TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 14, No. 1, April 2015, pp. 140 ~ 1
4
6
DOI: 10.115
9
1
/telkomni
ka.
v
14i1.746
1
140
Re
cei
v
ed
Jan
uary 4, 2015;
Re
vised Ma
rch 11, 2015; A
c
cepted Ma
rch 25, 2015
Hopf Bifurcation in Numerical Approximation for the
Generalized Lienard Equation with Finite Delay
Guangy
u
Zhao*
1
, Yanchu
n Li
2
1
School of Scie
nce, Cha
ngc
hu
n Univ
ersit
y
of Scienc
e an
d T
e
chn
o
lo
g
y
,
Cha
ngch
un, 13
002
2, Chi
n
a
2
Mathematics
Group, T
he Second Prim
ar
y
Schoo
l
Attache
d
to Northeast
Normal U
n
iv
ersit
y
,
Cha
ngC
hu
n, 1300
12, Ch
ina
*Corres
p
o
ndi
n
g
author, e-ma
i
l
:zg
y
w
s
hz
yz@
126.com
A
b
st
r
a
ct
T
he nu
merica
l
appr
oxi
m
ati
o
n
of the g
e
n
e
ral
i
z
e
d
Lie
nar
d
e
quati
on is
co
n
s
ider
ed usin
g del
ay
a
s
para
m
eter. F
i
rst, the delay
difference e
quati
on o
b
ta
in
ed by usi
ng
Euler
meth
od
is w
r
itten as a
ma
p.Accord
ing
to the theories
of bi
furcatio
n for discrete dy
n
a
mical syst
e
m
s,the conditi
on
s to guara
n
tee
th
e
existenc
e of H
opf bifurc
atio
n
for nu
mer
i
cal
appr
oxi
m
at
i
o
n
are g
i
ven. T
h
e rel
a
tions
of
Hopf b
i
furcati
o
n
betw
een th
e c
ontin
uo
us a
nd
the discr
ete ar
e disc
u
sse
d. T
hen w
h
en th
e
gen
eral
i
z
e
d
Li
enar
d e
quati
o
n
h
a
s
Hopf bifurc
atio
ns at
0
rr
, the numerical a
ppr
oxi
m
ati
on al
s
o
ha
s Hopf bifurcat
ions at
0
()
h
rr
o
h
is
proved. At las
t, the text listed an ex
am
ple of num
er
ic
al simulation, the resu
lt show
s that system (8)
discreti
z
e
d
by
Euler ke
eps th
e dyna
mic char
acterist
ic of former syste
m
(1)
,
and the theor
y is proved.
Ke
y
w
ords
: the
gener
ali
z
e
d
L
i
enar
d eq
uatio
n
,
Euler method,
Hopf bifurcati
o
n, nu
meric
a
l a
pprox
imatio
n
Copy
right
©
2015 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
In rece
nt years, the gene
ral
i
zed Li
ena
rd
equatio
n:
(
)
((
)
)
(
)
((
)
)
0
xt
f
x
t
x
t
g
xt
r
(1)
The be
havior of its solutio
n
attracte
d at
tenti
on of ma
ny schola
r
s.
Delay
s
are th
e key to
cau
s
e
differe
nce
s
betwee
n
del
ay differential e
quatio
n an
d
ordi
nary differential
equatio
n, so
use
delays a
s
pa
rameter to stu
d
y Hopf bifurcation
i
s
mea
n
ingful. Many
sch
olars hav
e done in
-de
p
th
resea
r
ch ab
o
u
t the Ho
pf bi
furcatio
n of system (1
)
13
. For exampl
e, in
1998, referen
c
e [1] u
s
e
s
delay
r
as parameter studi
ed
Ho
pf
bifu
rcatio
n of sy
stem (1), pr
o
v
ed the exist
ence of Ho
pf
bifurcation a
nd form
ula t
o
co
unt Ho
pf bi
furcation
wa
s given.
Referen
c
e
[2] use
s
-
D
partitionin
g
m
e
thod
of in
de
x polynomi
a
l
to disc
u
s
s th
e Hop
bifurca
t
ion of
syste
m
(1
)
usi
n
g
k
as
a param
eter.
