Procedia Engineering 29 (2012) 1264 – 1269

1877-7058 © 2011 Published by Elsevier Ltd.doi:10.1016/j.proeng.2012.01.124

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Procedia Engineering 00 (2011) 000–000

ProcediaEngineering

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2012 International Workshop on Information and Electronics Engineering (IWIEE)

A Novel Four-Wing Hyper-Chaotic System and Its Circuit Implementation

Xue Weia*, Fang Yunfeib, Li Qiangc

aDepartment of Automation，Tianjin University of Science and Technology,Tianjin300222, China bDepartment of Automation，Tianjin University of Science and Technology,Tianjin300222, China

cCollege of Science, China Agricultural University, Beijing 100083, China

Abstract

This paper presents a new continuous-time four-dimensional autonomous hyper-chaotic system based on the generalized augmented Lü system. This system can generate a four-wing hyper-chaotic attractor over a large parameter range. It possesses abundant dynamics characteristics. The existence of this hyper-chaotic attractor is verified through theoretical analysis, numerical simulation and circuit implementation.

© 2011 Published by Elsevier Ltd.

Keywords: hyperchaos; four-wing attractor; augmented Lü system; circuit implementation;

1. Introduction

In 1979, Rössler proposed the first autonomous hyper-chaotic system obtained by computer simulation [1]. Because orbits of hyper-chaotic attractor extend in more directions, its dynamic behavior is more complex and unpredictable. It has a great potential of application in engineering. Generating and implementation of new hyper-chaotic system has become a hotspot, plenty new systems has been introduced [2-5]. However, systematic method of generating hyper-chaos has not been found yet. Wang J Zh,Chen Z Q et al.[6] summarizes some methods used in generating hyper-chaos currently, one of which is by introducing a condition feedback controller to a continuous three-dimensional autonomous system. A considerable amount of research has been done in this area [7-13]. However, it is still not easy to obtain a hyper-chaotic system with simple structure and complex dynamic behaviors. This paper generates a new

* Corresponding author. E-mail address: kdxuewei@126.com

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continuous four-dimensional autonomous hyper-chaotic system based on the generalized augmented Lü system. The new system is investigated via Lyapunov exponents,bifurcation diagram and phase portraits. It has not only complex dynamics characteristics but also large parameter regions of hyper-chaotic. A four-wing hyper-chaotic attractor is observed numerically,and an analog circuit is built to realize the new system.

2. Generation of a four-wing hyper-chaotic system

Qiao and Bao[14] introduced a condition feedback controller to the third equation of augmented Lü system, constructing a new three-dimensional autonomous system which can generate a four-wing chaotic attractor. The mathematical model is listed as:

⎪⎩

⎪⎨

⎧

++=+=

−+−=

cxxybzzxzayy

yzxbaabx

&

&

& )]/([

(1)

The system has similar dynamics characteristics when parameter c varies in positive and negative interval. When 0=c , the system degrades to the augmented Lü system. So it is called the generated augmented Lü system. Here by adding a state feedback w to the second equation, and a xw to the third equation of system (1), a new four-dimensional system is constructed as follows:

⎪⎪⎩

⎪⎪⎨

⎧

=+++=

++=−+−=

dywxwcxxybzz

wxzayyyzxbaabx

&

&

&

& )]/([

(2)

Where x, y, x, z are state variables, a, b, c, d are the parameters. When given proper parameters, one obtains a four-wing hyper-chaotic attractor .It possesses abundant dynamics characteristics. Theoretical analysis and numerical simulation are given and an analog circuit is built to implement the system.

3. Basic characteristics analysis of system (2)

3.1. Dissipativity

The dissipativity of system (2) is described as:

babaabzz

yy

xxV +++−=

∂∂

+∂∂

+∂∂

=Δ )/(&&&

(3)

So, when a, b satisfy a+b<ab/(a+b), the system is dissipative. By solving the equilibrium equation of system (2), it can be easily observed that the system has only

one equilibrium point )0,0,0,0(=S .By linearizing system (2) at S, we obtain the Jacobian

1266 Xue Wei et al. / Procedia Engineering 29 (2012) 1264 – 1269 Author name / Procedia Engineering 00 (2011) 000–000 3

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡+

−

=

000

00

100

000

dbc

aba

ab

J (4)

According to the Jacobian (3), the characteristic equation is given as

0))()(( 2 =−−−+

+ dabba

ab λλλλ (5)

The corresponding eigenvalues are:

24

24

2

4

2

3

2

1

daa

daa

bba

ab

+−=

++=

=+

−=

λ

λ

λ

λ

(6)

Obviously, the engenvalues are independent from parameter c. When 0,0,0 <<< dba , Equilibrium S is unstable. It is possible to generate chaos or hyper-chaos in system (2)

4. Numerical simulation of system (2)

Fixing parameters 1114 -, d, c-b === , Fig.1 shows the Lyapunov exponents spectrum and bifurcation diagram of system (2) with ]20-200[ ，−∈a (the smallest LE, always less than -20, is neglected in the figure).

