TELKOM
NIKA
, Vol. 11, No. 8, August 2013, pp. 45
9
4
~4
601
e-ISSN: 2087
-278X
4594
Re
cei
v
ed Fe
brua
ry 10, 20
13; Re
vised
Ma
y 18, 20
13
; Accepte
d
May 29, 20
13
A New Regularity to Gene
rate High-dimensional
Hyperchaotic System
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
sy
stem pl
ays an importa
nt
rol
e
in
infor
m
ati
o
n
communic
a
tio
n
,
electric
al eq
u
i
p
m
e
n
t,
computer crypt
ogra
phy a
nd s
o
on. In this pa
per,
four new
high-
di
me
nsio
na
l
complex hype
rchaotic
syste
m
s
are found. A new overlayi
ng
regularity to generate a new high-d
imensional com
p
lex hyperchaotic system
is found
by ov
erlayi
ng a s
e
ri
es of low
-
di
me
nsio
nal
ch
aotic
system w
i
th the Du
ffin
g
cha
o
tic system. Th
e
regularity to generate
high-dim
ensi
onal complex hy
perc
h
aotic system
is
analy
z
e
d. The
features of chas
e
space maps and Lya
pun
ov
expo
ne
nt
s
’
ma
ps ar
e a
n
a
l
y
z
e
d
. T
h
e
re
sults of the
o
r
e
tical
an
alysis
an
d
exper
iment sh
ow
that new
systems
h
a
vin
g
strong cha
o
tic features.
Ke
y
w
ords
:
e
lectrical equipment, cryptography, co
mplex ch
aos, hig
h
-di
m
e
n
sio
nal ch
aos
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 the
o
ry
of relativity, the ch
aotic ph
en
omeno
n a
nd
the qu
antum
mech
ani
cs a
r
e thre
e
importa
nt sci
entific discov
erie
s in the 2
0
th
centu
r
y [1]. The cha
o
tic
phen
omen
on
is wid
e
spre
ad
in
informatio
n
communi
catio
n
field [2]. T
h
ere
often
we
re some
noi
ses i
n
the
ele
c
tri
c
al
equip
m
ent
and
com
m
uni
cation [3]. Those noi
se
s
were some
uncertai
n m
e
ssy
output
wav
e
form [4]. In
the
past, they
we
re
gene
rally
consi
dered to
be d
ue to
th
e
ci
rcuit to g
e
n
e
rate
self-excited o
scill
atio
n
and noi
se
s. In fact, in ma
ny ca
se
s, the circuit wa
s i
n
a ch
aotic
st
ate. T
herefore, to unde
rst
and
the cha
o
tic p
henom
eno
n and its produ
ced regul
arity
in electri
c
al
equipm
ent an
d comm
uni
ca
tion
has imp
o
rtan
t signifi
can
c
e
[5]. Beca
use
the
output
s
of ch
aotic
system i
s
ve
ry
sen
s
itive to
the
cha
nge
s of initial condition
s and ha
s th
e andrand
om
characte
risti
cs that en
cry
p
tion req
u
ire
d
, it
has b
e
come
an impo
rtant bran
ch of resear
c
h
ing for the information s
e
c
u
rity [6].
To any
cha
o
tic info
rmation
encryption, th
e high
er di
me
nsio
n it ha
s, the bette
r security it
has.
