Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
7,
No.
6,
December
2017,
pp.
3436
ā
3445
ISSN:
2088-8708
3436
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
Modiļ¬ed
Pr
ojecti
v
e
Synchr
onization
of
Chaotic
Systems
with
Noise
Disturbance,
an
Acti
v
e
Nonlinear
Contr
ol
Method
Hamed
T
irandaz,
Hamidr
eza
T
a
v
ak
oli
and
Mohsen
Ahmadnia
Electrical
and
Computer
Engineering
F
aculty
,
Hakim
Sabze
v
ari
Uni
v
ersity
,
Sabze
v
ar
,
Iran
Article
Inf
o
Article
history:
Recei
v
ed:
Jan
15,
2017
Re
vised:
Jul
25,
2017
Accepted:
Aug
9,
2017
K
eyw
ord:
Acti
v
e
method
Nonlinear
control
Modiļ¬ed
Projecti
v
e
synchronization
(MPS)
Stability
theorem
ABSTRA
CT
The
synchronization
problem
of
chaotic
systems
using
acti
v
e
modiļ¬ed
projecti
v
e
non-
linear
control
method
is
rarely
addressed.
Thus
the
concentration
of
this
study
is
to
deri
v
e
a
modiļ¬ed
projecti
v
e
c
ontroller
to
synchronize
the
tw
o
chaotic
systems.
Since,
the
parameter
of
the
m
aster
and
follo
wer
systems
are
considered
kno
wn,
so
acti
v
e
methods
are
emplo
yed
instead
of
adapti
v
e
methods.
The
v
alidity
of
the
proposed
con-
troller
is
studied
by
means
of
the
L
yapuno
v
stability
theorem.
Furthermore,
some
numerical
simulations
are
sho
wn
to
v
erify
the
v
alidity
of
the
theoretical
discussions.
The
results
demonstrate
the
ef
fecti
v
eness
of
the
proposed
method
in
both
s
peed
and
accurac
y
points
of
vie
ws.
Copyright
c
ī
2017
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Hamed
T
irandaz
Hakim
Sabze
v
ari
Uni
v
ersity
Electrical
and
Computer
Engineering
F
aculty
,
Hakim
Sabze
v
ari
Uni
v
ersity
,
Sabze
v
ar
,
Iran.
Phone
+98051-4401288
Email:
tirandaz@hsu.ac.ir
1.
INTR
ODUCTION
Master
-sla
v
e
synchronization
of
chaotic
systems
is
strik
ely
nonlinear
,
since
the
aperiodic
and
nonre
g-
ular
beha
vior
of
chaotic
systems
and
their
sensiti
vity
to
the
initial
conditions.
Chaotic
beha
vior
may
appear
in
man
y
ph
ysical
systems.
So,
chaos
synchronization
subject
has
recei
v
ed
a
great
deal
of
attention
in
the
last
to
decades,
due
to
its
potential
applications
in
ph
ysics,
chemistry
,
electrical
engineering,
secure
communication
and
so
on[1].
Up
to
no
w
,
man
y
types
of
controling
methods
are
re
v
ealed
and
in
v
estig
ated
for
control
and
syn-
chronization
of
chaotic
systems.
Acti
v
e
method[2,
3,
4,
5,
6],
adapti
v
e
method
[7,
8,
9],
linear
feedback
method
[10,
11],
nonlinear
feedback
method
[12,
14,
15],
sliding
mode
method
[16,
17,
18],
impulsi
v
e
method
[19],
phase
method
[20],
generalized
method
[21],
rob
ust
synchronization
[13]
and
project
i
v
e
method
[22,
23,
24]
are
some
of
the
introduced
methods
by
the
researchers.
Among
these
methods,
synchronization
with
some
types
of
projecti
v
e
methods
are
e
xtensi
v
ely
in
v
estig
ated
in
the
l
ast
decades,
since
the
f
aster
synchronization
due
to
its
synchronization
scaling
f
actors,
which
master
and
sla
v
e
chaotic
sys
tems
w
ould
be
synchronized
up
to
a
proportional
rate.