Refere
nce [3] discu
sse
s
H
opf bifurcati
on of system
(1) usi
ng
b
as a para
m
ete
r
,
and give
s the
Hopf bifurcat
ion diag
ram i
n
the
rb
param
eter plan
e.
This text di
scussed
the
Ho
pf bifurcation
in
num
eri
c
al
approximatio
n of the
sy
ste
m
(1
) by
cho
o
si
ng r a
s
the bifurcatio
n paramete
r
, usin
g t
he Eul
e
r metho
d
. T
he refe
ren
c
e
4 to 7 took th
e
lead in study
ing the Hopf
bifurcatio
n in nume
r
ic
al approximatio
n of the
finite delay Logi
stic
equatio
n and
got satisfied
result
s. Wha
t
is call
ed th
e numeri
c
al
approximatio
n is to examining
wheth
e
r it
s n
u
meri
cal
sol
u
tion ca
n mai
n
tain the dy
n
a
m
ic cha
r
a
c
teristic
of
th
e system while using
the nume
r
ical
method to achieve
the discretizatio
n of system.
2. The Existe
nce of
Hopf
Bifurc
ation
for the Ge
ner
a
lized Lienar
d Equation
A
s
t
o
sy
st
e
m
(1),
set
d
e
lay
0
r
as
co
nstant,
2
,,
f
gC
and
()
gx
sat
i
sf
y
i
n
g
(0
)
0
,
(
)
0
.
gx
g
x
Set
(0
)
,
(
0
)
,
f
ag
b
and
0,
0
.
ab
System (1) i
s
equivalent to
the followi
n
g
se
con
d
-o
rd
er-finite-d
e
lay system.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Hopf Bifurcati
on in Num
e
ri
cal App
r
o
x
im
ation for the
Gene
rali
zed
Liena
rd…
(G
uang
yu Zh
ao
)
141
()
(
)
()
(
(
)
)
()
(
(
)
)
xt
y
t
yt
f
x
t
y
t
g
x
t
r
，
，
(2)
Let
x
y
，
then d
o
the time conve
r
si
on
tr
s
，
and still n
o
te
()
,
(
)
x
rs
y
r
s
as
()
,
(
)
x
ty
t
, therefore Eq
uation (2
) can
be tran
sf
orm
ed into its eq
uivalent syste
m
.
()
,
()
(
(
)
)
()
()
(
(
1
)
)
,
xt
r
y
t
yt
r
f
x
t
yt
yt
r
g
x
t
(3)
Its linear pa
rt is:
()
,
()
(
1
)
(
)
,
xt
r
y
t
y
ta
r
x
t
b
r
y
t
(4)
The ch
aracte
ristic e
quatio
n of (4) Is:
22
0
ar
b
r
e
(5)
Lemma 1:
Set
r
a
s
a
para
m
eter,
so when
0
rr
, Equation
(3
)
exists
Ho
pf
bifurcation
，
a
nd
0
r
sat
i
sf
ie
s f
o
llowin
g
co
nd
it
ions:
1
0
0
0
1
2
22
2
0
1
si
n
(
)
,
1
4,
2
a
r
b
ab
a
(6)
a) Equatio
n (5) ha
s a p
a
i
r
of co
njugat
e com
p
lex ro
ots
1,
2
()
()
ri
r
，
and the
,
here a
r
e real
numbe
rs, whi
l
e
00
0
()
0
,
()
0
rr
.
b) Th
e roots of equ
ation
(5) in
0
rr
all
have st
rictly
negative
real
part
s
, exce
pt
00
()
,
(
)
rr
.
c)
0
Re
(
)
0
rr
dr
dr
.
3. Hopf
Bifur
cation in Nu
merical Appr
oxim
ation fo
r the Gen
e
ra
lized Lienard
Equation
Usi
ng the
[4
]
Eu
l
e
r
M
et
h
o
d
1
,)
hm
Z
m
（
, we get the num
erical sol
u
tion
of Equation (3).
