anti-phase synchronization of the yu-wang and … › uploadfile › 2015 › 0710 ›...

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


©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)



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)


' 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)


' 2 2 2

1 1 2 2 3 3( ) 0; , 0

iV e e e e (41)



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



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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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.



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