Projecti
v
e
lag
method
[25],
modiļ¬ed
projecti
v
e
synchronization
(MPS)
[26,
27,
28],
function
projecti
v
e
synchronization
(FPS)[29],
modiļ¬ed
function
projecti
v
e
synchronization
[30,
28],
general-
ized
function
projecti
v
e
synchronization
[31,
32]
and
modiļ¬ed
projecti
v
e
lag
synchronization[33,
34]
are
some
generalized
schemes
of
projecti
v
e
method,
which
utilize
some
type
of
scaling
f
actors.
When
the
parameters
of
a
chaotic
system
are
kno
wn
beforehand,
acti
v
e
related
methods
are
preferably
chosen
than
adapti
v
e
methods.
Acti
v
e
synchronization
problem
of
tw
o
chaotic
systems
with
kno
wn
parameters
are
v
astly
in
v
estig
ated
by
the
researchers.
F
or
e
xample,
in
[5,
3,
35],
the
acti
v
e
controlling
method
is
studied
for
synchronization
of
tw
o
typical
chaotic
systems.
And
also,
in
[2],
an
acti
v
e
method
for
controling
the
beha
vior
of
a
uniļ¬ed
chaotic
system
is
presented.
Chaos
s
yn
c
hronization
of
comple
x
Chen
and
Lu
chaotic
systems
are
addressed
in
citeMahmoud,
with
designing
an
acti
v
e
control
m
ethod.
Furthermore,
in
[36]
acti
v
e
J
ournal
Homepage:
http://iaesjournal.com/online/inde
x.php/IJECE
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
,
DOI:
10.11591/ijece.v7i6.pp3436-3445
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
3437
synchronization
of
tw
o
dif
ferent
fractional
order
chaotic
system
is
studied.
Consequently
,
the
modiļ¬ed
projecti
v
e
synchronization
of
tw
o
chaotic
syst
ems
with
kno
wn
sys
tem
parameters
by
acitv
e
control
method
are
rarely
in
v
estig
ated
by
the
researchers.
Therefore,
in
the
present
study
,
the
modiļ¬ed
projecti
v
e
synchronization
problem
is
achie
v
ed
by
means
of
acti
v
e
nonlinear
control
method.
An
appropriate
feedback
controller
is
designed
to
control
the
beha
vior
the
state
v
ariables
of
the
follo
wer
system
to
track
the
trajectories
of
the
leader
system
state
v
ariables.
In
Section
2,
the
problem
of
chaos
synchronization
is
discussed.
In
addition,
the
v
alidity
of
the
proposed
synchronization
method
is
v
eriļ¬ed
by
means
of
L
yapuno
v
stability
theorem.
Then,
in
Secti
on
3,
some
e
xperiments
are
deri
v
ed
to
sho
w
the
ef
fecti
v
eness
of
the
proposed
method.
Moreo
v
er
,
some
simulati
ons
are
carried
out.
Finally
,
some
concluding
remarks
are
gi
v
en
in
Section
4.
2.
SYNCHR
ONIZA
TION
A
wide
v
ariety
of
chaotic
systems
can
be
represented
as
follo
ws:
_
x
=
f
(
x
)
ī
+
F
(
x
)
+
ī
(1)
Where
x
=
(
x
1
;
x
2
;
ī
ī
ī
;
x
n
)
T
is
the
state
v
ariables
v
ector
of
the
system
(1).
ī
=
(
ī
1
;
ī
2
;
ī
ī
ī
;
ī
n
)
T
2
R
n
ī
1
and
ī
=
(
ī
1
;
ī
2
;
ī
ī
ī
;
ī
n
)
T
2
R
n
ī
1
are
tw
o
v
ectors
denoting
the
unkno
wn
parameter
v
ector
of
the
system
and
the
e
xternal
distrib
uti
v
e
no
i
se
of
t
he
system,
respecti
v
ely
.
f
(
x
)
2
R
n
ī
n
and
F
(
x
)
2
R
n
ī
1
stand
for
the
linear
and
nonlinear
matrix
of
functions,
respecti
v
ely
.
Let
the
dynamical
system
(1)
as
the
leader
system.
Then
the
follo
wer
system
can
be
gi
v
en
by
another
chaotic
function
as
follo
ws:
_
y
=
g
(
y
)
^
ī
+
G
(
y
)
+
u
(2)
Where
y
=
(
y
1
;
y
2
;
ī
ī
ī
;
y
n
)
T
presents
the
state
v
ariables
v
ector
of
the
follo
wer
system
(2).