1
1
nn
n
nn
n
m
n
xx
r
h
y
y
y
br
hx
ar
hy
(7)
Introdu
cing n
e
w vecto
r
11
(,
,
,
,
,
)
T
nn
n
n
n
n
m
n
m
Xx
y
x
y
x
y
，
we can exp
r
e
s
s (7
) as:
1
(,
)
nn
X
FX
r
(8)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 14, No. 1, April 2015 : 140 – 14
6
142
The
01
()
(
,
,
,
)
T
m
Fx
F
F
F
is a vector-val
ued fun
c
tion with
2(
1
)
m
dimensi
o
n
s
, i.e.
0
1
nn
nn
m
n
k
n
n
xr
h
y
k
y
b
rhx
a
rh
y
F
x
km
y
Expand the Equation (8) at
00
（，
）
，
1
(,
)
(
,
,
)
n
n
nn
nn
n
XA
X
B
X
X
C
X
X
X
(9)
Its linear pa
rt is :
1
nn
X
AX
(10)
In which,
00
00
0
00
0
00
0
AB
I
A
I
I
I
is a se
con
d
orde
r unit ma
trix,
10
0
,
01
0
rh
AB
ar
h
b
r
h
The ch
aracte
ristic e
quatio
n of
A
is
:
22
2
2
2
(,
,
)
(
1
)
(
1
)
0
mm
m
m
d
z
rh
z
z
a
r
h
z
z
b
rh
z
(11)
In order to f
a
cilitate th
e
discu
ssi
on
a
bout
the
bifu
rcatio
n
pro
b
l
e
m of th
e
n
u
meri
cal
solutio
n
in Equation (3
), we
introdu
ce eq
uation:
22
2
2
2
(,
,
)
(
)
(
)
0
Dr
h
e
g
h
a
r
e
g
h
b
r
e
(12)
In which
x
e
x
g
x
1
)
(
, pro
v
iding
1
)
0
(
g
J
u
s
t
lik
e the lemma 4.1 in literature [8],
we can get lemma 2.
Lemma 2:
if cha
r
a
c
teri
stic (5) satisfie
s con
d
ition (6
), then
(,
,
)
0
Dr
h
sat
i
sf
ie
s
:
a)
(,
,
)
0
Dr
h
has a pair
of conju
gate
compl
e
x root
s
1,
2
()
()
ri
r
；
b) The
r
e exist
s
0
()
h
rr
o
h
，
()
0
,
(
)
0
hh
rr
；
c)
()
0
h
rr
dr
dr
；
d) Th
ere exist
0
(nothing
to d
o
with
r, h) t
o
ma
ke for
N
m
m
h
,
1
. There exist
s
0
(,
)
(
,
0
)
rh
N
r
and
(,
)
(
,
)
(,
,
)
0
Re
rh
i
r
h
Dr
h
.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Hopf Bifurcati
on in Num
e
ri
cal App
r
o
x
im
ation for the
Gene
rali
zed
Liena
rd…
(G
uang
yu Zh
ao
)
143
Proof
：
(a
-
c
)B
eca
u
s
e
(,
,
0
)
(
,
)
Dr
d
r
, s
o
00
(,
,
0
)
(
,
)
Di
r
d
i
r
.In
00
(,
,
0
)
ir
,
00
0
00
((
)
,
)
()
((
)
,
)
r
dr
r
r
dr
r
, therefore
00
(,
)
0
di
r
. By the implicit function t
heorem, in the
neigh
borhoo
d
of
0
(,
0
)
r
, there
exists o
n
ly one functio
n
(,
)
,
(,
)
rh
r
h
making
1,
2
()
()
ri
r
. Because
00
(,
0
)
0
,
(,
0
)
0
rr
, there exists
h
rr
making
0
()
0
,
(
)
,
(
)
0
hh
h
rr
r
o
h
r
. By the implicit fun
c
t
i
on theo
rem
again, in
the
neigh
borhoo
d
of
0
(,
0
)
r
,
()
0
h
rr
dr
dr
. If
(,
,
)
0
Dr
h
,then
(,
,
)
0
Dr
h
, s
o
there exist
s
a neig
hbo
rho
od of
0
r
, mak
i
ng
(,
)
0
dr
ha
s only o
ne root
1
()
r
, s
a
tis
f
ying to
0
r
,there
is
11
Re
(
(
)
)
,
I
m
(
(
)
)
0
,
rr
and
(,
,
)
0
Dr
h
also h
a
s simil
a
r
cha
r
act
e
r.
