Foundation and Synchronization of the Dynamic Output Dual Systems

International Journal of Trend in Scientific Research and Development (IJTSRD)
Volume 3 Issue 6, October 2019
@ IJTSRD | Unique Paper ID – IJTSRD29256
Foundation
Dynamic Output Dual Systems
Professor, Department of Electrical Engineering, I
ABSTRACT
In this paper, the synchronization problem of the dynamic output dual
systems is firstly introduced and investigated. Based on the time
approach, the state variables synchronization of such dual systems can be
verified. Meanwhile, the guaranteed exponential convergence rate can be
accurately estimated. Finally, some numerical simulations are provided to
illustrate the feasibility and effectiveness of the obtained result.
KEYWORDS: Dynamic output dual systems, time
chaotic system; exponential synchronization, exponential convergence rate
1. INTRODUCTION:
In recent years, various types of chaotic systems have
been widely explored and studied by experts and scholars;
see, for instance, [1-6] and the references therein.
This is not only due to the mystery of its theory, but also to
its various applications in dynamic systems. As we know,
chaos is often one of the factors that cause system
instability and oscillation.
In the past decades, the synchronization design of various
systems has been developed and explored and has quite
good results; see, for instance, [7-11] and the references
therein. Moreover, synchronization design frequently
exists in various fields of application, such as master
chaotic systems, system identification, ecological systems,
and secure communication. Due to the unpredictability of
chaotic systems, this feature becomes more challenging
for the synchronization design of chaotic dual systems.
On the other hands, a variety of dual systems have been
proposed, investigated and studied in depth; see, for
instance, [12-14]. A variety of methodologies have been
proposed for analyzing dual systems, such as separation
principle, passivation of error dynamics, sliding
approach, frequency domain analysis, and Chebyshev
neural network (CNN). In particular, the synchronizat
analysis and design of dynamic systems with chaos is in
general not as easy as that without chaos. On the basis of
the above-mentioned reasons, the synchronization
2019 Available Online: www.ijtsrd.com e
29256 | Volume – 3 | Issue – 6 | September
Foundation and Synchronization of
Dynamic Output Dual Systems
Yeong-Jeu Sun
f Electrical Engineering, I-Shou University, Kaohsiung, Taiwan
In this paper, the synchronization problem of the dynamic output dual
systems is firstly introduced and investigated. Based on the time-domain
approach, the state variables synchronization of such dual systems can be
anteed exponential convergence rate can be
accurately estimated. Finally, some numerical simulations are provided to
illustrate the feasibility and effectiveness of the obtained result.
Dynamic output dual systems, time-domain approach, hyper-
chaotic system; exponential synchronization, exponential convergence rate
How to cite this paper
"Foundation and Synchronization of the
Dynamic Output Dual Systems"
Published in
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Journal of Trend in
Scientific Research
and Development
(ijtsrd), ISSN: 2456
6470, Volume
Issue-6, October
2019, pp.898
https://www.i
9256.pdf
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International Journal of T
Scientific Research and Development
Journal. This is an Open Access article
distributed under
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by/4.0)
In recent years, various types of chaotic systems have
been widely explored and studied by experts and scholars;
6] and the references therein.
This is not only due to the mystery of its theory, but also to
applications in dynamic systems. As we know,
chaos is often one of the factors that cause system
In the past decades, the synchronization design of various
systems has been developed and explored and has quite
11] and the references
therein. Moreover, synchronization design frequently
exists in various fields of application, such as master-slave
chaotic systems, system identification, ecological systems,
npredictability of
chaotic systems, this feature becomes more challenging
for the synchronization design of chaotic dual systems.
On the other hands, a variety of dual systems have been
proposed, investigated and studied in depth; see, for
4]. A variety of methodologies have been
proposed for analyzing dual systems, such as separation
principle, passivation of error dynamics, sliding-mode
approach, frequency domain analysis, and Chebyshev
neural network (CNN). In particular, the synchronization
analysis and design of dynamic systems with chaos is in
general not as easy as that without chaos. On the basis of
mentioned reasons, the synchronization
analysis and design of chaotic dual systems is actually
crucial and meaningful.
In this paper, the synchronization problem for a class of
chaotic dual systems will be considered. Based on the
time-domain approach with differential inequality, the
state variables synchronization of such dual systems will
be verified. In addition, the guara
convergence rate can be accurately estimated. Several
numerical simulations will also be provided to illustrate
the use of the obtained results. In what follows,
denotes the Euclidean norm of the vector
2. PROBLEM FORMULATION AND MAIN RESULTS
In this paper, we consider the following
systems.
Master chaotic system:
( ) ( ) ( )
( ) ( ) ( )( ) ( )
( )
( ) ( ) ( ) ( )(
( ) ( ) ( ) ( ) ((
( ) ( )