^
ī
=
(
^
ī
1
;
^
ī
2
;
ī
ī
ī
;
^
ī
n
)
2
R
n
ī
1
denotes
the
estimation
of
leader
system
parameters
v
ector
ī
.
Moreo
v
er
,
g
(
y
)
2
R
n
ī
n
and
G
(
y
)
2
R
n
ī
1
are
the
linear
and
nonlinear
matrix
of
functions,
respecti
v
ely
.
In
the
proposed
acti
v
e
nonlinear
control
method,
an
appropriate
controller
u
is
designed
which
the
states
of
leader
system
(1)
are
synchronized
with
their
corresponding
states
at
the
follo
wer
chaotic
sytem
(2),
base
on
the
modiļ¬ed
projecti
v
e
synchronization
error
that
is
deļ¬ned
as
follo
ws:
e
=
y
ī
ī
x
(3)
Where
ī
=
diag
f
ī
1
;
ī
2
;
ī
ī
ī
;
ī
n
g
represents
the
modiļ¬ed
scaling
f
actors
and
e
=
(
e
1
;
e
2
;
ī
ī
ī
;
e
n
)
T
2
R
n
ī
1
stands
for
synchronization
error
v
ector
.
Then
the
dynamical
synchronizaton
error
can
be
obtatined
as
follo
ws:
_
e
=
_
y
ī
ī
_
x
=
g
(
y
)
ī
+
G
(
y
)
+
u
ī
f
(
x
)
ī
ī
ī
F
(
x
)
ī
ī
ī
(4)
Where
ī
ī
denotes
the
estimation
of
noise
distrubance
ī
.
Deļ¬nition
1.
F
or
the
leader
system
(1)
and
the
follo
wer
system
(2),
the
chaos
synchronization
w
ould
be
achie
v
ed
if
an
appropriate
control
is
designed
to
force
the
stat
e
v
ariables
of
the
follo
wer
system
to
track
the
trajectories
of
the
leader
one,
meanly
,
the
synchronization
error
v
ector
(3)
con
v
er
ges
to
zero,
as
time
goes
to
inļ¬nity
,i.
e:
lim
t
!1
k
e
(
t
)
k
=
0
which
k
:
k
denotes
2-norm.
Chaos
synchroni
zation
can
be
achie
v
ed
by
deri
ving
an
appropriate
feed-
back
controller
,
which
is
the
subject
of
the
follo
wing
theorem.
Theor
em
1.
The
leader
system
(1)
with
the
state
v
ariables
v
ector
x
and
the
follo
wer
system
(2)
with
the
state
v
ariables
v
ector
y
,
the
parameters
v
ector
ī
and
an
y
noise
disturbance
v
ector
ī
,
w
ould
be
synchronized
for
an
y
initial
state
v
ariables
x
(0)
and
y
(0)
,
if
the
acti
v
e
feedback
control
la
w
is
deļ¬ned
as
follo
ws:
u
=
ī
ī
g
(
y
)
ī
f
(
x
)
ī
ī
ī
ī
G
(
y
)
ī
ī
F
(
x
)
ī
+
ī
ī
ī
ī
K
e
(5)
Where
ī
ī
can
be
estimated
dynamically
as
follo
ws:
_
ī
ī
=
ī
ī
e
ī
ī
(
ī
ī
ī
ī
)
(6)
Modiļ¬ed
Pr
ojective
Sync
hr
onization
of
Chaotic
Systems
with
...
(Hamed
T
ir
andaz)
Evaluation Warning : The document was created with Spire.PDF for Python.
3438
ISSN:
2088-8708
Where
K
=
diag
f
k
1
;
k
2
;
ī
ī
ī
;
k
n
g
and
ī
=
diag
f
1
;
2
;
ī
ī
ī
;
n
g
are
tw
o
diagonal
matrix
with
positi
v
e
v
alues
for
their
main
diagonal
elements.
Pr
oof
.
Let
the
L
yapuno
v
stability
function
as
follo
ws:
V
=
1
2
ee
T
+
1
2
(
ī
ī
ī
ī
)(
ī
ī
ī
ī
)
T
(7)
It
is
ob
vious
that
the
L
yapuno
v
function
deļ¬ned
in
(7)
is
positi
v
e
deļ¬nite.
