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Anti-Phase Synchronization of the Yu-Wang and
Burke-Shaw Chaotic Dynamic Systems via
Nonlinear Controllers
Edwin A. Umoh Department of Electrical Engineering Technology, Federal Polytechnic, Kaura Namoda, Nigeria
Email: [email protected]
Abstract—In this paper, anti-phase synchronization of two
non-identical autonomous chaotic systems is presented.
These two systems are neither diffeomorphic nor
topologically equivalent, but possessed chaotic properties
that ease synchronization and antisynchronization based on
different control strategies. The Yu-Wang possessed a cross-
product quadratic term and a nonlinear hyperbolic term in
its algebraic structure while the Burke-Shaw has two
nonlinear terms which adds complexity to the system's
dynamic evolutions. Nonlinear active controllers were
designed to regulate the two exponentially divergent chaotic
trajectories of the coupled system to achieve anti-phase
synchronization in finite times, while Lyapunov stability
theory was employed to test for local and global
convergence of the error dynamics to the origin. The results
of the various numerical simulations via MATLAB software
demonstrates the effectiveness of the coupling scheme and
the applicability of the antisynchronized signals in
modelling and design of electrical and communications
systems that are critical to secure communication and
electrical power outage minimization.
Index Terms—yu-wang chaotic system, burke-shaw chaotic
system, antisynchronization, synchronization, active
controllers
I. INTRODUCTION
Interest in chaotic dynamics has continued to increase
every year due to the discovery of these dynamics in
multitude of systems cutting across most disciplines.
Research continues to spurn out new and novel chaotic
attractors whose dynamics are subsequently subjected to
practical and simulative scrutiny using known methods.
Chaos is a feature of nonlinear deterministic systems
which have pronounced sensitivity to disturbances in
their parameters and initial conditions. Chaotic regimes
have been found to be extensive in nature and many man-
made systems including power systems [1], neurology
and medicine [2], electronics circuits [3] radar systems
[4], food [5], and economics and finance [6] among
others. Chaos synchronization occurs when two
dissipative chaotic systems are coupled such that, in spite
of the exponential divergence of their state vector
trajectories, synchrony is achieved in their chaotic
Manuscript received February 11, 2014; revised February 10, 2015.
behaviours in finite time. Several conditions such as
coupling strength, parameter region of the systems and
their degrees of parametric and initial conditions and their
stabilizability play crucial role in achieving mutual
couplings. Since the Pecora-Caroll breakthrough in the
1990s [7], riveted attention has been focused on the use
of chaos antisynchronization and synchronization in
security enhancement of communication channels and
information systems such as chaos masking, chaos
switching, chaos modulation using simple cost-effective
circuits and employed in masking transmitted signals
over public channels that are susceptible to third party
interception with attendant security risks. The broadband
spectrum of chaos-based communication systems allows
for effective spectra assimilation of a message by the
chaos carrier while the high sensitivity feature has acted
as effective encryption keys [8]-[11]. Recently, owing to
increasing understanding of the principles of
synchronization and antisynchronization (anti-phase),
engineers have focused attention on their use in power
systems such as in management of power outage [12],
[13]. In the same vein, several methods have been used to
(anti)synchronize chaotic systems. These methods
include linear control [14], hybrid feedback control [15],
active control [16], fuzzy control [17], feedback control
[18], sliding mode control [19] among others. Essentially,
as new systems continue to evolve, the challenges of
evaluating their controllability and synchronizability
using existing and new methods remains an open problem.
This paper examines the antisynchronizability of two
non-identical chaotic systems using the method of
nonlinear active controller design [16].
II. SYSTEMS DESCRIPTION
A. The Yu-Wang and Burke-Shaw Systems
The YU-WANG autonomous chaotic system [20] is a
three-dimensional system that possesses a quadratic
cross-product and a nonlinear term in the form of a
hyperbolic sine or cosine function in its system equations.
