TELKOM
NIKA
, Vol. 11, No. 8, August 2013, pp. 45
3
0
~4
538
e-ISSN: 2087
-278X
4530
Re
cei
v
ed Ma
rch 5, 2
013;
Re
vised
Ma
y 15, 2013; Accepted Ma
y 27
, 2013
High-Dimensional Chaotic System and its Circuit
Simulation
Jianming Liu
Heb
e
i Ke
y L
a
b
of Industrial C
o
mput
er C
ontr
o
l Eng
i
ne
eri
ng,
Yansha
n Un
iv
ersit
y
, Qin
hua
n
gda
o, Chi
n
a
e-mail: ppkkkk
@126.com
A
b
st
r
a
ct
T
he ch
aotic
s
ystem pl
ays
a
n
i
m
portant
ro
le
in
el
ectroni
c
an
d electric
al eq
uip
m
ent, compute
r
cryptography, computer comm
unic
ation and so on. In
this paper, we established
three new six-dim
e
nsional
complex chaot
ic system
s and one new ten-dim
e
nsi
onal com
p
lex chaotic system
. The regularity
to
generate
high-
d
im
ensional c
h
aotic system
is
also found by
overlaying a
s
e
ries
of low-dim
e
nsional chaotic
system
s with Duffing chaotic system
. The circ
iuts of
the new
high-dimensional com
p
lex chaotic system
ar
e
desi
gne
d. T
h
e
simulati
on
ex
peri
m
e
n
ts of the h
i
gh-
di
me
n
s
ion
a
l co
mplex
chaotic c
i
rcuit
s
are tested.
T
h
e
results
of theoretical analysis
an
d ex
per
im
ent show that new high-d
imensional c
o
m
p
lex
chaotic systems
and the
i
r circiut
s
have the hy
p
e
rcha
otic char
acteristics.
Ke
y
w
ords
:
e
lectrical e
qui
p
m
ent, cryptogra
p
h
y, c
haotic circ
iut, high-
di
me
n
s
ion
a
l cha
o
s
Copy
right
©
2013 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
The cha
o
tic system,
the
quantum
me
cha
n
ics
a
nd
the theo
ry of
relativity are thre
e
importa
nt sci
entific di
scov
erie
s in
the
20th
cent
u
r
y
[
1
]
.
The
cha
o
t
i
c
sy
st
em i
s
wide
sp
rea
d
in
electri
c
al
and
elect
r
oni
c e
quipme
n
ts [2]
,
commu
nication [3], astro
physi
cs, a
e
ro
spa
c
e,
weath
e
r
forecast, com
puter
cryptog
r
aphy [4] and
powe
r
net
wo
rk [5]. To any
chaoti
c
en
cryption syste
m
,
the highe
r di
mensi
on it ha
s, the better
secu
rity
it has
[6]. By adding the st
ate fee
dba
ck
co
ntrol
l
er
on thre
e-dim
ensi
onal
cha
o
tic syste
m
, some fi
ve-di
m
ensi
onal
ch
aotic sy
stem
s are gen
era
t
ed.
For exa
m
ple:
in 200
9, Hu
aqing
Li ad
d
ed state
fe
e
dba
ck
co
ntrol
l
er on
the th
ree
-
dime
nsi
o
nal
Lore
n
z
cha
o
tic system to g
enerate a five-dim
en
siona
l Loren
z ch
ao
tic system [7]. In 2010, Feng
Han
ad
ded
st
ate feed
ba
ck
controlle
r o
n
the three-
dim
ensi
onal
Lu
chaotic sy
ste
m
to g
ene
rat
e
a
five-dimen
sio
nal
L
u
cha
o
tic system
[8]. In
201
1,
Lu
Hua
ng
add
ed
state
feed
ba
cks
co
ntroll
er on
the three
-
dim
ensi
onal
Che
n
ch
aotic
syst
em to
gen
era
t
e a five-dime
n
sio
nal Chen
cha
o
tic
syste
m
[9]. In this p
aper,
we
will
study
how to ge
nerate a
six-di
men
s
io
nal
chaoti
c
system, a ten
-
dimen
s
ion
a
l
cha
o
tic
syste
m
and a
cha
o
tic sim
u
latio
n
circuit. The
n
, we will
exp
l
ore the
reg
u
l
a
rity
to gen
erate
high-dimen
s
i
onal
ch
aotic syste
m
. Th
e
re
sult of
this study wi
ll
ha
s
p
r
a
c
tical
signifi
can
c
e i
n
electri
c
al a
nd ele
c
troni
c
circ
uits,
chaot
ic cryptograp
hy and com
m
unication.