Set
,,
mm
m
rh
to mak
e
0
(,
,
)
0
,
(
,
)
(
,
0
)
,
l
i
m
0
mm
m
m
m
m
m
Dr
h
r
h
N
r
h
, s
o
m
is unifo
rmly boun
ded. So
there exi
s
ts
j
m
, to mak
e
00
,,
0
jj
j
mm
m
rr
h
. By the
contin
uity of
00
(,
,
0
)
0
Dr
, there
exist
s
00
0
,
h
ir
r
. So:
(,
)
(
,
)
(,
,
)
0
Re
rh
i
r
h
Dr
h
Lemma 3:
W
h
e
n
1
h
m
, the ne
ce
ssary
and
sufficie
n
t con
d
ition of
(,
,
)
0
Dr
h
has
the root
is (11) ha
s the ro
ot
m
Z
e
.
Proof
：
Subs
titute
m
e
for Z in (11).
22
2
2
2
()
()
0
eg
h
a
r
e
g
h
b
r
e
So the lemma 3 is prove
d
.
Lemma 4:
0
h
rr
dz
dr
Proof
：
m
Z
e
，
1
h
m
，
2
zz
z
，
so there exist
s
:
2
()
(,
)
2
hh
h
h
h
dz
dz
dz
d
d
d
r
h
z
z
he
e
h
e
e
he
dr
dr
dr
dr
dr
dr
,
Bec
a
us
e
(,
)
0
h
rr
dr
h
dr
，
s
o
0
h
rr
dz
dr
.
Theorem 1:
If differential
Equation
(3
) ha
s
Hopf
bifurcation i
n
0
rr
,
so wh
en st
e
p
siz
e
h
is sufficiently sm
all, differential Equati
on
(8
)will
p
r
o
duce
Hopf
bifurcation
in
0
()
h
rr
o
h
.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 14, No. 1, April 2015 : 140 – 14
6
144
Proof
：
We ca
n learn by le
mma 3 and
4 that to the
step si
ze
1
h
m
0
()
mm
, in
the
neigh
borhoo
d
of
0
r
, if charact
e
risti
c
equ
ation (5) h
a
s ro
ot,
m
Z
e
is the root of (11). If (5) have
a pair
of sim
p
le conju
gate
com
p
lex ro
o
t
s
0
i
, while
oth
e
r roots
have
stri
ctly real
parts.
So the differe
ntial Equation
(8)
have a
p
a
ir of
conju
g
a
te com
p
lex
roots
h
i
m
e
in
0
()
h
rr
o
h
1
()
h
m
, and
1
h
i
m
e
, while other roots’ m
odule
s
le
ss th
an 1, and
0
h
rr
dz
dr
.
4. Numerical
Simulation
This sectio
n gives an exa
m
ple of nume
r
ical
si
mulat
i
o
n
of
sy
st
em (
1
).
The re
sult
sho
w
s
that system
(8) di
screti
zed
by Euler
ke
eps th
e
dy
na
mic c
h
a
r
a
c
t
e
r
i
st
ic of
f
o
r
m
e
r
sy
st
e
m
(1
),
and the theo
ry is proved.
Set
1
(0
)
0
.
8
,
(
0
)
1
.
fa
g
b
and the system turne
d
into:
()
()
()
0
.
8
(
)
(
)
xt
y
t
y
ty
t
x
t
r
，
，
(
1
3
)
System (13) exists only equilibrium point
*
(0
,
0
)
.
E
Acco
rdi
ng to the theorem 4
.
1 of referen
c
e [3], it’s easy to get:
0
0. 378 316 029 857 13,
r
So system (1
3) gen
erates
Hopf bifurcati
on at
0
rr
.
