Journal of Robotics, Networking and Artificial Life, Vol. 3, No. 3 (December 2016), 158-162

Unidirectionally coupled synchronization and circuitdesign of a novel chaotic system
Wei Xue 1, Xue Li 2
1

Department of Automation , Tianjin University of Science and Technology,
1038 Dagunanlu Road, Hexi Distric, Tianjin 300222, PR China
E-mail: xuewei@tust.edu.cn

2

Department of Automation , Tianjin University of Science and Technology,
1038 Dagunanlu Road, Hexi Distric, Tianjin 300222, PR China
E-mail: lixueTJKJ@163.com

Abstract
In this paper, a novel three-dimensional autonomous chaotic system is proposed. Based on Lyapunov stability
theory, a kind of unidirectionally coupled synchronization method is investigated. When coupling coefficient isselected properly, the response system can be synchronized completely with drive system. On this basis, the analog
synchronization circuit of the system is designed and realized by using Multisim software. The obtained circuit
simulation results are consistent with the numerical simulation results, which can further confirm the effectivenessand feasibility of the presented synchronization scheme.
Keywords: Novel chaotic system; Unidirectionally coupled synchronization; Numerical simulation; Circuit design.

1. Introduction
In 1963, American meteorologist Lorenz discoveredchaos in a simple system of three autonomous ordinary
differential equations, called the Lorenz system.1 As thefirst chaotic model, the Lorenz system has become aparadigm of chaos research. Recently, there has been
increasing enthusiasm in finding new chaotic attractors,
such as Chen system,2 Lü system,3 and so on.4-5 In thispaper, a novel three-dimensional autonomous chaotic
system is introduced. Numerical analysis shows thatthe proposed chaotic attractor is a new attractor, which
is not topologically equivalent to the original chaoticsystem, or the Lorenz system, or the Chen system, oreven the Lorenz system family.
Since the ideal of synchronizing chaotic systems wasproposed by Pecora and Carroll,6 it has been an active

research topic, mainly due to their practical applicationsin the fields of secure communication, image encryption,
electrical engineering and so on. Many synchronization
approaches have been proposed in order to synchronize
the chaotic systems. For example,
activesynchronization,7 adaptive synchronization,8 projective
synchronization,9 and sliding mode synchronization.10
A kind of unidirectionally coupled synchronization of
the novel three-dimensional chaotic system isinvestigated. The suitable coupling coefficient isselected for the response system to realize theunidirectionally coupled synchronization. By means oftheoretical analysis and numerical simulation verify theeffectiveness of the presented synchronization scheme.
On this basis, the analog synchronization circuit of thesystem is designed.

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2. The novel chaotic system
A novel three-dimensional autonomous chaotic system
is proposed, which can be expressed as the followingform: x = ay − ax − yz(1) y = bx + y + xz
2 z = − dz + cy
where the parameters a =33, b= 50, c=3, and d=15 are
fixed, the phase portraits of system(1) are shown inFig.1. The Lyapunov exponents of system (1) arecalculated as L1 = 5.7673 , L2 = 0.0034 , and L3 = −7.8236 .
The Lyapunov exponent spectrum of the system(1)varies with the increase of d is given in Fig.2.

 y1 = ay2 − ay1 − y2 y3 − k ( y1 − x1 ) y 2 = by1 + y2 + y1 y3 − k ( y2 − x2 ) y 3 = − dy3 + cy2 y2 − k ( y3 − x3 )(3)Where e1 = y1 − x1 , e2 = y2 − x2 , ande3 = y3 − x3 .