2. Ne
w
Duffing-Lor
en
z Chaotic Sy
stem
2.1. The Desi
gn of Du
ffin
g
-Lor
enz
Ch
aotic Sy
stem
Duffing
ch
aot
ic
system
is o
ne of th
e
co
mmonl
y u
s
e
d
model
s i
n
si
gnal tran
smi
s
sion
and
freque
ncy tra
n
sform field [10]. The form
of the Duffing cha
o
tic sy
stem is:
wt
e
x
dy
y
y
x
cos
3
(1)
D a
nd e
a
r
e
real
co
nsta
nts. The
cha
r
a
c
teri
st
ic
prop
erties of the
Lore
n
z sy
ste
m
as the
first di
scovered p
h
ysi
c
al
chaotic
sy
ste
m
are
fo
und
in
mo
re and
more
field
s
. The
fo
rm of the
three
-
dime
nsi
on Loren
z ch
aotic sy
stem i
s
as the follo
wing:
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TELKOM
NIKA
e-ISSN:
2087
-278X
High
-Dim
en
si
onal Chaoti
c
System
and it
s Circuit Sim
u
lation (Jian
m
ing Liu)
4531
cz
xy
z
y
xz
bx
y
x
y
a
x
(2)
The pa
ramet
e
rs of a~c a
r
e re
al con
s
t
ants.
The external ex
citation part of the origin
al
Duffing
syste
m
is repla
c
e
d
by an aut
onomo
u
s
part. By the bridge of the a
u
tonomo
u
s
p
a
rt,
Equation
(1
) and
Equ
a
tion
(2) a
r
e
combine
d
into
a
ne
w
six-dimen
s
ion
a
l
compl
e
x
cha
o
tic
sy
st
em.
fxv
w
w
e
u
dv
v
v
u
cz
xy
z
w
y
xz
bx
y
gv
x
y
a
x
)
cos(
)
(
3
(3)
The pa
ramet
e
rs of a
~
g are real con
s
ta
nts.
2.2. L
y
apuno
v
Exponent Analy
s
is
Whe
n
the initi
a
l co
ndition
s
are a
=
1
0
, b=55, c=8/
3, d
=
0.6, e=-3, f=1
,
x=1, y=1, z=1, w=1
and dt
=0.00
5
,
the Lyapu
n
o
v expone
nts are
1.399,
0
.
852, 0.039,
-0.097,
-1.45
3
and
-1
4.97
7.
Since the
r
e a
r
e thre
e po
sitive Lyapunov
expone
nts,
the system i
s
in
the hypercha
o
tic state [11]
.
3.
Ne
w
Duffing-Chen Ch
aotic
Sy
stem
3.1. The Desi
gn of Du
ffin
g
-Chen Cha
o
tic Sy
stem
The form of Chen chaoti
c
system is:
bz
xy
z
cy
xz
x
a
c
y
x
y
a
x
)
(
)
(
(4)
Equation
(1
)
and E
quation
(4
)
are
ove
r
laid
into
a
n
e
w
six-dimen
s
ion
a
l
Duffin
g
-Chen
compl
e
x hyperchaoti
c
syst
em:
fxv
w
w
e
u
dv
v
v
u
cz
xy
z
w
hy
xz
x
a
b
y
gv
x
y
a
x
)
cos(
)
(
)
(
3
(5)
The pa
ramet
e
rs of a
~
h a
r
e real con
s
ta
nts.
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TELKOM
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Vol. 11, No
. 8, August 2013: 4530 –
4538
4532
3.2. L
y
apuno
v
Exponent Analy
s
is
Whe
n
the pa
rameters are
a=1
0
,
b
=
55, c=8/3,
d
=
0.6,
e=-3, f=1, g
=
3, h=1, x=1, y
=
1,
z=1
,
u=1, v
=
1,
w=1 an
d dt
=0.0
05, t
he
Lyap
unov exp
one
nts a
r
e
0.97
0
,
3.108,
-1.46
8
, -0.4
61, -3.574
and
-10.8
29.