W
ith
calculating
its
time
deri
v
ati
v
e,
we
ha
v
e:
_
V
=
_
ee
T
+
_
ī
ī
(
ī
ī
ī
ī
)
T
(8)
Then,
substituting
the
dynamical
representation
of
synchronization
error
v
ector
(4)
and
consequently
consider
-
ing
the
proposed
feedback
controller
(5)
and
the
noise
estimation
(6),
one
can
get:
_
V
=
ī
K
ee
T
ī
ī
(
^
ī
ī
ī
)(
^
ī
ī
ī
)
T
(9)
Therefore,
deri
v
ati
v
e
of
V
is
ne
g
ati
v
e
deļ¬nite,
when
K
and
ī
are
diagonal
matrix
with
positi
v
e
elements
on
their
primary
diagonal
elements.
In
the
follo
wing
section,
some
numerical
results
are
gi
v
en
to
sho
w
the
ef
fecti
v
eness
of
the
proposed
synchronization
method.
3.
NUMERICAL
SIMULA
TIONS
This
section
is
de
v
oted
to
the
synchronization
of
tw
o
dif
ferent
chaotic
or
h
yperchaotic
systems.
In
the
follo
wing
subsection,
chaos
synchronization
between
tw
o
chaotic
systems,
Zhang
chaotic
system
and
Lorenz
chaotic
system
is
addressed.
Then,
the
synchronizaton
problem
between
tw
o
h
yperchaotic
system
as
Chen
h
yperchaotic
system
and
Lorenz
h
yperchaotic
system
is
studied
in
the
last
subsection
3.1.
chaotic
systems
Chaos
synchronization
between
Zhang
chaotic
system
[14]
and
the
L
ĀØ
u
chaotic
system
[37]
is
addressed
in
this
subsection.
The
Zhang
chaotic
system
is
gi
v
en
by
a
three
simple
inte
ger
-based
and
nonlinear
dif
ferential
equations
that
depends
on
the
three
positi
v
e
real
parameters
as
follo
ws
_
x
1
=
a
(
x
2
ī
x
1
)
ī
x
2
x
3
_
x
2
=
bx
1
ī
x
2
1
(10)
_
x
3
=
ī
cx
3
+
x
2
2
Where
x
1
;
x
2
and
x
3
are
the
state
v
ariables
of
the
system
and
a,
b,
and
c
are
the
three
constant
parameters
of
the
system.
When
a
=
10
;
b
=
30
and
c
=
6
,
the
beha
viour
of
the
system
is
chaotic.
The
phase
portraits
of
the
system
is
sho
wn
in
Fig.
1,
with
initial
state
v
ariables
x
1
(0)
=
5
;
x
2
(0)
=
2
and
x
3
(0)
=
30
.
In
addition,
the
L
ĀØ
u
chaotic
system
can
be
described
as
follo
ws:
_
y
1
=
ī
1
(
y
2
ī
y
1
)
_
y
2
=
ī
2
y
2
ī
y
1
y
3
(11)
_
y
3
=
y
1
y
2
ī
ī
3
y
3
Where
y
1
;
y
2
and
y
3
are
the
state
v
ariables
of
the
system
and
ī
1
;
ī
2
and
ī
3
are
the
parameter
of
the
system.
The
chaotic
beha
vior
of
the
L
ĀØ
u
system
is
sho
wn
in
Fig.
2,
with
system
parameters
as:
ī
1
=
2
:
1
;
ī
2
=
30
and
ī
3
=
0
:
6
,
and
state
v
ariables
initial
v
alues
as:
x
1
(0)
=
4
:
3
;
x
2
(0)
=
7
:
2
and
x
3
(0)
=
5
:
8
.
The
Zhang
chaotic
system
(10)
can
be
re
written
based
on
the
leader
system
(1)
as
follo
ws:
_
x
1
=
a
(
x
2
ī
x
1
)
ī
x
2
x
3
+
ī
1
_
x
2
=
bx
1
ī
x
2
1
+
ī
2
(12)
_
x
3
=
ī
cx
3
+
x
2
2
+
ī
3
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V
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6,
December
2017:
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ā
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IJECE
ISSN:
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3439
Figure
1.
Phase
portraits
of
hte
Zhang
chaotic
system
Figure
2.
Phase
portraits
of
hte
Lu
chaotic
system
Modiļ¬ed
Pr
ojective
Sync
hr
onization
of
Chaotic
Systems
with
...