The resulting complex dynamics formed by these
nonlinearities can be manipulated to evolve two- and
four-wing attractors. Detailed structural and parametric
analyses have been reported in [20]. The governing
equation of the system can be expressed as:
International Journal of Electronics and Electrical Engineering Vol. 3, No. 6, December 2015
©2015 International Journal of Electronics and Electrical Engineering 438doi: 10.12720/ijeee.3.6.438-444
'
'
'
( )
( )
ym ym ym ym
ym ym ym ym ym ym
ym ym ym
x y x
y x x z
z f t z
(1)
where , ,ym ym ymx y z are state variables,
, , , 0ym ym ym ym are positive constants and ( )f t is a
changeable nonlinear hyperbolic function of the form
sinh( )ym ymx y or cosh( )ym ymx y . For values of
10, 30, 2, 2.5ym ym ym ym and the sinh
hyperbolic function, the phase portrait of the system is
given in Fig. 1. Linearizing the system at 0,0,0
J produced
the following eigenvalues 1 2 36, 5, 2.5 ,
which indicates the system is unstable.
-4-2
02
4
-5
0
50
20
40
60
XymYym
Zzm
Figure 1. Phase portrait of the Yu-Wang system
B. The Burke-Shaw Chaotic System
The Burke-Shaw (BS) chaotic system [21] is a three-
dimensional system with two quadratic nonlinear terms in
its system equations and is algebraically, but non-
topologically equivalent to the Lorenz system. The main
departure in the two systems is mainly in their
organization in the z-plane [22]. The set of equations
describing the system is given as
'
'
'
bs bs bs bs bs
bs bs bs bs bs
bs bs bs bs bs
x x y
y y x z
z x y
(2)
where , ,bs bs bsx y z are state variables, , 0bs bs are
positive constants. As bs is varied within a bounded set
1 15bs , two distinct attractors can be evolved for
values of 4.272 and 13, with portraits depicted in Fig. 2.
-2-1
01
2
-4-2
02
4-2
-1
0
1
2
XBSYBS
ZB
S
(a)
-6-4
-20
24
-10-5
05
10-10
-5
0
5
10
XBSYBS
ZB
S
(b)
Figure 2. Phase portrait of the Burke-Shaw System. For (a) 4.272BS
(b) 13BS
III. DESCRIPTION OF CONTROL OBJECTIVES
Given two dissipative chaotic systems described by the
following equations
' ( , , , )
' ( , , , ) ( , , , )
x p t x y z
y q t x y z F t x y z
(3)
where , ,x y z are state variables, p, q are vector field that
model the systems, F is the nonlinear control function. p,
q are the drive and response systems respectively. The
control objective is to design the nonlinear controller F
such that the time series evolutions of the two systems are
synchronized in finite time, while satisfying the condition
lim ( ) 0; (0); 1,2,3it
e t e i
(4)
where the antisynchronization error vector states are
1 2 3
T d r d r d re e e e x x y y z z (5)
IV. ANTISYNCHRONIZATION OF THE TWO SYSTEMS
A. Case 1: Antisynchronization of Identical Yu-Wang
System
To study the antisynchronization of identical Yu-Wang
system, the system in (1) serves as both drive and
response systems. Let the drive Yu-Wang system be
represented as '
'
' '
( )
( )
d d
ym ym ym ym
d d d
ym ym ym ym ym ym
d
ym ym ym
x y x
y x x z
z f t z
(6)
The response system is given as
'
'
( )
( )
r r
ym ym ym ym
r r r
ym ym ym ym ym ym
r r r
ym ym ym
x y x
y x x z
z f t z
(7)
By adding (6) to (7) and using the relationship (5), the
antisynchronization error dynamics becomes
International Journal of Electronics and Electrical Engineering Vol. 3, No. 6, December 2015
©2015 International Journal of Electronics and Electrical Engineering 439
' 1
1
' 2
2
' 3
3
( )
( ) ( )
( ) ( ) ( )
r d r r
ym ym ym ym ym as
r d r r d d
ym ym ym ym ym ym ym ym as
r d r d
ym ym ym as
e y y x x F
e x x x z x z F
e f t f t z z F
(8)
where i
asF are the nonlinear control inputs. Eq. (8) can
also be represented as
' 1
1 2 1
' 2
2 1
' 3
3 3
( )
( )
( ) ( )
ym as
r r d d
ym ym ym ym ym ym as
r d
ym as
e e e F
e e x z x z F
e e f t f t F
(9)
The nonlinear control inputs 1
asF , 2
asF , 3
asF can be
defined as follows
1 1
2 2
3 3
( )
( ) ( )
( ) ( ) ( )
as ym
r r d d
as ym ym ym ym ym ym
r d
as ym
F G t
F x z x z G t
F f t f t G t
(10)
By inserting (10) into (9), the error dynamics becomes
' 1
1 2 1
' 2
2 1
' 3
3 3
( ) ( )
( )
( )
ym ym
ym ym
ym ym
e e e G t
e e G t
e e G t
(11)
Eq. (11) is reduced to a linear system with control
inputs i
ymG as functions of the error vector states, which
can be represented in the following form
1
1
2
2
3
3
ym
ym
ym
G e
G P e
G e
(12)
P is a 3×3 matrix which is chosen to ensure that (11) is
asymptotically stable in finite time and is given by
0
0 0
0 0
ym ym
ym
ym
(13)
For (11) to be asymptotically stable, the eigenvalues of
(13) must lie in the negative real part. Making the matrice
1
2
3
0
0
0 0
ym
ym
ym
P
(14)
The controller coefficients 1 0 ,
2 0 , 3 0 .