And k is the coupling coefficient, it can make system(2)and system(3) compose a coupling system.
From Eq.(2) and Eq.(3), the error system is obtained
which can be expressed as:e1 = ae2 − ae1 − y2 e3 − y3e2 + e2 e3 − ke1e2 = be1 + e2 + y1e3 + y3e1 − e1e3 − ke2e3 = − de3 + cy2 e2 + cy2 e2 − ce2 e2 − ke3(4)

Eq.(4) utilizes Laplace transform, the error system isprovided which is described by:sE1 ( s) − e1 (0) = aE2 ( s) − aE1 ( s) − L{y2 e3 }− L{y3e2 }+ L{e2 e3 }− kE1 ( s)sE2 ( s) − e2 (0) = bE1 ( s) + E2 ( s) + L{y1e3 }(5)+ L{y3e1}− L{e1e3 }− kE2 ( s )sE3 ( s ) − e3 (0) = −dE3 ( s ) + cL{y2 e2 }+ cL{y2 e2 }− cL{e2 e2 }− kE3 ( s)From Eq.(5) , the error system is obtained as follow:

Fig.1. Phase portraits of system (1) with a = 33, b = 50,c = 3 and d = 15 .

Fig.2. Lyapunov exponent spectrum of system (1) whend ∈ (0,30)

3. Unidirectionally coupled synchronization of
the chaotic system
In order to realize the unidirectionally coupled synchro-nization of the system, the drive system is defined asfollow:

 x1 = ax2 − ax1 − x2 x3
 x 2 = bx1 + x2 + x1 x3
 x3 = − dx3 + cx2 x2(2)and the response system can be described in the form of: E ( s ) = aE2 ( s ) − e1 (0) − L{y2 e3 }
 1s+k +a s+k +a s+k +a
L{y3e2 } L{e2 e3 }−+s+k +a s+k +a
L{y1e3 }bE1 ( s )e (0) E2 ( s ) =+ 2+s + k −1 s + k −1 s + k −1L{y3e1} L{e1e3 }+−s + k −1 s + k −1
e (0)
cL{y2 e2 } cL{y2 e2 }++
 E3 ( s ) = − 3s+k +d s+k +d s+k +d
cL{e2 e2 }−s+k+d

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Unidirectionally coupled synchronization andBased on the final value theorem of Laplacetransform, the error system is described in the form of:sL{y2 e3 }alim e1 (t ) = lim sE1 ( s ) =lim e2 (t ) − limt →∞s →0s →0 s + k + as + k + a t →∞+ lims →0

sL{y3e2 }sL{e2 e3 }+ lims + k + a s →0 s + k + a(7)sL{y1e3 }blim e2 (t ) = lim sE2 ( s ) =lim e1 (t ) + lim
0t →∞s →0t→∞s→s + k −1s + k −1s →0sL{y3e1}sL{e1e3 }− lim
s + k − 1 s →0 s + k − 1=0
lim e3 (t ) = lim sE3 ( s ) = c limt →∞s →0− c lims →0s →0sL{e2 e2 }s+k +dt →∞system(2) can synchronize with system(3) completely.
4. Numerical simulation=0+ lim

From Eq.(7), Eq.(8), and Eq.(9), the error system will
be asymptotically stable, which means that
lim ei (t ) = 0 (i = 1,2,3) . Theoretical analysis demonstrate(8)sL{y2 e2 }sL{y2 e2 }+ c lims →0 s + k + d
s+k +d(9)=0Fig.3. Variation of state variables with time: (a) ( x1 , y1 ) ;(b) ( x2 , y2 ) ; (c) ( x3 , y3 ) .

In following, numerical simulation of unidirectionallycoupled synchronization through designing theappropriate coupling coefficient is illustrated. The
system parameters are chosen as a=33, b=50, c=3, andd=15. The initial values of the drive system(2) and theresponse system(3) are taken as x1 (0) = −3, x2 (0) = 0,x3 (0) = 2, y1 (0) = −3, y2 (0) = −0.6, and y3 (0) = −1.6,respectively. When the coupling coefficient is taken
as k = 200, it can be observed that the drive systemand the response system are in asymptotic
synchronization in Fig.3. In addition, curves of statevariable are displayed in Fig.4. From Fig.4, we canobserve that the curve is a straight line through theorigin and xi = yi (i = 1,2,3) . The obtained numericalsimulation results agree with theoretical analysis results.

