodic o bi and he coo dina e by0a e analy ic unc ions a 0
in he a iable
γ
1/3. Namely, o
γ
T<0and |
γ
|su icien ly small we ha e
App =2+(3
π
)2/3
2
1+a
(T2a)1/3
γ
2/3+O(
γ
4/3),
P=2
π
+2(3
π
)1/3a−1
(a2T)1/3
γ
1/3−2
π
15
(15a2−12D+T2−a(3D+T2))
a2T
γ
+O(
γ
4/3),
ρ
=1−2(3
π
)1/3T
a2/3
γ
1/3+2(3
π
)2/3T
a4/3
γ
2/3+
+2
π
15
15a2+12D−31T2
a2
γ
+O(
γ
4/3),
and
by0=3
π
a
T1/3
γ
1/3+
π
2T
3a1/3
γ
2/3+
π
15a2+3D+T2
15aT
γ
+O(
γ
4/3).
I should be no ed ha he sign o he pa ame e Dis no ele an o he bi u ca-
ion and hus ou esul co e s he cases he ocus-an isaddle case (D>0), he case
o D=0 (pa abolic case) and he saddle case (D<0). O cou se, he uniqueness
o he bi u ca ing limi cycle is e e ed o a neighbo hood o he mos ex e nal pe-
iodic o bi o he linea cen e exis ing o
γ
=0, and so i is a local uniqueness.
The case T=0 is explici ly excluded om ou esul because hen we should ha e
The Focus-Cen e -Limi Cycle Bi u ca ion in D2PWLS2wi hou Sliding 5
a global cen e o he c i ical alue
γ
=0; he possible bi u ca ion o limi cycles
o such non-gene ic case equi es di e en echniques, as conside ed in [7].
Undoing he changes o a iables in oduced in he p oo o P oposi ion 1, i is
easy now o ge a simila esul o sys em (2) wi h b=0 and adequa e hypo heses.
Theo em 2. Conside sys em (2) unde he hypo heses T2
−−4D−<0, a−<0,
a+<0, T+6=0and assume ha in he zone x >0we ha e an in isible angency a
he o igin, ha is b =0. The linea cen e con igu a ion es ic ed o he zone x ≤0,
ha exis s o T−=0gi es place o a unique pe iodic oscilla ion o T−T+<0and
|T−|su icien ly small.
Mo e p ecisely, o T+<0 he limi cycle bi u ca es o T−>0and i is s able,
while o T+>0 he limi cycle bi u ca es o T−<0and i is uns able. I we deno e
by bx0= (0,by0)T he lowe c ossing poin o he bi u ca ing limi cycle, he peak-
o-peak ampli ude App in x, he pe iod P, he cha ac e is ic mul iplie
ρ
and by0
a e analy ic unc ions a 0in he a iable T1/3
−. Namely, o T−T+<0and |T−|
su icien ly small we ha e
App =−2a−
D−−1
23
π
22/3a++a−
D−a−
a+1/3T−
T+2/3
+O(T−)4/3,(4)
P=2
π
√D−−(12
π
)1/3a−−a+
a1/3
−a2/3
+√D−T−
T+1/3
+
+
π
15 15a2
+D−−3D+(4a2
−+a−a+)+ T2
+a−(a−−a+)
a2
+D3/2
−
T−
T+
+O(T−)4/3,(5)
ρ
=1+(12
π
)1/3
√D−T+a−
a+2/3−T−
T+1/3
+(18
π
2)1/3T2
+
D−a−
a+4/3T−
T+2/3
+
+
π
15
45a2
++a2
−(12D+−31T2
+)
a2
+D3/2
−
T−+O(T−)4/3,
and
by0=3
π
21/3a1/3
+a2/3
−
√D−−T−
T+1/3
−
π
2
121/3a4/3
−T+
a1/3
+D−T−
T+2/3
−
−
π
30
15a2
+D−+a2
−(3D++T2
+)
a+D3/2
−
T−
T+
+O(T−)4/3.
(6)
This heo em is an ex ension o he esul s ob ained in [3], whe e only he con i-
nuous case was analyzed. Consequen ly,i we pu in he abo eexp essions a+=a−
he con inuous case is eco e ed. The exp essions (4)-(7) ha e been compu ed wi h
he help o symbolic compu a ion sys ems (bo h Ma hema ica [8] and Maple [9])
and only he i s coe icien s o he se ies a e explici ly shown. Mo e e ms can be
compu ed wi h he same echniques, i equi ed.