Consequently, the closed loop system (11) is
asymptotically stable. By inserting (14) into (12), (10)
and (11) becomes
1
1 1 2
2
1 2 2
3
3 3
( )
( )
( ) ( ) ( )
as ym ym
r r d d
as ym ym ym ym ym ym
r d
as ym
F e e
F e x z x z e
F f t f t e
(15)
'
1 1 1
'
2 2 2
'
3 3 3
e e
e e
e e
(16)
Theorem 1: The identical Yu-Wang systems will anti-
synchronized for any initial conditions
, ,r d r d r dx x y y z z provided the error dynamics
converges asymptotically at the origin as t .
Proof: Adopt a Lyapunov function candidate
2 2 2
1 2 3
1( ) ( )
2 ym
V e e e e
(17)
' ' ' '
1 1 2 2 1 3
1( ) ( )
ym
V e e e e e e e
(18)
By inserting (16) in (18),
' 2 2 231 21 2 3( ) 0; 0i
ym ym ym
V e e e e
(19)
Thus, the error dynamics converges to the origin the
state trajectories achieves anti-synchrony in finite time.
The simulated results are depicted in Fig. 3 (a)-Fig. 3(c)
and Fig. 4 respectively.
B. Case 2: Antisynchronization of Identical Burke-Shaw
Systems
In this case, (2) is used as the identical equations for
the drive and response systems respectively. Applying the
nomenclature in case I, we can rewrite (2) in the
following forms '
'
'
d d
bs bs bs bs bs
d d d
bs bs bs bs bs
d d
bs bs bs bs bs
x x y
y y x z
z x y
(20)
The response system becomes
'
'
'
r r
bs bs bs bs bs
r r r
bs bs bs bs bs
r r
bs bs bs bs bs
x x y
y y x z
z x y
(21)
The error dynamics becomes
' 1
1 1 2
' 2
2 2
' 3
3
( )
( )
( ) 2
bs
r r d r
bs bs bs bs bs
r r d r
bs bs bs bs bs bs
e e e H
e e x z x z H
e x y x y H
(22)
The nonlinear control functions are defined as
1 1
2 2
3 3
( )
( ) ( )
( ) 2 ( )
bs
r r d r
bs bs bs bs bs bs
r r d r
bs bs bs bs bs bs
H L t
H x z x z L t
H x y x y L t
(23)
This reduces to a linear system with control inputs
represented in the form
1
1
2
2
3
3
L e
L S e
L e
(24)
International Journal of Electronics and Electrical Engineering Vol. 3, No. 6, December 2015
©2015 International Journal of Electronics and Electrical Engineering 440
S is a 3×3 matrice whose eigenvalues must lie in the
negative real part such that (22) is asymptotically stable
in finite time and is given by
1
2
3
0
0 1 0
0 0
ym ym
(25)
With 1 2 30; 0; 0 , the close loop system (22)
is asymptotically stable. By inserting (25) in (24), (22)
becomes '
1 1 1
'
2 2 2
'
3 3 3
e e
e e
e e
(26)
Theorem 2: The identical Burke-Shaw systems will
anti-synchronized for any initial conditions
, ,r d r d r dx x y y z z provided the error dynamics
converges asymptotically at the origin as t .