Fig.4. curves of state variable (a) ( x1 , y1 ) ; (b) ( x2 , y2 ) ;(c) ( x3 , y3 ) .

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5. Synchronization circuit design
In order to design the synchronous circuit, LF347N isselected as the amplifier, AD633JN is chosen as themultiplier and the power supply is ± 15 V. Because of
the restriction in the operating voltage of amplifier andmultiplier, the state variable are decreased by 10 times,
which means ( x1 , x2 , x3 ) → (10 x2 ,10 x2 ,10 x3 ) and( y1 , y2 , y3 ) → (10 y2 ,10 y2 ,10 y3 ) .
For the drive system, the circuit equation is as follows:111( − x2 ) −
x2 x3
x1 −
 x1 = −10R
CRCR
3C12 11 1(10)1
11( − x2 ) −(− x1 ) x3(− x1 ) −
 x2 = −10 R6C2
R5C2R4C2
11
 x3 = − R C x3 − 10 R C (− x2 ) x2
8 3
7 3

The circuit equation of the response system is as follow:1111( − y2 ) −(− x1 ) −y1 −y 2 y3
 y1 = −
10R
CR
C
R
C
R
15 4
16 4
17 4
18C4

(11)
1111

(− y1 ) −( − x2 ) −(− y1 ) y3y2 − y 2 = −10 R22C5
R19C5
R20C5
R21C5

111
 y3 = − R C y3 − R C (− x3 ) − 10 R C (− y2 ) y223 624 625 6
When a =33, b=50, c=3 and d=15, the circuit schematicof synchronization system is given in Fig.5. Where
R5 = 1000 kΩ, R7 = 66.67 kΩ, R15 = 4.29 kΩ,R23 = 4.65 kΩ, R4 = R20 = 20 kΩ, R8 = R25 = 3.33 kΩ,R1 , R2 , R16 = 30.3 kΩ,R17 , R19 , R21 , R24 = 5 kΩ,R3 , R6 , R9 , R10 , R11 , R12 , R13 , R14 , R18 , R22 , R26 , R27 , R28 , R29=10kΩ, C1 , C2 , C3 , C4 , C5 , C6 = 10nF . The value ofR17 , R21 and R24 of circuit schematic in respond systemcorrespond to coupling coefficient k . The values of
R1 , R4 , R8 and R7 of circuit schematic in drive systemcorrespond to the value of a, b, c and d , while we also
can change the values of a, b, c and d by adjusting
R15 , R20 , R25 and R23 of circuit schematic in respondFig.5. Circuit diagram of synchronization systemsystem. The synchronization circuit result is shown inFig.6 through the oscilloscope TDS2014B is consistent
with the numerical simulation result. From Fig.6, we
can observe that the curve is a straight line through theorigin and xi = yi (i = 1,2,3) , indicating that system(2)and system(3) achieve synchronization completely.

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Unidirectionally coupled synchronization andAcknowledgement
This work is supported by the Young Scientists Fund of
the National Natural Science Foundation of China(Grant No.11202148).
References1.

Fig.6. circuit simulation diagram (a) ( x1 , y1 ) ; (b) ( x2 , y2 ) ;(c) ( x3 , y3 ) .

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6. Conclusion
In this paper, a novel three-dimensional autonomouschaotic system is discussed. The synchronization of theproposed chaotic system through selecting suitablecoupling coefficient is investigated. The presented
synchronization method is simple and the coupling
coefficient is adjusted easily. On this basis, the analog
synchronization circuit of the system is designed andrealized by Multisim software. The circuit simulation
results are coincide well with the theoretical analyses
and the numerical simulation results, which can further
verify the effectiveness and feasibility of the proposedsynchronization scheme.

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