Fig.1.(a) Lyapunov exponent spectrum with a varies(b)bifurcation diagram in y-direction with a varies

1267Xue Wei et al. / Procedia Engineering 29 (2012) 1264 – 12694 Author name / Procedia Engineering 00 (2011) 000–000

In Fig.1 (a), we observe that there are two positive Lyapunov exponents over a large range of parameters, which implies that the system is hyper-chaotic over a broad range. Generally, chaotic and hyper-chaotic systems are distinguished by the Lyapunov exponents. For a continuous-time four-dimensional autonomous system, among its four Lyapunov exponents, if one of them is zero, the other three are negative, the system is periodic; if two are zero, the other two are negative, it is quasi-periodic; if one of them is positive, one is zero and the other two are negative, it is chaotic; if two of them are positive, one is zero, one is negative, the system is hyper-chaotic. From Fig.1 we can except:

when ]2.73,120[ −−∈a and ]66,4.69[ −−∈a , system (2) is a periodic attractor; when ]61,66[ −−∈a ,it is quasi-periodic attractor; when ]21,4.22[ −−∈a ,it is chaotic attractor; when ]4.22,61[ −−∈a and ]20,21[ −−∈a ,it is hyper-chaotic attractor. To observe the orbits of system (2), we give some typical attractors of system (2) by

selecting 223572110 -, a-a-, a-a ==¬== .The Phase portraits are shown in Fig.2.

-100 -80 -60 -40 -20 0 20 40 60 80 100-60

-40

-20

0

20

40

60

x

y

0 10 20 30 40 50 60 70 80 90 100-2.5

-2

-1.5

-1

-0.5

0

0.5

z

w

-200-100

0100

200

-100

0

100

200-150

-100

-50

0

50

100

150

xy

z

-150 -100 -50 0 50 100 150-100

-50

0

50

100

150

x

y

(a) a=-110,periodic orbit

-80 -60 -40 -20 0 20 40 60 80-50

-40

-30

-20

-10

0

10

20

30

40

50

x

y

0 10 20 30 40 50 60 70 80-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

z

w

-100 -50 0 50 100 150-150

-100

-50

0

50

100

150

y

z

-15 -10 -5 0 5 10 15-150

-100

-50

0

50

100

150

w

z

(b)a=-72,quasi-periodic attractor (c)a=-35,hyper-chaotic attractor

-150 -100 -50 0 50 100 150-150

-100

-50

0

50

100

x

y

-150 -100 -50 0 50 100 150-15

-10

-5

0

5

10

15

z

w

(d)a=-22,chaotic attractor

Fig.2.Phase portraits of system (2)

Lyapunov exponent spectrum, bifurcation diagram and phase portraits demonstrate that the system can generate complex dynamics behaviors. A four-wing hyper-chaotic attractor is observed over a wide range of parameters

5. Circuit implementation

An analog circuit has been designed to realize system (2).The circuit diagram is shown in Fig.3,where KΩ,R,R,R,R,R,R,R,R,R,RR 102224232019121110542 = , KΩ,R,R,R,R,R,RR 100252117153176 = ,

KΩ,R,R,RR 1181691 = , KΩR 253 = , Ω= KR 1258 , KΩ.R 1714 = , F1C~C 41 μ= . Select LF347N as the

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amplifier, AD633 as the multiplier respectively. The equations should be linearly transformed properly, considering the maximum allowable voltage for LF347 is 15± V. Here the output is decreased by 20 times. Let )20,20,20,20(),,,( wzyxwzyx → .The resistance R8 corresponds to parameter a, R14

corresponds to b, R3 corresponds to b)ab/(a +− .Changing values of b)/(a a, b, -ab + can be achieved by

regulating values of ,R,R R 3148 respectively.

Fig.4 shows the experimental output observed on an oscilloscope. Compared with numerical simulation in Fig.2, it is easy to find that they are basically the same, verifying the hyper-chaotic attractor on physical level.

Fig.3. Circuit diagram of system (2)

Fig.4. phase portraits of system(2) observed on the oscilloscope

6. conclusions

This paper proposes a new continuous-time four-dimensional autonomous hyper-chaotic system based on the generalized augmented Lü system. A four-wing hyper-chaotic attractor is obtained. Numerical simulation and circuit implementation proves the existence of the hyper-chaotic attractor. This system performs good properties such as bigger positive Lyapunov exponent, rich in dynamics characteristics etc.

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This system adds to the current hyper-chaotic system library, and is anxiety to be applied in engineering like communication encryption.

Acknowledgement

This work is supported by the grants: the National Natural Science Foundation of China (Grant Nos.10772135, 60874028), the Scientific Research Staring Foundation for the Returned Overseas Chinese Scholars, Ministry of Education of China.

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