(Hamed
T
ir
andaz)
Evaluation Warning : The document was created with Spire.PDF for Python.
3440
ISSN:
2088-8708
Where
ī
1
;
ī
2
and
ī
3
are
the
three
noise
disturbance
corresponding
to
the
state
v
ariables
x
1
;
x
2
and
x
3
,
respec-
ti
v
ely
.
Then,
the
L
ĀØ
u
chaotic
system
(11)
can
be
represented
as
the
follo
wer
system
as
follo
ws:
_
y
1
=
a
(
y
2
ī
y
1
)
+
u
1
_
y
2
=
by
2
ī
y
1
y
3
+
u
2
(13)
_
y
3
=
y
1
y
2
ī
cy
3
+
u
3
According
to
the
proposed
control
la
w
(5)
and
noise
disturbance
estimation
(6),
we
deļ¬ne
the
follo
wing
feed-
back
controller
as:
u
1
=
ī
ay
2
+
ī
1
ax
1
+
ae
1
ī
ī
1
x
2
x
3
+
ī
1
ī
ī
1
ī
k
1
e
1
u
2
=
ī
by
2
+
y
1
y
3
+
ī
2
(
bx
1
ī
x
2
1
)
+
ī
2
ī
ī
2
ī
k
2
e
2
(14)
u
3
=
ī
y
1
y
2
+
ce
3
+
ī
3
x
2
2
+
ī
3
ī
ī
3
ī
k
3
e
3
;
and
the
noise
disturbance
estimation
as:
_
ī
ī
1
=
ī
ī
1
e
1
ī
1
(
ī
ī
1
ī
ī
1
)
_
ī
ī
2
=
ī
ī
2
e
2
ī
2
(
ī
ī
2
ī
ī
2
)
(15)
_
ī
ī
3
=
ī
ī
3
e
3
ī
3
(
ī
ī
3
ī
ī
3
)
Assume
the
parameter
of
the
Zhang
chaotic
system
as
a
=
10
;
b
=
30
and
c
=
6
and
the
initial
v
alues
for
the
dri
v
e
chaotic
system
(12)
are
tak
en
as,
x
1
(0)
=
12
;
x
2
(0)
=
5
,
and,
x
3
(0)
=
6
:
5
.
In
additiion,
the
initial
v
alues
of
the
response
L
system
(3)
are
selected
as:
y
1
(0)
=
2
;
y
2
(0)
=
15
and
y
3
(0)
=
0
.
Consider
the
nosie
disturbance
v
alues
as
ī
1
=
0
:
8
;
ī
2
=
0
:
6
and
ī
3
=
0
:
3
and
also
their
corresponding
estimation
ititial
v
alues
as
ī
ī
1
=
0
:
15
;
ī
ī
2
=
0
:
2
and
ī
ī
3
=
0
:
1
.
Let
the
g
ain
constants
as
k
1
=
2
;
k
2
=
2
;
k
3
=
2
;
ī
1
=
1
:
5
;
ī
2
=
1
:
5
and
ī
3
=
1
:
5
.
The
v
alidify
of
the
proposed
synchronization
method
for
contorling
the
beha
vior
of
the
Lu
chaotic
system
(13)
to
track
the
motion
trajectories
of
the
Zhang
chaotic
system
(12)
and
the
noise
disturbance
estima-
tion
are
sho
wn
in
Figure
3
and
4,
respect
i
v
ely
.
Figure
3
sho
ws
that
the
state
v
ariables
of
the
sys
tem
(13)
track
ef
fecti
v
ely
the
motion
trajectories
of
the
leader
chaotic
system.
In
addition,
in
Figure
4
e
xhibit
that
the
distance
between
noise
disturbance
and
its
estimation
v
alues
con
v
er
ge
to
zero.
0
1
2
3
4
5
6
7
8
-20
-10
0
10
20
0
1
2
3
4
5
6
7
8
-20
0
20
40
0
1
2
3
4
5
6
7
8
9
10
-10
0
10
20
30
Figure
3.
T
ime
responce
of
the
dri
v
e
Zhang
chaotic
system
and
the
response
Lorenz
chaotic
system
3.2.
Hyper
chaotic
systems
In
this
subsection,
the
synchronization
between
tw
o
h
yperchaotic
systems
as
Chen
h
yperchaotc
system
and
Lorenz
h
yperchaotic
system
is
in
v
esti
v
ated
via
the
proposed
control
method.