6 En ique Ponce, Ja ie Ros and El´ısabe Vela
The es o he pape is o ganized as ollows. An applica ion o Theo em 2 o he
s udy o he dynamics o an elec onic ci cui is gi en in Sec . 2. The p oo s o main
esul s along wi h o he echnical lemmas appea inally in Sec . 3.
2 Applica ion o he dynamics o a Wien b idge oscilla o
C2
R1C1
Vo
R2
+
+
-
-
+
-
RSRF
OA
Fig. 2 Scheme o he Wien b idge oscilla o . The powe supply needed o bias he ope a ional
ampli ie is no shown.
In [10] a ealis ic model o an ope a ional ampli ie is p oposed o illus a e he
jump ansi ion o oscilla ion in he Wien b idge oscilla o . The p oblem leads o
a idealized symme ic model ha can be se led down in a con inuous ec o ield
con ex , see [3]. He e, we will assume a mo e gene al si ua ion in he sense ha
he nonlinea cha ac e is ic o he elec onic de ice is allowed o ha e some lack o
symme y. This assump ion implies ha in he bi h o oscilla ions only wo linea
zones a e in ol ed, and we can disca d one o he mos ex e nal egions. Thus we
assume ha he ci cui model is gi en by
¨ +2
Γ
˙ + =0, >1,
¨ −2
Γε
˙ + =0, ≤1,
whe e is he dimensionless inpu ol age in he ope a ional ampli ie ,
Γ
is a posi-
i e cons an pa ame e which depends on he passi e elemen s o he ci cui , and
ε
is aken as he bi u ca ion pa ame e , see [10] o mo e de ails.
The abo e equa ions, a e he ansla ion =x+1, can be w i en in he o m
˙x=y,˙y=−x−2
Γ
y−1,i x>0,
−x+2
εΓ
y−1,i x≤0.(7)
Sys em (8) is in he o mula ion (1.1)gi en in [2] and we can apply he P oposi ion
3.1 o ha pape in o de o pu sys em (8) in Li´ena d o m. The homeomo phism
The Focus-Cen e -Limi Cycle Bi u ca ion in D2PWLS2wi hou Sliding 7
ex=1 0
−2
Γ
−1xi x>0,ex=1 0
2
εΓ
−1xi x≤0,
ans o ms sys em (8) in
˙x=−2
Γ
x−y,i x>0,
2
εΓ
x−y,i x<0,˙y=x+1,(8)
whe e he ildes ha e been d opped o he sake o simplici y.
Now, we can apply Theo em 2 o sys em (9) by aking T+=−2
Γ
,T−=2
εΓ
,
D+=D−=1, a+=a−=−1 and b=0, ob aining ha he bi u ca ion akes place
o
ε
=0, he limi cycle exis s o su icien ly small
ε
>0 and i is s able.
The peak- o-peak ampli ude, he pe iod and he cha ac e is ic mul iplie o he
limi cycle a e
App =2+3
π
22/3
ε
2/3−
π
4/3(99+52
Γ
2)
40·181/3
ε
4/3+O(
ε
5/3),
P=2
π
+4(18
π
5)1/3
5
ε
5/3+O(
ε
2),
ρ
=1−2(12
π
)1/3
Γε
1/3+4(18
π
2)1/3
Γ
2
ε
2/3+2
π
15
Γ
(27−124
Γ
2)
ε
+
+4
54
π
91/3
Γ
(34
πΓ
3−27
πΓ
−5)
ε
4/3+O(
ε
5/3).
We no e ha he o de
ε
5/3 e m in he las exp essiono he pe iod Pis no explici ly
shown in Theo em 1. I has been compu ed wi h he same p ocedu e o he p oo o
Theo em 1, inc easing he numbe o e ms in (18).
We emphasize ha he abo e se ies a e e y use ul o desc ibe accu a ely all he
ea u es o he nonlinea oscilla ion o pa ame e s nea he c i ical alue co espon-
ding o he bi u ca ion.O cou se, we could ha e ob ained he same esul s applying
P oposi ion 1 and Theo em 1.
3 P oo s o main esul s
We gi e i s he p oo o P oposi ion 1.
P oo (P oo o P oposi ion 1). Clea ly, he assump ion on igh angency a he
o igin, ha is ˙x|(x,y)=(0,0)=0, educes o b=0, and his angency will be in isible
o a+<0.
Unde he hypo heses, i we de ine
ω
>0 such ha
ω
2=D−−T2
−/4 and
α
=T−/2, he eigen alues o he linea pa a S−in (2) a e
α
±i
ω
. We make
i s he change X=
ω
x,Y=y,
τ
=
ω
o he a iables in he hal plane S−, wi -
hou al e ing a iables and ime in S+. No e ha we do no change he coo dina e y,
so ha pe iodic o bi s using bo h hal planes a e p ese ed. Then, we ge
8 En ique Ponce, Ja ie Ros and El´ısabe Vela
dX
d
τ
=1
ω
dX
d =dx
d =X
ω
T−−Y,
dY
d
τ
=1
ω
dY
d =1
ω
dy
d =1
ω
D−X
ω
−a−=D−
ω
2X−a−
ω
.
In oducing he pa ame e
γ
=
α
/
ω
, we see ha T−=2
γω
and D−= (
γ
2+1)
ω
2.
Making in he whole plane a homo he yo ac o k obe de ined below and emo ing
he ac o kin he equa ions, we ha e o x<0,
dx
d
τ
=kdX
d
τ
=k2
γ
x
k−y
k=2
γ
x−y,
dy
d
τ
=kdY
d
τ
=kh(
γ
2+1)x
k−a−
ω
i= (
γ
2+1)x−ka−
ω
,
while o x>0 we ha e ˙x=Tx −y, and ˙y=Dx−ka+. By imposing now ha
k=−(
γ
2+1)
ω
/a−>0, we ge he exp essions gi en in he s a emen .
Be o e gi ing he p oo o Theo em 1, we need i s some echnical esul s.
Le us conside sys em (1). We assume ha he o bi o ec o ield F+s a ing
a a poin bx0= (0,by0)wi h F+
1(bx0)>0, lies in S+and e en ually comes back o he
discon inui y line a i ing ans e sally a he poin bx1= (0,by1), whe e F+
1(bx1)<0.
Lemma 1. I we deno e as b
δ
+( ) = (x( ,bx0),y( ,bx0)) he solu ion o ˙
x=F+(x)
sa is ying b
δ
+(0) =bx0, hen a alue
τ
+>0exis ssuch ha x( ,bx0)>0 o 0< <
τ
+
and x(
τ
+,bx0) = 0, y(
τ
+,bx0) = by1. Then we can de ine a igh Poinca ´
e map PRin a
neighbo hood o he poin bx0such ha PR(by0) = by1and he i s de i a i e o map
PRa by0is gi en by
P′
R(by0) = F+
1(bx0)
F+
1(bx1)expZ
τ
+
0di F+.
P oo . Le us conside he di e en ial sys em ˙
x=F+(x)de ined in R2, and le
us deno e as
δ
+( ) = (x( ,x0),y( ,x0)), he solu ion sa is ying
δ
+(0) = x0whe e
x0= (x0,y0). Le us in oduce he equa ion Z( ,x0,y1) = 0, whe e
Z( ,x0,y1) = x( ,x0)
y( ,x0)−y1,
and om ou hypo heses we ha e Z(
τ
+,bx0,by1) = 0and F+
1(0,by1)6=0. Then he
Jacobian ma ix
∂
(Z1,Z2)
∂
( ,y1)(
τ
+,bx0,by1) = 