Since
the
r
e are two
positive
Lya
punov
expo
nents, th
e
system is in
th
e
hypercha
o
tic state.
4. The Exploration of
Re
gularit
y
By the an
alysis of the
ab
ove mentio
ne
d tw
o
ki
nd
s o
f
the ne
w
six-dimen
s
ion
a
l
compl
e
x
hypercha
o
tic system
s, the regul
arity to gene
rate
the high
-dim
e
n
sio
nal comp
lex hypercha
o
tic
system
s is di
scovere
d
. The exte
rnal excitation pa
rt of the Du
ffing system is
repla
c
e
d
by an
autonom
ou
s part. The p
o
sitive feedback is ad
ded. T
hen, by the b
r
idge of the a
u
tonomo
u
s p
a
rt,
the Duffing chaotic
syste
m
and the lo
wer-dim
e
n
s
io
nal ch
aotic
system are
co
mbined into a
new
six
-
dim
e
n
s
ion
a
l compl
e
x
hy
per
cha
o
t
i
c
sy
st
em.
Th
e new ove
r
l
a
ying reg
u
la
rity of comp
lex
hypercha
o
tic
system i
s
sh
o
w
n in Figu
re
1.
2
:
3
:
2
1
)
(
:
1
Feedback
part
system
chaotic
Duffing
part
Feedback
system
chaotic
Another
or
Lorenz
part
Figure 1. The
Overlaying Regula
r
ity
The ab
ove proce
s
s can be
sub
d
ivided i
n
to three
ste
p
s: (1
) Th
e p
a
ram
e
ters
of positive
feedba
ck an
d the form of
feedba
ck
are asce
rtai
ne
d; (2) T
he a
u
tonomo
u
s
p
a
rt of the Du
ffing
cha
o
tic syst
e
m
is ascertai
ned; (3) T
h
e
two c
haoti
c
systems a
r
e
combin
ed in
to a new hig
h
-
dimen
s
ion
a
l complex hype
rcha
otic sy
ste
m
.
5. The Verif
y
of Regula
r
ity
and its Circuit Simulation
5.1. The Desi
gn of Du
ffin
g
-Lu Comple
x Chao
tic Sy
stem
Lu ch
aotic
system is a
s
th
e followin
g
:
dz
xy
z
cy
xz
y
x
y
a
x
)
(
(6)
Be based o
n
the reg
u
larit
y
to generate
high-
dimen
s
i
onal compl
e
x
cha
o
tic syst
em,
the
Duffing ch
aot
ic system
of Equation
(1
) and
th
e Lu
chaotic sy
ste
m
of Eq
uatio
n (6) are
com
b
ined
into a new
six-dime
nsi
onal
Duffi
ng-Lu co
mplex hyperchaotic
system
.
gyz
w
w
f
u
ev
v
v
u
dz
x
z
cy
xz
y
bw
x
y
a
x
)
cos(
)
(
3
2
(7)
The pa
ramet
e
rs of a
~
g are real con
s
ta
nts.
5.2. L
y
apuno
v
Exponent Analy
s
is
When the ini
t
ial conditio
n
s
are
a=3
6
, b=1, c=2
0
, d=3, e=0.6, f=3, g=
1, x=1, y=1, z=1,
u=1, v=1,
w=1 and dt
=0.0
05, the
Lyap
unov expon
e
n
t are 1.22,
0.16,
-0.38, -0.54, -1.17 a
nd -
Evaluation Warning : The document was created with Spire.PDF for Python.
TEL
K
18.8
5
hype
r
5.3.
T
K
OM
NIKA
5
. Since
the
r
r
cha
o
t
i
c st
at
e
T
he Circ
uit
S
Be base
d
SC2 is
o
p
Figure
The outp
1
V/
V 0
V
Y
X
V
=0
.
3
*
c
o
s
V(1
)
V(2
)
V(3
)
V(4
)
I(
V
5
I(
V
6
High
-Di
m
r
e are two
p
e
.