≥∀=
=
+−=
+
+=
+=
,0,
,,,
,
,
,
,
1
432144
21333
4
12122
1121
ttxty
xtxtxtxftx
txtxftdxtx
tcx
xtbxtxtxftx
xftxatx
β
&
&
&
&
Slave system:
e-ISSN: 2456 – 6470
6 | September - October 2019 Page 898
f the
Kaohsiung, Taiwan
How to cite this paper: Yeong-Jeu Sun
"Foundation and Synchronization of the
Dynamic Output Dual Systems"
Published in
International
Journal of Trend in
Scientific Research
and Development
(ijtsrd), ISSN: 2456-
6470, Volume-3 |
6, October
2019, pp.898-901, URL:
https://www.ijtsrd.com/papers/ijtsrd2
Copyright © 2019 by author(s) and
International Journal of Trend in
Scientific Research and Development
Journal. This is an Open Access article
distributed under
the terms of the
Creative Commons
Attribution License (CC BY 4.0)
http://creativecommons.org/licenses/
analysis and design of chaotic dual systems is actually
his paper, the synchronization problem for a class of
chaotic dual systems will be considered. Based on the
domain approach with differential inequality, the
state variables synchronization of such dual systems will
be verified. In addition, the guaranteed exponential
convergence rate can be accurately estimated. Several
numerical simulations will also be provided to illustrate
the use of the obtained results. In what follows, x
denotes the Euclidean norm of the vector .n
x ℜ∈
PROBLEM FORMULATION AND MAIN RESULTS
In this paper, we consider the following master-slave dual
( )
))
( )),
,
3
t
tx
(1)
IJTSRD29256

International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
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( ) ( )
( ) ( ) ( )( )
( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( )( ) ( ) ( )[ ]











≥∀
−−=
+−=
−=
=
.0
,,
1
,,
,
11
,
3121224
21333
112
1
t
tztbztztzftz
c
tz
tztzftzdtz
tzf
a
ty
a
tz
ty
tz
&
&
&
β
β
(2)
where ( ) ( ) ( ) ( ) ( )[ ] 4
4321: ℜ∈=
T
txtxtxtxtx is the state
vector of master chaotic system, ( ) ℜ∈ty is the system
output, ( ) ( ) ( ) ( ) ( )[ ] 4
4321: ℜ∈=
T
tztztztztz is the state
vector of slave system, β,,,, dcba are the system
parameters with 0>d and 0≠βac . For the existence and
uniqueness of the system (1), we assume that all the
functions () { }4,3,2,1, ∈∀⋅ ifi , are sufficiently smooth.
Remark1. It is noted that the modified hyper-chaotic Pan
system [1] is the special cases of system (1) with
,
3
8
,1,1,10 ==−== dcba
( ) ( ) ,28,,10 1212111 xxxfxxf =−=
( ) ( ) 24321421213 10,,,,, xxxxxfxxxxf −== .
The time response of the modified hyper-chaotic Pan
system is depicted in Fig. 1-Fig. 4.
For brevity, let us define the synchronous error vector as
( ) ( ) ( ) ( ) ( )[ ]
( ) ( )tztx
tetetetete
T
−=
=
:
: 4321
(3)
The precise definition of exponential synchronization is
given as follows.
Definition1. The slave system (2) exponentially
synchronizes the master chaotic system (1) provided that
there are positive numbers k and α such that
( ) ( ) 0,exp ≥∀−≤ ttkte α .
In this case, the positive number α is called the
exponential convergence rate.
Now we present the main result for the master-slave dual
systems (1) and (2).
Theorem1. The slave system (2) exponentially
synchronizes the master chaotic system (1). Besides, the
guaranteed exponential convergence rate is given by .d
Proof. From (1)-(3), it can be readily obtained that
( ) ( ) ( )
( ) ( ) ;0,0
111
≥∀=−=
−=
t
tyty
tztxte
ββ
(4)
( ) ( ) ( )
( ) ( )
( ) ( )( )





−−






−=
−=
tzf
a
ty
a
xf
a
tx
a
tztxte
11
111
222
11
11
&
&
β
( ) ( )
( ) ( )( )