No
w, onl
y a few of five-dim
en
siona
l comp
l
e
x ch
aotic
system
s have be
en f
ound. By add
ing
the state fe
edba
ck co
ntroller o
n
low-dimen
s
ion
a
l cha
o
tic syste
m
s,
so
me
five-dime
n
si
on
al
hypercha
o
tic system
s
a
r
e gene
rat
ed. F
o
r exam
ple: in 200
9, Hua
q
ing Li a
dde
d state fee
d
b
a
ck
on the th
ree
-
dimen
s
ion
a
l
Lore
n
z
syste
m
to gene
rat
e
a five-dim
e
n
sio
nal L
o
re
nz hyp
e
rcha
otic
system [7]. In 201
0, Fen
g
Ha
n ad
de
d state
fe
ed
back o
n
the
three
-
dime
n
s
ion
a
l Lu
ch
aotic
system
to ge
nerate
a five
-dimen
sion
al
Lu hyp
e
rcha
otic
system [
8
]. In 201
1,
Lu
Hua
ng a
d
ded
state feed
ba
ck on the th
re
e-dim
e
n
s
iona
l Chen
ch
aoti
c
sy
stem to
gene
rate a fi
ve-dime
n
si
on
al
Che
n
hyperchaotic
syste
m
[9]. In this pape
r, we
will study how
to gene
rate a
six-dime
nsi
o
nal
compl
e
x hyp
e
rchaoti
c
sy
stem an
d ex
plore th
e la
w to ge
nera
t
e high-dime
nsio
nal
com
p
lex
hypercha
o
tic system.
The result
of
this study will furt
her reveal th
e operation
mech
ani
sm o
f
th
e
high-dimen
s
i
onal
hyp
e
rch
aotic oscillati
on circuits.
T
he result of this
re
sea
r
ch
will ha
s
pra
c
t
i
cal
signifi
can
c
e i
n
cryptog
r
a
p
h
y
, communi
cation, electro
n
ic an
d ele
c
trical eq
uipme
n
ts.
2. Ne
w
Duffing-Lor
en
z Chaotic Sy
stem
2.1. The For
m of the Du
ffing-Lo
ren
z
Chao
tic Sy
stem
Duffing
syste
m
ha
s
rich
nonlin
earity
dynamics ch
ara
c
teri
stics
[10]. It is o
n
e
of th
e
comm
only used syste
m
in informatio
n transmit fi
eld. Duffing sy
ste
m
is as the fo
llowing:
wt
e
x
dy
y
y
x
cos
3
(1)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
e-ISSN:
2087
-278X
A New Regul
arity to Ge
nerate High
-dim
ensi
onal
Hyp
e
rchaoti
c
System
(Jianm
ing Liu)
4595
D and e a
r
e real co
nsta
nts. The form of
the Lore
n
z
system is a
s
th
e followin
g
:
cz
xy
z
y
xz
bx
y
x
y
a
x
(2)
The pa
ram
e
ters
of a~c a
r
e real con
s
tant
s. Equatio
n (1
) and E
q
uation (2) a
r
e
overlai
d
into a new
Du
ffing-Lo
ren
z
complex hype
rcha
otic sy
ste
m
:
gx
w
w
f
u
ev
v
v
u
bz
xy
z
y
xz
cx
y
w
dzu
x
y
a
x
)
cos(
)
(
3
(3)
The pa
ramet
e
rs of a
~
g are real con
s
ta
nts.
2.2. Phase Space
The p
hase
sp
ace
s
of
Duffing Lo
ren
z
co
mplex
hype
rchaotic syste
m
ar
e sh
own in
Figu
re
1.
(a)
(b)
(c
)
(d)
(e)
Figur
e 1. Phase Spa
c
e Pict
ure
s
(a
) x
-
y
,
(b) x
-
z, (
c
) x
-
u
,
(d) x
-
v
,
(e) x
-
w
2.3. L
y
apuno
v
Exponent Analy
s
is
In the b
a
selin
e pa
ram
e
ters of a
=
10,
b=8
/
3, c=
28, d
=
-2.5, e=0.6, f=-8, g
=
9.7, x
=
1, y=1,
z=1, u=1, v=1
,
w=1
and
dt=0.
005, the Ly
apun
ov expo
nents [1
1] wi
t
h
the pa
ram
e
ters
ch
ang
e a
r
e
s
h
ow
n
in
F
i
gu
r
e
2(
a
)
-
(
g
)
.