S
imulation
d
on Multisi
m
p
po
sition cir
c
(a)
3. The Inter
n
ut
s o
f
the si
m
1
V/V 0
V
Y
X
s
(
1
0*
V
(
1)
+
p
i
)
)
)
)
)
5
)
6
)
e-I
m
en
si
onal
C
p
ositive Lya
p
m
7, the sim
u
Figure 2.
c
uit. SC1 is
p
n
al Co
n
necti
o
m
ulation circ
u
1
V/V
Y
X
1
V/V
Y
X
16
6.
7k¦
¸
1k
¦
¸
10
0k
¦
¸
1
V
Y
X
SSN: 2087
-
2
C
haoti
c
Syst
e
p
unov expon
u
l
a
tion
circui
t
The
Simula
t
p
rop
o
rti
on a
m
o
n Pictures
o
u
it ar
e
s
h
ow
n
2.
8k
¦
¸
2.
8k
¦
¸
10
0k¦
¸
10
k¦
¸
5k
¦
¸
0 V
10
k¦
¸
R1
7
33
.
3
k
¦
¸
0 V
R
10
0
10k
¦
¸
V
/V
0
V
2
78
X
e
m
and its C
i
ent
s,
t
h
e sy
t
of Equatio
n
t
ion Circuit
m
plification
c
o
f the Sub-c
i
n
in Figure
4
X1
SC1
IO
1
IO
1
IO
X3
SC
1
IO
1
IO
1
X
5
S
C
1
IO1
IO1
X
7
S
C
1
IO1
IO1
R
24
0
k¦
¸
X
9
S
C
1
IO1
IO1
X
1
S
C
1
IO1
IO1
i
rc
uit Simula
t
st
em of Eq
u
n
(7
) is s
h
ow
n
c
ir
cuit
.
T
h
e
ci
r
(b)
i
rcuit.(a)S
C
1
4
.
IO
1
IO
1
IO
1
IO
1
I
O
IO
1
O
2
IO2
IO
2
IO
2
1
IO
2
IO
2
1
IO
2
IO
2
IO
1
1
IO
2
IO
2
I
O
1
1
IO
2
IO
2
I
O
t
ion (Jianm
i
n
u
ation (7) i
s
n
in Figure 2
r
cuit
pict
u
r
e
s
.(b)S
C
2
X2
SC2
IO2
IO2
X4
SC2
IO2
IO2
X6
SC2
O
1I
O
2
IO2
X8
SC
2
IO
1
I
O
2
IO
2
X10
SC
2
IO1
O
1
IO
2
IO
2
X12
SC
2
IO1
O
1
IO
2
IO
2
n
g Liu)
4533
in the
.
s
are:
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TELKOM
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Vol. 11, No
. 8, August 2013: 4530 –
4538
4534
(a)
(b)
(c)
(d)
(e)
Figure 4. The
Outputs of the Simulation
Circ
uit (a
)x
-y
, (b)x
-
z
,(
c)x
-
u, (d)x
-v
,(e
)
x
-
w
6. Anoth
e
r Verif
y
of Regularit
y
and its Circuit Simulation
6.1. The Desi
gn of Du
ffin
g
-Lor
enz-Sp
rott J
Compl
ex Cha
o
tic S
y
stem
The form of the three
-
dim
ensi
onal Sp
ro
tt J chaotic
system is:
2
y
y
x
z
z
by
y
az
x
＝
＝
＝
。
。
。
(8)
The pa
ramet
e
rs a a
nd b a
r
e re
al co
nsta
nts.
Be ba
sed
on
the me
ntione
d regula
r
ity of
the
high
-dim
ensi
onal
com
p
lex chaoti
c
system,
the extern
al
excitation p
a
rt of the ori
g
i
nal Du
ffing
system is
re
pl
ace
d
by a
n
a
u
tonomo
u
s p
a
rt.