Proof: Adopt a Lyapunov function candidate,
2 2 2
1 2 3
1( ) ( )
2 bs
V e e e e
(27)
' ' ' '
1 1 2 2 1 3
1( ) ( )
bs
V e e e e e e e
(28)
By inserting (16) in (18),
' 2 2 231 21 2 3( ) 0; 0i
bs bs bs
V e e e e
(29)
Thus, the error dynamics converges to the origin and
the trajectories of the coupled systems achieves anti-
synchrony in finite time. The simulated results are
depicted in Fig. 5 and Fig. 6 respectively.
C. Case 3: Anti-Synchronization of Non-Identical Yu-
Wang and Burke-Shaw Systems
Using (6) and (21), the drive and response systems are
given as '
'
' '
( )
( )
d d
ym ym ym ym
d d d
ym ym ym ym ym ym
d
ym ym ym
x y x
y x x z
z f t z
(30)
'
'
'
r r
bs bs bs bs bs
r r r
bs bs bs bs bs
r r
bs bs bs bs bs
x x y
y y x z
z x y
(31)
By adding (30) to (31), the error dynamics becomes
' 1
1
' 2
2
' ' 3
3
( )
( )
r r d d
bs bs bs bs ym ym ym as
r r r d d d
bs bs bs bs ym ym ym ym ym as
r r d
bs bs bs bs ym ym as
e x y y x U
e y x z x x z U
e x y f t z U
(32)
For the following values 10ym
, 30ym
, 2ym
,
, 10ym bs
; 4.272bs , (32) transforms to
' 1
1 1
' 2
2 2
'
3 3
3
10( )
10 30 2
2.5 10 4.272 sinh( )
2.5
r d
bs ym as
r r r d d d
ym bs bs ym ym ym as
r r d d
bs bs ym ym
r
bs as
e e y y U
e e y x z x x z U
e e x y x y
z U
(33)
The nonlinear control functions are defined as
1 1
2 2
3
3
10( ) ( )
10 30 2 ( )
10 4.272 sinh( )
2.5 ( )
r d
as bs ym
r r r d d d
as ym bs bs ym ym ym
r r d d
as bs bs ym ym
r
bs
U y y x t
U y x z x x z x t
U x y x y
z x t
(34)
where 1
1
2
2
3
3
x e
x W e
x e
(35)
1 0 0
0 1 0
0 0 2.5
W
(36)
The eigenvalues of the matrice (36) are chosen such
that it is Hurwitz. This leads to the following resolutions
1
2
3
1 0 0
0 1 0
0 0 2.5
W
(37)
1 2 30; 0, 0
By inserting (37) in (35) and (34), the error dynamics
(33) reduces to '
1 1 1
'
2 2 2
'
3 3 3
e e
e e
e e
(38)
Theorem 3: The Yu-Wang and Burke-Shaw systems
will anti-synchronized for any initial conditions
, ,r d r d r d
x x y y z z provided that the state
trajectories of the error dynamics converges
asymptotically at the origin as t .
Proof: Adopt a Lyapunov function candidate,
2 2 2
1 2 3( ) ( )
2bs
V e e e e
(39)
' ' ' '
1 1 2 2 1 3( ) ( )
bs
V e e e e e e e
(40)
By inserting (16) in (18),
' 2 2 2
1 1 2 2 3 3( ) 0; , 0
iV e e e e (41)
International Journal of Electronics and Electrical Engineering Vol. 3, No. 6, December 2015
©2015 International Journal of Electronics and Electrical Engineering 441
Thus, the error dynamics converges to the origin and
the trajectories of the coupled systems achieves anti-
synchrony in finite time. The resulting plots are depicted
in Fig. 7 and Fig. 8 respectively.