-3
0
-2
0
-10
0
10
20
30
-3
0
-2
0
-1
0
0
10
20
30
x
y
-30
-20
-1
0
0
10
20
30
0
10
20
30
40
50
60
x
z
-30
-2
0
-10
0
10
20
30
-2
-1
0
1
2
3
x
u
-3
0
-20
-1
0
0
10
20
30
-5
0
5
x
v
-30
-2
0
-10
0
10
20
30
-15
0
-10
0
-5
0
0
50
10
0
15
0
x
w
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e-ISSN: 2
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TELKOM
NIKA
Vol. 11, No
. 8, August 2013: 4594 –
4601
4596
(a)
(b)
(c
)
(d)
(e)
(f)
(g)
Figure 2. Lyapunov Pictu
r
e
s
(a
) a ch
ang
e, (b) b c
han
ge, (c) c chan
ge, (d) d
cha
nge, (e
) e ch
ange,
(f) f chan
ge, (g) g ch
ang
e
The
re
sults o
f
the expe
rim
ent sho
w
the
stea
dy state
,
the chaoti
c
state a
nd th
e
hyper
cha
o
tic state
whe
n
the pa
rameters chan
ge of
the Du
ffing-Lo
ren
z
chaotei
c syste
m
. When the
r
e
are
two
po
siti
ve Lyapu
nov
expone
nts i
n
the same
time, the
system
of Duffing-L
o
ren
z
co
mple
x
cha
o
tic is in t
he hyperch
ao
tic state.
2.4. Po
w
e
r S
p
ectr
u
m Ana
l
y
s
is
Figure 3. System Time-d
o
m
ain Wavefo
rm
0
5
10
15
20
25
30
35
40
-3
5
-3
0
-2
5
-2
0
-1
5
-1
0
-5
0
5
a
Ly
a
p
un
ov
e
x
pone
nt
s
D
y
n
a
mi
c
s
of
Ly
ap
uno
v
ex
po
nen
t
s
LE1
LE2
LE3
LE4
LE5
LE6
0
5
10
15
20
-2
5
-2
0
-1
5
-1
0
-5
0
5
b
Ly
a
p
u
nov
e
x
p
on
e
n
t
s
Dy
n
a
m
i
c
s
of
Ly
a
punov
ex
p
onent
s
LE
1
LE
2
LE
3
LE
4
LE
5
LE
6
0
5
10
15
20
25
30
35
40
-1
6
-1
4
-1
2
-1
0
-8
-6
-4
-2
0
2
c
Ly
a
p
u
nov
e
x
p
on
e
n
t
s
D
y
nami
c
s
of
Ly
apu
nov
ex
pon
ent
s
LE1
LE2
LE3
LE4
LE5
LE6
-1
0
-5
0
5
10
-1
6
-1
4
-1
2
-1
0
-8
-6
-4
-2
0
2
d
Ly
a
punov
e
x
pone
nt
s
D
y
na
m
i
c
s
o
f
Ly
apu
no
v
e
x
p
one
nt
s
LE
1
LE
2
LE
3
LE
4
LE
5
LE
6
0
1
2
3
4
5
-1
6
-1
4
-1
2
-1
0
-8
-6
-4
-2
0
2
e
Ly
a
punov
e
x
pone
nt
s
D
y
na
m
i
c
s
o
f
Ly
apu
no
v
e
x
p
one
nt
s
LE
1
LE
2
LE
3
LE
4
LE
5
LE
6
-1
0
-5
0
5
10
-1
6
-1
4
-1
2
-1
0
-8
-6
-4
-2
0
2
f
Ly
a
puno
v
e
x
pone
nt
s
Dy
nam
i
c
s
of
L
y
apu
nov
ex
p
onent
s
LE
1
LE
2
LE
3
LE
4
LE
5
LE
6
0
5
10
15
20
25
30
-1
6
-1
4
-1
2
-1
0
-8
-6
-4
-2
0
2
g
Ly
a
pun
ov
e
x
po
ne
nt
s
D
y
na
m
i
c
s
o
f
Ly
apu
no
v
e
x
p
one
nt
s
LE
1
LE
2
LE
3
LE
4
LE
5
LE
6
0
5
10
15
20
25
-5
0
0
50
t/
s
x
0
5
10
15
20
25
-1
0
0
0
10
0
t/
s
y
0
5
10
15
20
25
0
10
0
20
0
t/
s
z
0
5
10
15
20
25
-2
0
2
t/s
u
0
5
10
15
20
25
-2
0
2
t/s
v
0
5
10
15
20
25
-1
0
0
10
t/s
w
Evaluation Warning : The document was created with Spire.PDF for Python.