The
Chen
h
yperchaotic
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0
1
2
3
4
5
6
7
8
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
noise disturbance errors
Figure
4.
T
ime
responce
of
the
noise
disturbance
estimation
system
is
introduced
in
[38],
as
an
e
xtention
of
a
three-dimensional
Chen
chaotic
system
as
follo
ws:
x
1
=
a
(
x
2
ī
x
1
)
+
x
4
x
2
=
dx
1
+
cx
2
ī
x
1
x
3
x
3
=
x
1
x
2
ī
bx
3
(16)
x
4
=
x
1
x
2
+
r
x
4
Where
x
1
;
x
2
;
x
3
and
x
4
are
the
state
v
ariables
and
a;
b;
c
and
d
are
the
parameter
of
the
sys
tem.
The
phase
prortait
of
the
system
(16)
is
sho
wn
in
Fig.
5,
with
state
v
ariables
x
1
(0)
=
;
x
2
(0)
=
;
x
3
(0)
=
and
x
4
(0)
=
and
the
parameters
as
a
=
35
;
b
=
3
;
c
=
12
;
d
=
7
and
r=0.5
.
As
it
can
be
seen
the
beha
vior
of
the
system
(16)
is
h
yperchaotic.
The
Lorenz
h
yperchaotic
system,
which
w
as
introduced
in
[39],
can
be
described
as
follo
ws:
y
1
=
ī
1
(
y
2
ī
y
1
)
+
y
4
y
2
=
ī
y
1
y
3
+
ī
3
y
1
ī
y
2
y
3
=
y
1
y
2
ī
ī
2
y
3
(17)
y
4
=
ī
y
1
y
3
+
ī
4
y
4
Where
y
1
;
y
2
;
y
3
and
y
4
are
the
state
v
ariabels,
a;
b;
c
and
d
are
parameter
of
the
system.
The
chaotic
beha
vior
of
the
Lorenz
h
yperchaotic
system
is
sho
wn
in
Fig.
6,
with
initial
v
alues
for
the
system
sta
te
v
ariables
as
x
1
(0)
=
;
x
2
(0)
=
;
x
3
(0)
=
and
x
4
(0)
=
and
the
system
parameters
as
ī
1
=
36
;
ī
2
=
3
;
ī
3
=
20
and
ī
4
=
1
:
3
The
leader
system
can
be
deļ¬ned
based
on
the
Chen
h
yperchaotic
system
(16)
as
follo
ws:
x
1
=
a
(
x
2
ī
x
1
)
+
x
4
+
ī
1
x
2
=
dx
1
+
cx
2
ī
x
1
x
3
+
ī
2
x
3
=
x
1
x
2
ī
bx
3
+
ī
3
(18)
x
4
=
x
1
x
2
+
r
x
4
+
ī
4
Where
ī
1
;
ī
2
;
ī
3
and
ī
4
are
the
noi
se
disturbances
of
the
system.
Then,
consider
the
Lorenz
h
yperchaotic
system
(17),
as
the
follo
wer
system
as
follo
ws:
y
1
=
a
(
y
2
ī
y
1
)
+
y
4
+
u
1
y
2
=
ī
y
1
y
3
+
dy
1
ī
y
2
+
u
2
y
3
=
y
1
y
2
ī
by
3
+
u
3
(19)
y
4
=
ī
y
1
y
3
+
cy
4
+
u
4
Modiļ¬ed
Pr
ojective
Sync
hr
onization
of
Chaotic
Systems
with
...
(Hamed
T
ir
andaz)
Evaluation Warning : The document was created with Spire.PDF for Python.
3442
ISSN:
2088-8708
Where
u
1
;
u
2
;
u
3
and
u
4
are
the
feedback
controller
of
the
system.