0 0.5 1 1.5 2 2.5 3-4
-2
0
2
4
t(s)
xd
,xr
xd
xr
(a)
0 0.5 1 1.5 2 2.5 3-5
0
5
10
t(s)
yd
,yr
yd
yr
(b)
0 0.5 1 1.5 2 2.5 3-60
-40
-20
0
20
40
t(s)
zd
,zr
zd
zr
(c)
Figure 3. Dynamics of the antisynchronized chaotic systems
0 0.5 1 1.5 2 2.5 3-5
0
5
10
t(s)
e1,e
2,e
3
e1
e2
e3
Figure 4. Asymptotic convergence of the error state vectors
V. SIMULATION RESULTS
A. Case 1: Identical Yu-Wang Systems
The identical Yu-Wang systems were simulated with
MATLAB software for the following initial conditions:
Drive system, (0), (0), (0)
[3, 2, 10]d d d
ym ym ymx y z
p and the
Response system, (0), (0), (0)
[1, 6, 5]r r r
ym ym ymx y z
q and error
system 1 2 3(0) 4, (0) 8, (0) 5e e e . The resulting
plots are depicted in Fig. 3 (a)-Fig. 3(c) and Fig. 4.
B. Case 2: Identical Burke-Shaw Systems
The identical Burke-Shaw systems were simulated
with MATLAB software for the following initial
conditions: Drive system, (0), (0), (0)
[1, 3, 10]d d d
ym ym ymx y z
p
and the Response system, (0), (0), (0)
[3, 6, 6]r r r
ym ym ymx y z
q and
error system1 2 3(0) 4, (0) 9, (0) 4e e e . The
resulting plots are depicted in Fig. 5 and Fig. 6.
0 1 2 3 4 5-4
-2
0
2
4
6
t(s)
xr,
xd
xr
xd
(a)
0 1 2 3 4 5-10
-5
0
5
10
15
t(s)
yr,
yd
yr
yd
(b)
0 1 2 3 4 5-15
-10
-5
0
5
10
t(s)
zr,
zd
zr
zd
(c)
Figure 5. Dynamics of the antisynchonized chaotic systems
0 1 2 3 4 5-5
0
5
10
t(s)
e1,e
2,e
3
e1
e2
e3
Figure 6. Asymptotic converged dynamics of the error state vectors
International Journal of Electronics and Electrical Engineering Vol. 3, No. 6, December 2015
©2015 International Journal of Electronics and Electrical Engineering 442
C. Case 3: Non-Identical Yu-Wang and Burke-Shaw
Systems
The two non-identical systems were simulated for the
same initial conditions as in case 1. The plotted are given
in Fig. 7 and Fig. 8.
0 1 2 3 4 5-3
-2
-1
0
1
2
3
t(s)
xr,
xd
xr
xd
(a)
0 1 2 3 4 5-4
-2
0
2
4
6
8
t(s)
yr,
yd
yr
yd
(b)
0 1 2 3 4 5-40
-20
0
20
40
t(s)
zr,
zd
zr
zd
(c)
Figure 7. Dynamics of the antisynchronized chaotic systems
0 1 2 3 4 5-10
-5
0
5
10
t(s)
e1,e
2,e
3
e1
e2
e3
Figure 8. Asymptotic convergence of the error state vectors
VI. CONCLUSION
The Yu-Wang and Burke-Shaw systems synchronized
in anti-phase via nonlinear active control strategy. The
positive constants and nonlinear hyperbolic functions of
the system structures increases spectral densities to the
systems and this can increase the complexity of the
chaotic encryption keys when utilized in chaos-based
secure communication scheme. The robustness of this
simple antisynchronization scheme can be evaluated
through observation of the antisynchronized dynamics in
each case under study.
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©2015 International Journal of Electronics and Electrical Engineering 443
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Edwin A. Umoh received the B.Eng (Electrical/Electronics) and M. Eng
(Electronics) degrees in 1995 and 2011 from Abubakar Tafawa Balewa University, Bauchi,
Nigeria. He is currently a Principal Lecturer in
the Department of Electrical Engineering Technology, Federal Polytechnic, Kaura
Namoda, Nigeria. Engr Umoh is a member of the IEEE Control System Society and IEEE
Computational Intelligence Society. His
research interests are in chaos control, fuzzy modelling and illumination
engineering.
International Journal of Electronics and Electrical Engineering Vol. 3, No. 6, December 2015
©2015 International Journal of Electronics and Electrical Engineering 444