TEL
K
wav
e
conti
n
has
a
(3) a
r
cha
r
a
3. N
e
3.1.
T
sy
st
e
Equ
a
3.2.
P
K
OM
NIKA
A Ne
w
The time
-
e
form of t
he
c
n
uou
s a
nd
s
a
nu
mb
er
o
f
r
e
s
h
ow
n
in
F
The wa
v
a
ct
eri
s
t
i
c
s
.
e
w
Duffing-
L
T
he For
m o
f
Lu ch
aoti
xy
z
xz
y
y
a
x
(
The pa
r
a
e
m i
s
rep
l
a
c
a
tion (1)
and
g
y
z
w
e
v
v
v
u
x
z
xz
y
y
a
x
(
2
The pa
ra
P
hase Spac
e
The pha
s
w
Regul
arity
-
do
main wa
v
c
ycle signal
i
s
mooth. Th
e
peaks. T
he
p
F
igure 3 a
n
d
Fi
g
v
eforms of
L
u Chao
tic
S
f
the Duffin
g
c sy
st
em
is
a
dz
cy
z
x
y
)
a
me
te
r
s
o
f
a
c
ed
by a
n
a
Equatio
n
(4
)
z
f
u
v
dz
cy
z
bw
x
y
c
o
)
3
met
e
rs
of a
~
e
s
e spaces o
f
e-I
to Ge
nerate
v
eform of ch
a
i
s discrete.
T
e
po
we
r
sp
e
c
p
ow
e
r
s
p
ec
t
d
Figu
re 4.
g
ure 4. The
P
Figu
re 3
S
ys
t
e
m
g
-Lu Cha
o
ti
c
a
s
the follo
w
a
~c a
r
e real
uto
nom
ou
s
)
a
r
e overl
a
i
d
w
)
o
s(
~
g are real
c
f
Duffing-Lu
SSN: 2087
-
2
High
-dim
en
s
a
ot
i
c
sy
st
e
m
T
he p
o
wer
s
p
c
trum wav
e
f
rum wav
e
fo
r
P
ower Spe
c
t
and Fi
gure
c
Sy
stem
w
ing:
con
s
t
ant
s.
T
p
a
rt and
t
h
d
into a new
c
ons
t
ants
.
co
mple
x
hy
p
2
78
X
s
i
onal Hyp
e
r
i
s
sim
ilar p
e
p
ectru
m
wa
v
f
orm of
ch
a
o
r
m and time
-
t
rum W
a
v
e
f
o
4
s
how
t
T
he externa
l
e p
o
sitive
f
e
Duffing-Lu c
p
erch
aoti
c
s
y
r
chaoti
c
Sys
t
e
riodic
i
ty. Th
v
ef
orm of
no
n
o
t
i
c sy
st
em
-
domain wa
v
o
rm
t
hat they
h
l
excitati
on
p
e
edb
ack i
s
ompl
ex hyp
e
y
ste
m
are:
t
em
(Jianm
i
n
e power
sp
e
n
-p
eriodi
c si
g
i
s
continu
o
u
v
e
f
orm of Eq
u
h
ave the c
(
p
art of the
D
i
n
trodu
ce
d.
e
rc
haot
i
c sy
s
(
n
g Liu)
4597
e
ct
r
u
m
g
nal is
s and
u
ation
ha
otic
(
4)
D
uffing
T
hen,
s
te
m
.
(
5)
Evaluation Warning : The document was created with Spire.PDF for Python.
TEL
K
4598
3.3.
L
u=1,
18.8
5
hype
r
4. N
e
4.1.
