Seven Dimension Chaotic System and Its Circuit Implementation

Jianliang Zhu1,a, Chunyu Yu2,b

1 College of Electrical & Electronic Engineering, Harbin University of Science and Technology,

Harbin, China

2 Electrical and Electric Training Center, Harbin University of Science and Technology, Harbin, China

aemail:zhu_jianliang@163.com,

bemail:honeychunyu@126.com

Keywords: seven dimension chaotic system; numerical simulation; circuit implementation

Abstract. In order to generate more complex chaotic attractors, a seven-dimensional chaotic system

is constructed, and relevant chaotic attractors can be obtained by Matlab numerical simulation.

Lyapunov exponents validate that the system is chaotic. Implementation circuit of this system is

designed, and circuit simulation can be done by using Multisim. Circuit simulation result is identical

to system simulation completely. Chaotic behavior of the system is proved farther. A new chaotic

signal source is provided for practical application based on chaos such as secrecy communication and

signal encryption fields.

Introduction

The research of multidimensional chaotic system is a advanced topic of nonlinear theory, and this

field has a great progress in recent years. The design of multidimensional chaotic system and its

circuit implementation problems are concerned [1-3]. Guangyi Wang and others constructed a

three-dimensional chaotic system and its circuit implementation [4], Fanzhen Wang and others gave a

four-dimensional chaotic attractor [5], Jianliang Zhu and others present five-dimensional and

six-dimensional chaotic system and relevant circuit implementation [6-7].

Seven-dimensional chaotic system and relevant circuit implementation is present, and Matlab

simulation result is identical to designed circuit simulation result completely. A new chaotic signal

source is provided for practical application based on chaos such as secrecy communication and signal

encryption fields.

Seven dimension chaotic system

The equation of seven dimension chaotic system is:

1 1 6 2 3 4 5 6 7

2 1 2 1 3 4 5 6 7

3 2 3 5 1 2 4 5 6 7

4 4 6 1 2 3 5 6 7

5 5 6 1 2 3 4 6 7

6 5 6 1 2 3 4 5 7

7 6 7 1 2 3 4 5 6

x ax bx cx x x x x x

x bx bx x x x x x x

x bx ax bx x x x x x x

x dx ex x x x x x x

x fx gx x x x x x x

x ex hx x x x x x x

x bx ax ix x x x x x

= − + + = − − −

= − − + = − − − = − + +

= + − = − +

�

�

�

�

�

�

�

(1)

Where a=15, b=20, c=0.785, d=5, e=10, f=30, g=19, h=3.9, i=1.5.

By analyzing Lyapunov exponents of seven-dimensional chaotic system, the result was shown in

Fig. 1.

LE1=8.78458, LE2= －5, LE3= －14.9014, LE4= －15.002， LE5= －15.067, LE6= －20.0311,

LE7= －34.8821, so the system is chaotic status.

Advanced Materials Research Vols. 588-589 (2012) pp 1251-1254Online available since 2012/Nov/12 at www.scientific.net© (2012) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMR.588-589.1251

All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP,www.ttp.net. (ID: 130.15.241.167, Queen's University, Kingston, Canada-14/09/13,09:28:15)

For system (1)

3 5 6 71 2 4

1 2 3 4 5 6 7

= 66x x x xx x x

V a b a d f g ax x x x x x x

∂ ∂ ∂ ∂∂ ∂ ∂∇ = + + + + + + − − − − + − = −∂ ∂ ∂ ∂ ∂ ∂ ∂

� � � �� � � (2)

That is, every volume element including system track converges with index rate e-66t. All system

track finally will be limited to a limit point aggregate with zero volume, and its asymptotic dynamic

behavior will be fixed on a attractor, which shows the existence of chaotic attractors.

Fig. 1 Lyapunov exponents of the system

Matlab Simulation of the System

By Matlab simulation, the seven-dimensional system has all two-dimensional and three-dimensional

chaotic attractors, some of attractors are shown in Fig. 2.

(a)x1-x2 attractor (b) x2-x3 attractor (c) x3-x4attractor (d) x4-x5 attractor

(e) x5-x6 attractor (f) x6-x7 attractor (g) x1-x2-x3 attractor (h) x2-x3-x5 attractor

(i) x2-x5-x6 attractor (j) x3-x4-x5 attractor (k) x5-x6-x7 attractor

Fig.2 Results of MATLAB simulation

1252 Advances in Mechanics Engineering

Circuit implementation and Simulation Result

Implementation circuit of seven-dimensional system is shown in Fig. 3. Reverse input operation

circuit is used, and element parameters are in accord with system parameters, the multiplier is

composed of five AD633.

Fig.3 Implementation circuit of the system

Multisim simulation results of the circuit are shown in Fig. 4.

(a) x1-x2 attractor (b) x2-x3 attractor (c) x3-x4 attractor

(d) x4-x5 attractor (e) x5-x6 plane attractor (f) x6-x7 attractor

Figure 4. Multisim simulation results of the circuit

Advanced Materials Research Vols. 588-589 1253

The results are identical to Matlab simulation completely. Time domain waveforms of system

parameters are shown in Fig. 5.

(a) (b) (c) (d)

(e) (f) (g)

Figure 5. Time domain waveforms of the system variables

((a) is x1 time domain waveform; (b) is x2 time domain waveform; (c) is x3 time domain waveform;

(d) is x4 time domain waveform; (e)is x5 time domain waveform; (f) is x6 time domain waveform;

(g) is x7 time domain waveform)

Conclusion

The seven-dimensional chaotic system introduced above have larger Lyapunov exponents and more

complex dynamic behavior. Numerical simulation results is identical to Circuit simulation results

completely, both present same chaotic attractors. Reverse input circuit is used in designed chaotic

circuit, and circuit parameters and system parameters are correspondence one by one. The designed

circuit structure is simple, easy to be realized, convenient for debugging. Chaotic system and chaotic

circuit, which present in the paper, can be regarded as a new source of chaotic signal to be applied in

chaotic communication and chaotic encryption fields.

Acknowledgement

This work was supported by the Research Foundation of Education Bureau of Heilongjiang Province

(Grant No. 12511093)

References

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1254 Advances in Mechanics Engineering

Advances in Mechanics Engineering 10.4028/www.scientific.net/AMR.588-589 Seven Dimension Chaotic System and its Circuit Implementation 10.4028/www.scientific.net/AMR.588-589.1251