The
proposed
chaos
synchronization
between
the
leader
Chen
h
yperchaotic
System
(18)
and
the
fol-
lo
wer
Lorenz
h
yperchaotic
system
(19)
can
be
achi
v
ed
by
designing
an
appropriate
control
la
w
and
noise
estimation
la
w
as
follo
ws:
u
1
=
ī
ay
2
+
ī
1
ax
1
+
ae
1
ī
e
4
+
ī
1
ī
ī
1
ī
k
1
e
1
u
2
=
+
y
1
y
3
ī
ī
2
x
1
x
3
ī
dy
1
+
ī
2
dx
1
+
y
2
+
cī
2
x
2
+
ī
2
ī
ī
2
ī
k
2
e
2
u
3
=
ī
y
1
y
2
+
ī
3
x
1
x
2
+
be
3
+
ī
3
ī
ī
3
ī
k
3
e
3
(20)
u
4
=
y
1
y
3
+
ī
4
x
1
x
2
ī
cy
4
+
ī
4
r
x
4
+
ī
4
ī
ī
4
ī
k
4
e
4
;
and,
_
ī
ī
1
=
ī
ī
1
e
1
ī
1
(
ī
ī
1
ī
ī
1
)
_
ī
ī
2
=
ī
ī
2
e
2
ī
2
(
ī
ī
2
ī
ī
2
)
(21)
_
ī
ī
3
=
ī
ī
3
e
3
ī
3
(
ī
ī
3
ī
ī
3
)
_
ī
ī
4
=
ī
ī
4
e
4
ī
4
(
ī
ī
4
ī
ī
4
)
No
w
,
some
numerical
results
related
to
the
proposed
synchronization
of
tw
o
h
yperchaotic
systems
are
gi
v
en.
Consider
the
parameter
of
the
leaer
Chen
h
yperchaotic
system
(18)
as
a
=
35
;
b
=
3
;
c
=
12
;
d
=
7
and
r
=
0
:
5
and
its
initial
v
alues
are
tak
en
as,
x
1
(0)
=
11
;
x
2
(0)
=
5
;
x
3
(0)
=
9
,
and,
x
4
(0)
=
13
.
In
additi
ion,
the
initial
v
alues
of
the
response
Lorenz
h
yperchaotic
system
(19)
are
selected
as:
y
1
(0)
=
1
;
y
2
(0)
=
11
;
y
3
(0)
=
2
and
y
4
(0)
=
3
.
Consider
the
nosie
disturbance
v
alues
as
ī
1
=
0
:
8
;
ī
2
=
0
:
6
;
ī
3
=
0
:
3
and
ī
4
=
0
:
5
.
Let
the
g
ain
constants
as
k
1
=
2
;
k
2
=
2
;
k
3
=
2
;
k
4
=
2
;
ī
1
=
1
:
5
;
ī
2
=
1
:
5
;
ī
3
=
1
:
5
and
ī
4
=
1
:
5
.
The
ef
fecti
v
eness
of
the
synchronization
method
for
the
contorling
beha
vior
of
the
Lorenz
h
yper
-
chaotic
system
(19)
to
track
the
motion
trajectories
of
the
Chen
h
yperchaotic
system
(18)
and
the
noise
distur
-
bance
estimation
are
illustrated
in
Figure
3
and
4,
respecti
v
ely
.
Figure
3
sho
ws
that
the
state
v
ariables
of
the
system
(19)
track
ef
fecti
v
ely
the
motion
trajectories
of
the
leader
chaotic
system(18).
In
addition,
in
Figure
4
e
xhibit
that
the
distance
between
noise
disturbance
and
its
estimation
v
alues
con
v
er
ge
to
zero.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
-20
0
20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
-20
0
20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
20
40
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
200
400
Figure
5.
T
ime
responce
of
the
dri
v
e
Zhang
chaotic
system
and
the
response
Lorenz
chaotic
system
4.
CONCLUSION
In
this
research,
some
results
related
to
the
modiļ¬ed
projecti
v
e
synchronization
of
kno
wn
chaotic/h
yperchaotic
systems
with
noise
disturbances
are
deri
v
ed.
Since
the
paramers
of
the
leader
system
is
considered
knonwn,
an
appropriated
acti
v
e
nonline
ar
feedback
control
la
w
with
designed
via
modiļ¬ed
projecti
v
e
synchronization
error
.
The
v
alidity
of
the
proposed
method
is
pro
v
ed
by
means
of
L
yapuno
v
stability
theorem.
Furtheremore,
IJECE
V
ol.
7,
No.
6,
December
2017:
3436
ā
3445
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
3443
0
0.5
1
1.5
2
2.5
3
3.5
4
-2
-1.5
-1
-0.5
0
0.5
1
1.5
Figure
6.