−−






−=
txf
a
tx
a
xf
a
tx
a
111
111
11
11
&
&
;0,0 ≥∀= t (5)
( ) ( ) ( )
( ) ( )[ ]
( ) ( ) ( )( )[ ]tztzftzd
xxftdx
tztxte
2133
2133
333
,
,
+−−
+−=
−= &&&
( ) ( )[ ] ( )
( )213
21333
,
,
zzf
xxftztxd
−
+−−=
( ) ( ) ( )2132133 ,, xxfxxftde −+−=
( ) .0,3 ≥∀−= ttde
This implies
( ) ( )[ ]
( ) ( ) ( ) ( )0exp0exp
0
exp
33
3
etdte
dt
tdted
=⇒
=
( ) ( ) ( ) .0,0exp 33 ≥∀−=⇒ tetdte (6)
In addition, from (1)-(6), one has
( ) ( ) ( )tztxte 444 −=
( ) ( ) ( )( ) ( ) ( )
( ) ( ) ( )( ) ( ) ( )






−−−






−−=
c
tztbz
c
tztzf
c
tz
c
txtbx
c
txtxf
c
tx
312122
312122
,
,
&
&
( ) ( ) ( )( ) ( ) ( )
( ) ( ) ( )( ) ( ) ( )






−−−






−−=
c
tztbx
c
txtxf
c
tx
c
txtbx
c
txtxf
c
tx
312122
312122
,
,
&
&
( ) ( )
c
tetbx 31
−=
( ) ( ) ( ) .0,
0exp 31
≥∀
−
−= t
c
etdtbx
(7)
As a result, from (4)-(7), we conclude that
( ) ( ) ( ) ( ) ( )
( ) ,0,exp
2
4
2
3
2
2
2
1
≥∀−⋅≤
+++=
ttdk
tetetetete
in view that ( )tx1 is a chaotic signal. This completes the
proof.
By Theorem 1 with Remark 1, it is straightforward to
obtain the following result.
Corollary1. The slave system of
( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )









≥∀+−=
+−=
+=
=
,0,28
,
3
8
,
10
1
,
31124
21
3
3
12
1
ttztztztztz
tztz
tz
tz
tztytz
ty
tz
&
&
&
β
β
exponentially synchronizes the modified hyper-chaotic
Pan system with the guaranteed exponential convergence
rate is given by .
3
8
=α

3. NUMERICAL SIMULATIONS
Consider the slave system:
( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )








≥∀+−=
+−=
+=
=
.0,28
,
3
8
,
20
1
,
2
31124
21
3
3
12
1
ttztztztztz
tztz
tz
tz
tztytz
ty
tz
&
&
&
(8)
By Corollary 1, we conclude that the system (8) is
exponentially synchronizes the following master chaotic
system
( )
( )
( )
( )
( )








≥∀=
−=
+
−
=
+−=
−=
,0,2
,10
,
3
8
,28
,1010
1
24
2133
43112
121
txty
xtx
xxxtx
xxxxtx
xxtx
&
&
&
&
(9)
with the guaranteed exponential convergence rate .
3
8
=α
The time response of error states for the systems (8) and
(9) is depicted in Fig. 5. From the foregoing simulations
results, it is seen that the system (8) is exponentially
synchronizes the master system of (9).
4. CONCLUSION
In this paper, the synchronization problem of the dynamic
output dual systems has been firstly introduced and
investigated. Based on the time-domain approach, the
state variables synchronization of such dual systems can
be verified. Besides, the guaranteed exponential
convergence rate can be accurately estimated. Finally,
some numerical simulations have been offered to show the
feasibility and effectiveness of the obtained result.
ACKNOWLEDGEMENT
The author thanks the Ministry of Science and Technology
of Republic of China for supporting this work under grant
MOST 107-2221-E-214-030. Furthermore, the author is
grateful to Chair Professor Jer-Guang Hsieh for the useful
comments.
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Figure 1: The chaotic signals of ).(1 tx
0 10 20 30 40 50 60
-15
-10
-5
0
5
10
15
20
25
t (sec)
x1(t)

Figure 5: The time response of error states.
0 10 20 30 40 50 60
-20
-15
-10
-5
0
5
10
15
20
25
30
t (sec)
x2(t)
0 10 20 30 40 50 60
0
10
20
30
40
50
60
t (sec)
x3(t)
0 10 20 30 40 50 60
-50
0
50
100
t (sec)
x4(t)
0 0.5 1 1.5 2 2.5 3 3.5 4
-60
-50
-40
-30
-20
-10
0
10
t
e1(t);e2(t);e3(t);e4(t)
e1=e2=0
e3 e4

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