Diag
ram
1 t
o
3 exp
r
e
ss wavefo
rm
s
and traje
c
tory diagram of
sol
u
tion
system (1
3)
before di
scre
tized. Dia
g
ra
m 4 to 6 expre
ss
wave
fo
rms a
nd traj
ectory dia
g
ra
m of system
(8)
discreti
zed
b
y
Euler. Th
e
diag
ram
1
sho
w
s that
whe
n
0
rr
, z
e
ro
s
o
lution of
sys
tem is
asymptoticall
y
stabled. The
diag
ram
2 sho
w
s that when
0
rr
, system experie
n
c
e
s
Ho
pf
bifurcation at
origin, and
stable bifurcati
ng
periodi
c solution
was produ
ced around equilibri
um
point. The di
agra
m
3
sho
w
s th
at wh
en
0
rr
, zero solutio
n
of syste
m
i
s
un
stabl
e. T
he dia
g
ra
m
4 to 6 sho
w
s that when
0
rr
, zero
solution
o
f
system (8
) i
s
asym
ptotically stabled,
and sta
b
le
perio
dic
sol
u
tion wa
s p
r
o
duced a
r
ou
n
d
0
rr
. When
0
rr
,
zero solut
i
o
n
of
sy
st
em (8
)
is
unsta
ble, whi
c
h me
an
s sy
stem (8) di
scretize
d
by
Euler
kee
p
s t
he dynami
c
cha
r
a
c
teri
stic of
former sy
ste
m
(1).
Figure 1. Wa
veform and p
hase orbit of system
（）
13
whe
n
0
0.2
rr
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TELKOM
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ISSN:
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Hopf Bifurcati
on in Num
e
ri
cal App
r
o
x
im
ation for the
Gene
rali
zed
Liena
rd…
(G
uang
yu Zh
ao
)
145
Figure 2. Wa
veform and p
hase orbit of system
（）
13
whe
n
0
rr
Figure 3. Wa
veform and p
hase orbit of system (13
)
whe
n
0
0.5
5
rr
Figure 4. Wa
veform and p
hase orbi
t of discrete
syste
m
(8)
when
0
0.2
,
0.02
rr
h
Figure 5. Wa
veform and p
hase orbi
t of discrete
syste
m
(8)
when
0
,0
.
0
2
rr
h
Figure 6. Wa
veform and p
hase orbi
t of discrete
syste
m
(8)
when
0
0.55
,
0
.02
rr
h
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ISSN: 23
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TELKOM
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KA
Vol. 14, No. 1, April 2015 : 140 – 14
6
146
Referen
ces
[1]
F
engj
un T
ang,
Z
hen
xu
n H
u
a
ng, Jio
ng
Ru
a
n
. Hopf B
i
furc
ation
of T
he g
ener
aliz
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Lie
nard E
q
u
a
tio
n
w
i
t
h
delay
as parameter.
Ch
in
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ls of
Mathe
m
atics
. 1
998; 19A(
4
): 4
69-4
76.
[2]
Suqi Ma, Qish
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opf Bifurcatio
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a
tio
n
w
i
t
h
De
la
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Journ
a
l of Ch
in
a Agricu
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a
l
Univers
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ty
. 200
3; 8(4): 1-4.
[3]
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ang. H
opf Bifurcati
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T
he gen
er
alize
d
L
i
en
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w
i
t
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F
i
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De
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de
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T
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[4]
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[5]
Kazari
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a
n
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B
i
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a
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ab
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Peri
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Sol
u
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of
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a
l-D
i
fferenc
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nd In
tegro-Differ
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a
l Eq
uati
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Journ
a
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Institute of
Mathe
m
atic
a
l
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iatio
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78; 21: 46
1-46
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[6]
Hale
j, Lun
el S
V
. Introduction
to F
unctiona
l D
i
fferentia
l Equ
a
t
ions. Ne
w
Yor
k
: Spring-Ver
l
a
g
. 1993.
[7]
Guckenh
eimer
J, Ho
lmes
P
J. No
Lin
ear
Oscilla
ti
ons,
D
y
n
a
mical
S
y
stem
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d B
i
furcati
on
of Vect
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r
F
i
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w
Y
o
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3.
[8] Neville
Ford,
Volker
W
u
lf. Numeric
a
l H
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for a
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D
i
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JCAM
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15: 601-
61
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