The p
o
sitive
feedba
ck is
adde
d. By the bri
dge
of the auto
nom
o
u
s
part i
n
th
e Duffing ch
aotic
system, Equ
a
t
ion (2
), Equa
tion (1
) an
d
Equati
on
(8)
are
com
b
ine
d
into a n
e
w t
en-di
men
s
ion
a
l
Duffing-Lo
ren
z
-Sp
r
ott J co
mp
lex hyperchaotic
system
:
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TELKOM
NIKA
e-ISSN:
2087
-278X
High
-Dim
en
si
onal Chaoti
c
System
and it
s Circuit Sim
u
lation (Jian
m
ing Liu)
4535
p
s
pr
s
q
q
p
r
kpr
r
jq
q
hr
p
gx
w
w
f
u
ev
v
v
u
bz
xy
z
y
xz
cx
y
w
dzs
x
y
a
x
＝
＝
＝
＝
＝
＝
＝
＝
＝
＝
。
。
。
。
。
。
。
。
。
。
2
3
cos
(9)
The pa
ramet
e
rs a
~
j a
r
e re
al con
s
tant
s.
6.2. L
y
apuno
v
Exponent Analy
s
is
Whe
n
the init
ial co
ndition
s are a
=
1
0
, b
=
8/3, c=2
8
, d
=
-2.5, e
=
0.6,
f=-8, g
=
28, h
=
2, j=2,
k=-2, x=1, y=1, z=1, u=1, v=
1, w=1, p=1, q=2,
r=1, s=1 and dt=0
.005,
the Lyapunov expon
ent
are 0.622, 0.366,
0.1
16,
0
.
033
，
-0.3
64,
-1.02
6
, -0.92
6
, -1.84
6
, -2.727
and
-1
1.107. Sin
c
e th
ere
are fou
r
po
sitive Lyapunov
expone
nts, the
system i
s
in
the hypercha
o
tic state.
6.3. The Circ
uit Simulation
Becau
s
e
of the limit of si
mulation exp
e
rime
n
t, the output of the
simulatio
n
circuit i
s
diminished 1
0
times. The
circuit of SC2 is oppo
si
tio
n
circuit. The
circuit of SC1 is prop
ortio
n
amplificatio
n circuit
an
d
integral circuit. The
ci
rcuits o
f
SC2 and S
C
1 a
r
e
sho
w
n in Figu
re 5
(
a)-
(b).
(a)
(b)
Figure 5. The
Internal Co
n
nectio
n
Pict
ure of the Sub-circuit. (a
)SC1,(b)S
C
2
W i
s
dimi
nish
ed 2
0
time
s.
P, q and
r
are en
han
ced
2 times. B
a
sed o
n
the
Mu
ltisim 7,
the simulatio
n
circuit of Equation (9
) is
shown in Figu
re 6.
50k
Ω
33.
33n
F
100k
Ω
IO1
IO
2
100k
Ω
100
k
Ω
IO1
IO2
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e-ISSN: 2
087-278X
TELKOM
NIKA
Vol. 11, No
. 8, August 2013: 4530 –
4538
4536
Figure 6. The
Simulation Circuit
sc2
IO1
IO
2
sc2
IO1
IO
2
sc2
IO1
IO
2
sc2
IO1
IO
2
sc2
IO1
IO
2
sc2
IO1
IO
2
sc1
IO1
IO
1
IO2
IO2
sc1
IO1
IO
1
IO2
IO2
sc1
IO1
IO
1
IO2
IO2
sc1
IO1
IO
1
IO2
IO2
sc1
IO1
IO
1
IO2
IO2
sc1
IO1
IO
1
IO2
IO2
50
k
Ω
10
k
Ω
80
k
Ω
10k
Ω
Y
X
13
k
Ω
1
0k¦
¸
Y
X
100
k¦
¸
V
=
co
s
(
20
*
V
(1))
V(1
)
V(2
)
V(3
)
V(4
)
I(V5)
I(V6)
7.
14
k¦
¸
Y
X
Y
X
10
0k
¦
¸
100
k
Ω
1
00k
Ω
1
0k¦
¸
37
.
5
k
Ω
Y
X
sc1
IO1
IO
1
IO2
IO2
sc1
IO1
IO
1
IO2
IO2
sc1
IO1
IO
1
IO2
IO2
sc1
IO1
IO
1
IO2
IO2
sc2
IO1
IO
2
sc2
IO1
IO
2
sc2
IO1
IO
2
sc2
IO1
IO
2
5
0k¦
¸
1
00k
¦
¸
1
00k
Ω
50
k
Ω
1
00k
¦
¸
2
00k
Ω
1
00k
¦
¸
10
0k
¦
¸
2
00k
Ω
1
00k
¦
¸
Y
X
Y
X
12
.