T
com
p
K
OM
NIKA
V
(a)
Fig
u
L
y
a
puno
v
E
x
When t
h
v=1, w=1
a
n
5
, Since
the
r
r
cha
o
t
i
c st
at
e
e
w
Duffing-
C
T
he For
m o
f
The form
xy
z
c
y
y
a
x
(
(
Equation
p
l
e
x hyperch
a
f
x
v
w
d
v
v
v
u
xy
z
b
y
y
a
x
(
(
The pa
ra
ol. 11, No
. 8
(d)
u
re 5. Phase
x
ponent A
n
h
e ini
t
ial con
d
n
d dt
=0.0
05
,
r
e are two
p
e
.
C
hen Ch
aot
i
f
the Duffin
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TEL
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4600
6. T
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6.1.
T
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posit
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Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
e-ISSN:
2087
-278X
A New Regul
arity to Ge
nerate High
-dim
ensi
onal
Hyp
e
rchaoti
c
System
(Jianm
ing Liu)
4601
7. Conclusio
n
No
w, the five-dime
n
si
onal
compl
e
x hyp
e
rchaot
i
c
syst
ems
have b
e
en built by a
d
d
ing the
feedba
ck con
t
roller to a three
-
dime
nsi
o
nal cha
o
tic system. In this pape
r, there are thre
e n
e
w
six-dim
e
n
s
ion
a
l com
p
lex
hypercha
o
tic sy
stems
and a n
e
w four-dimen
sion
al co
mp
lex
hypercha
o
tic
system to be
found.
A new overlaying re
gularity is fou
nd by overlay
i
ng a se
rie
s
of
low-dime
nsio
nal chaotic
syst
em a
nd
the Duffing
cha
o
tic
syste
m
to gene
ra
te a new
hi
gh-
dimen
s
ion
a
l compl
e
x cha
o
tic syste
m
. The re
sult
s o
f
theoretical a
nalysi
s
and e
x
perime
n
t
reveal
the relatio
n
sh
ip between th
e low-dime
nsi
onal
chaot
i
c
system a
nd t
he hig
h
-dim
e
n
sio
nal
compl
e
x
hypercha
o
tic syste
m
. Th
e result of t
h
is
re
sea
r
ch
ha
s p
r
a
c
tical si
gnifican
c
e to a
nalyze
and
desi
gn the
hi
gh-di
men
s
ion
a
l ch
aotic
sy
stem in
co
m
m
unication, e
l
ectri
c
al
equi
pment, comp
uter
cryptog
r
a
phy and so on.
Ackn
o
w
l
e
dg
ments
The work was supp
orte
d
by the 201
2 Natu
ral S
c
ien
c
e F
oun
dation of th
e Heb
e
i
Province, Chi
na ( F20
122
0
3088 ).
Referen
ces
[1]
W
e
ijia
n R
en,
Cha
oha
i K
ang,
Yin
g
y
in
g
Li, L
i
ying
Gon
g
. C
haotic
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br
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a
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edd
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y
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h
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r
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OMNIKA
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[4]
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g, Yan H
ua, Ya
ng
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he Rese
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ar
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adi
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K
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hang W
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he Electrom
ag
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d
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w
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ng Devic
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OMNIKA
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[6]
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ui S
un, Ge
ng Z
h
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ubl
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Encr
yptio
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g
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OMNIKA
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[7]
Li Hu
aqi
ng, Lu
o Xi
ao
hu
a, Da
i Xi
an
ggu
an
g. A h
y
perch
aotic
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y
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d its s
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.
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a
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a
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e
n
g
, T
a
n
g
Jiash
i
. D
y
na
mical Be
havi
o
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i
ve dimensio
nal C
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haotic S
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a
l
of dyna
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.
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ng
Lu, T
ang Jias
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i
rcuit R
ealiz
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n a
n
d
Contro
lli
ng M
e
thod
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i
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ensi
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l of Hai
n
an Nor
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i
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[10]
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un
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o
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y
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e
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a Pres
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.
[11]
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. Experi
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Evaluation Warning : The document was created with Spire.PDF for Python.