∂
x
∂
∂
x
∂
y1
∂
y
∂
∂
y
∂
y1


(
τ
+,bx0,by1)
=F+
1(0,by1)0
F+
2(0,by1)−1
is nonsingula .
The Focus-Cen e -Limi Cycle Bi u ca ion in D2PWLS2wi hou Sliding 9
Then, we conclude om he implici unc ion heo em he exis ence o wo un-
c ions
ϕ
(x0),
ψ
(x0)de ined in a neighbo hood o x0, such ha
x(
ϕ
(x0),x0) = 0,y(
ϕ
(x0),x0)−
ψ
(x0) = 0,(9)
wi h
ϕ
(bx0) =
τ
+,
ψ
(bx0) = by1.
Thus, we can de ine he igh Poinca ´e map PRin a neighbo hoodo he poin x0
as y1=PR(y0) =
ψ
(0,y0), which sa is ies by1=PR(by0) =
ψ
(bx0).
To compu e he i s de i a i e o Poinca ´e map, we ake de i a i es wi h espec
o (x0,y0)in equa ions (10) o ge
∂
Z
∂
(
ϕ
(x0),x0,
ψ
(x0))·Dx
ϕ
(x0)+ B(x0)−0
Dx
ψ
(x0)=0,(10)
whe e
∂
Z
∂
(
ϕ
(x0),x0,
ψ
(x0)) = 


∂
x
∂
(
ϕ
(x0),x0)
∂
y
∂
(
ϕ
(x0),x0)

=F+(x1)
and
B(x0) = DxZ(
ϕ
(x0),x0,
ψ
(x0)) = 


∂
x
∂
x0(
ϕ
(x0),x0)
∂
x
∂
y0(
ϕ
(x0),x0)
∂
y
∂
x0(
ϕ
(x0),x0)
∂
y
∂
y0(
ϕ
(x0),x0)


sa is ies he equali y B(x0)F+(x0) = F+(x1),(11)
due o he elemen a y p ope ies o a ia ional equa ions.
By igh -mul iplying (11) by he ec o (0,1)T, we ge
ϕ
y(x0)F+(x1)+B(x0)0
1−0
ψ
y(x0)=0,
and so
B(x0)0
−1=F+
1(x1)
ϕ
y(x0)
F+
2(x1)
ϕ
y(x0)−
ψ
y(x0).(12)
Taking in o accoun (12) and (13) we can w i e
F+
1(x1)0
F+
2(x1)−11
ϕ
y(x0)
0
ψ
y(x0)=B(x0)F+
1(x0)0
F+
2(x0)−1
and aking de e minan s we a i e a he ela ion
F+
1(x1)
ψ
y(x0) = F+
1(x0)expZ
ϕ
(x0)
0di (F+)d ,(13)