5
k¦
¸
x
y
z
u
v
w
p
q
r
s
Y
X
Evaluation Warning : The document was created with Spire.PDF for Python.
TEL
K
6.4.
T
K
OM
NIKA
The outp
Figure 7.
T
T
he Time-d
o
The time
-
(a)
High
-Di
m
ut
s o
f
the si
m
(a)
(c
)
T
he
Outputs
o
main Test
-
d
o
m
ain wa
v
Figure
e-I
m
en
si
onal
C
m
ulation circ
u
of the Simul
v
efo
r
ms of t
h
8. System
T
SSN: 2087
-
2
C
haoti
c
Syst
e
u
it ar
e
s
h
ow
n
(e)
ation Circuit
h
e sim
u
lation
(b)
T
ime-d
o
m
ain
s
2
78
X
e
m
and its C
i
n
in Figure
7
(a
) x
-
y
,
(b) z
-
ci
rcuit are
s
s
(a)t
-x
,
(b)t
-
i
rc
uit Simula
t
7
(
a
)-
(e)
.
(b)
(d)
-
u,(c) p-q,(d
)
s
ho
wn in
Fig
u
u, (c
)t-p
t
ion (Jianm
i
n
)
r-
s,(e
)
v
-
w
u
re 8(
a)
-(c)
.
(c
)
n
g Liu)
4537
Evaluation Warning : The document was created with Spire.PDF for Python.
e-ISSN: 2
087-278X
TELKOM
NIKA
Vol. 11, No
. 8, August 2013: 4530 –
4538
4538
7. Conclusio
n
No
w, the p
opula
r
re
se
a
r
chi
ng di
re
ction of ch
aoti
c
syst
em is to desig
n highe
r-
dimen
s
ion
a
l hypercha
o
tic system. Th
e five-dim
en
sion
al co
mpl
e
x hypercha
o
tic syst
ems have
been b
u
ilt by adding
state feedba
ck controller.
I
n
this pa
per, there are three n
e
w
six-
dimen
s
ion
a
l
compl
e
x hyp
e
rchaoti
c
sy
stems a
nd
a new
te
n-di
me
nsio
nal comp
lex
hypercha
o
tic
system to b
e
found.
Wh
en combini
n
g the lo
w-
di
mensi
onal chaotic syste
m
and
the Duffing
cha
o
tic
syste
m
to ge
nerate high
-dim
en
sion
al
compl
e
x hyperch
ao
tic sy
stem, th
e ne
w
combi
n
ing
regul
arity is found. T
he
re
sult of thi
s
stu
d
y w
ill lay the
foundatio
n to
desi
gn
a vari
ety of new
hi
gh-
dimen
s
ion
a
l complex hype
rcha
otic sy
ste
m
s in the futu
re.
The exp
e
rim
ents of th
e si
mulation
circuits
a
r
e d
e
si
g
ned a
nd te
sted. The
outp
u
ts of the
simulatio
n
ex
perim
ents
co
nfirmed the
perfo
rman
ce
and effective
ness of the
desi
gne
d hig
h
-
dimen
s
ion
a
l compl
e
x
hyp
e
rchaoti
c
system
s.
The
resea
r
ch o
n
high
-dim
en
sion
al
compl
e
x
hyperchaotic system
and its
circuit i
m
plementa
tion will
have i
m
portant si
gnificance for the
desi
gn of communi
catio
n
equip
m
ent
, electro
n
ic
equipm
ent, electri
c
al
system and ch
aotic
encryption sy
stem in the fu
ture.
Ackn
o
w
l
e
dg
ments
The autho
rs wish to thank
the engin
eers of
the Key L
aboratory of Indu
strial Co
mputer
Control Engin
eerin
g of Heb
e
i Province. This
work wa
s
sup
porte
d by the 2012
Nat
u
ral Sci
e
n
c
e
Found
ation of
the Hebei Province, China
.
( F2012 20
3
088 ).
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W
e
ijia
n R
en,
Cha
oha
i K
ang,
Yin
g
y
in
g
Li, L
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Immune
Genetic
H
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br
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Al
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Wu Zhu, Qi
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aqi
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