T
ime
responce
of
the
noise
disturbance
estimation
its
ef
fecti
v
eness
is
v
eriļ¬ed
by
some
numerical
simulations
of
the
chaotic
and
h
yperchaoti
c
systems.
Finally
,
some
ļ¬gures
are
sho
wn
to
v
eri
fy
the
accurac
y
of
the
theorical
discussions.
As
it
can
be
seen
from
these
results,
the
motion
trajectories
of
the
leader
chaotic
systems
containing
noise
disturbances
can
ef
fecti
v
ely
track
by
the
state
v
ariables
of
the
follo
wer
chaotic
systems
state
v
ariabels,
which
af
fected
by
proposed
control
method.
REFERENCES
[1]
G.
Chen,
X.
Y
u,
Chaos
control:
theory
and
applications,
V
ol
.
292,
Springer
Science
&
Business
Media,
2003.
[2]
C.-M.
Lin,
M.-H.
Lin,
R.-G.
Y
eh,
Synchronization
of
uniļ¬ed
chaotic
system
via
adapti
v
e
w
a
v
elet
cere-
bellar
model
articulation
controller
,
Neural
Computing
and
Applications
23
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(2013)
965ā973.
[3]
M.-c.
Ho,
Y
.-C.
Hung,
Synchronization
of
tw
o
dif
ferent
systems
by
using
generalized
acti
v
e
control,
Ph
ysics
Letters
A
301
(5)
(2002)
424ā428.
[4]
E.
A.
Umoh,
Adapti
v
e
Hybrid
Synchronization
of
Lore
nz-84
System
with
Uncertain
P
arameters,
TELK
OMNIKA
Indonesian
Journal
of
Electrical
Engineering,
v
ol.12,
no.7,
pp.
5251-5260,
2014
[5]
H.
Agiza,
M.
Y
ass
en,
Synchronization
of
rossler
and
chen
chaotic
dynami
cal
systems
using
acti
v
e
control,
Ph
ysics
Letters
A
278
(4)
(2001)
191ā197.
[6]
G.
M.
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IJECE
ISSN:
2088-8708
3445
BIOGRAPHIES
OF
A
UTHORS
Hamed
T
irandaz
recei
v
ed
the
B.Eng.
De
gree
i
n
Applied
Mathematics
from
the
Uni
v
e
rsity
of
Sabze
v
ar
T
arbiat
Moallem
Uni
v
ersity
,
Sabze
v
ar
,
Iran,
in
2006,
and
PhD.
de
gree
in
Mechatronics
Engineering
from
Semnan
Uni
v
ersity
,
Semnan,
Iran,
in
2009.
He
is
been
w
orking
as
a
Lecturer
at
Hakim
Sabze
v
ari
Uni
v
ersity
since
2010.
His
research
interests
include
mainly
Chaos
control
and
synchronization.
He
has
published
se
v
eral
papers
in
the
abo
v
e
mentioned
area.
Email:
tirandaz@hsu.ac.ir
Mohsen
Ahmadnia
has
recei
v
ed
his
PhD
de
gree
in
Po
wer
Engineering
at
the
P
eopleā
s
Friendship
uni
v
ersity
of
Russia.
He
is
currently
with
the
F
aculty
of
Electrical
and
Computer
Engineering
at
Hakim
Sabze
v
ari
Uni
v
ersity
,
Sabze
v
ar
,
Iran.
His
research
interests
are
in
nonlinear
dynamics.
Email:
m.ahmadnia@hsu.ac.ir
Hamidr
eza
T
a
v
ak
oli
has
obtained
his
PhD
de
gree
in
Electrical
Engineering
at
the
Iran
Uni
v
ersity
of
Science
and
T
e
chnology
and
Ryerson
Uni
v
ersity
in
T
oronto,
Canada.
He
is
currently
with
the
F
aculty
of
Electrical
and
Computer
Engineering
at
H
akim
Sabze
v
ari
Uni
v
ersity
,
Sabze
v
ar
,
Iran.
His
research
interests
are
in
Chaos
and
nonlinear
dynamics.
Email:
ta
v
ak
oli@hsu.ac.ir
Modiļ¬ed
Pr
ojective
Sync
hr
onization
of
Chaotic
Systems
with
...
(Hamed
T
ir
andaz)
Evaluation Warning : The document was created with Spire.PDF for Python.