Chaos, Solitons and Fractals 21 (2004) 69–74

www.elsevier.com/locate/chaos

Experimental confirmation of a new chaotic attractor

Fengling Han a,c,*, Yuye Wang b, Xinghuo Yu a,c, Yong Feng b

a School of Electrical and Computer Engineering, Royal Melbourne Institute of Technology, GPO Box 2476V, Melbourne,

Vic. 3001, Australiab Department of Electrical Engineering, Harbin Institute of Technology, Harbin 150001, PR China

c Faculty of Informatics and Communication, Central Queensland University, Rockhampton, Qld 4702, Australia

Accepted 19 September 2003

Abstract

This letter reports the experimental confirmation of a new chaotic attractor, which is a transition system between the

Lorenz and the Chen systems. It is noticed that this realization is different from that of Chen’s system. By adjusting an

adjustable resistor in this simple circuit, both Lorenz and Chen attractors can be observed by oscilloscope. Experi-

mental results verify the effectiveness of the new chaotic attractors.

� 2003 Published by Elsevier Ltd.

1. Introduction

Since the discovery of Chua’s circuit, chaos in electronic systems has been studied extensively [1]. In recent

years, anti-control of chaos, or chaotification [2] has attracted increasing attentions. It is known that chaos can be

generated from various systems via different methods. The RC realization of Chua’s circuit based on the simplest

possible models for second-order RC sinusoidal oscillators was designed in [3]. The improved implementation of

Chua’s chaotic oscillator using the current feedback operational amplifiers was proposed in [4]. There was a

systematic implementation for hysteresis chaotic generators reported [5]. Yalcin et al. designed some simple circuits

for generating n-scroll attractors and scroll grid attractors [6,7]. Chen’s attractor was produced in [8], and its

bifurcation analysis was studied in [9]. Chen’s attractor, is topologically more complex than the Lorenz’s attractor,

belongs to another canonical family of chaotic systems. The circuitry implementation and synchronization of

Chen’s system was also reported [10]. Linear feedback technique and switching manifold approach were also

used in generating chaos [11,12]. L€u and Chen [13] reported a new chaotic system, for statement convenience,

named L€u system in this letter, which bridges the gap between the Lorenz and the Chen systems. L€u system is a

transition system between the Lorenz system and the Chen’s system. The dynamical behavior was studied in

their late work [14]. In [15] the parameter identification and backstepping control of uncertain L€u system was

done.

In this letter, the L€u system is experimental confirmed by an electronic circuit. This design is different with that of

Chen’s system. Furthermore, it is easier than the implementation of Chen’s attractor in [10], and can demonstrate both

Lorenz and Chen’s attractors in the same circuit by change the value of an adjustable resistor. Experimental results also

verify the effectiveness.

* Corresponding author. Address: School of Electrical and Computer Engineering, Royal Melbourne Institute of Technology, GPO

Box 2476V, Melbourne, Vic. 3001, Australia. Tel.: +61-3-992-553-60; fax: +61-3-992-543-43.

E-mail addresses: s3044014@student.rmit.edu.au, f.han@cqu.edu.au (F. Han), x.yu@rmit.edu.au (X. Yu), yfeng@hope.hit.edu.cn

(Y. Feng).

0960-0779/$ - see front matter � 2003 Published by Elsevier Ltd.

doi:10.1016/j.chaos.2003.09.045

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Fig. 1. The x–y–z trajectories of the L€u system: (a) c ¼ 13:0, (b) c ¼ 20:0, (c) c ¼ 28:0.

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-1

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Fig. 2. x–y trajectories of the L€u system: (a) c ¼ 13:0, (b) c ¼ 20:0, (c) c ¼ 28:0.

70 F. Han et al. / Chaos, Solitons and Fractals 21 (2004) 69–74

F. Han et al. / Chaos, Solitons and Fractals 21 (2004) 69–74 71

2. The new chaotic attractor

In this section, the model of the new chaotic attractor––L€u system was reviewed. The relationship among L€u system,

the Lorenz and Chen attractors is further discussed.

Lorenz system is described by the following equation:

_x ¼ aðy � xÞ;_y ¼ cx� xz� y;_z ¼ xy � bz;

8<: ð1Þ

and the Chen system was obtained from the equation

_x ¼ aðy � xÞ;_y ¼ ðc� aÞx� xzþ cy;_z ¼ xy � bz;

8<: ð2Þ

while the mathematical model of L€u system can be described by the following three-dimensional autonomous system:

_x ¼ aðy � xÞ;_y ¼ �xzþ cy;_z ¼ xy � bz;

8<: ð3Þ

where x, y and z are the state variables; a, b and c are constants.

Fig. 3. The circuit implementation of the L€u system.

Fig. 4. Oscilloscope observations for the x–y trajectories of the L€u system: (a) c ¼ 13:0, (b) c ¼ 20:0, (c) c ¼ 28:0.

Fig. 5. Oscilloscope observations for the x–z trajectories of the L€u system: (a) c ¼ 13:0, (b) c ¼ 20:0, (c) c ¼ 28:0.

72 F. Han et al. / Chaos, Solitons and Fractals 21 (2004) 69–74

F. Han et al. / Chaos, Solitons and Fractals 21 (2004) 69–74 73

It has been found that the Lorenz system (1) and the Chen system (2) are classified as two opposite classes of chaotic

system [16]: for the linear part A ¼ ½aij�, a12a21 > 0 for Lorenz system; while Chen’s system satisfies a12a21 < 0. However,

L€u system (3) satisfies a12a21 ¼ 0. L€u system is one of the simplest chaotic systems that bridge the gap between the

Lorenz and the Chen systems.

When the parameters a ¼ 36 and b ¼ 3 are fixed while parameter c varies, one can observe that the attractors

generated by L€u system (3) is similar to the Lorenz attractor in Eq. (1) for 12:7 < c < 17:0; it has a transitory shape

when 18:0 < c < 22:0, and then with the increase of parameter c, it becomes similar to Chen’s attractor in Eq. (2) when

23:0 < c < 28:5. The simulation results of x–y–z trajectories and the x–y trajectories of these three cases corresponding

three different parameters c ¼ 13:0, 20.0 and 28.0 are depicted in Figs. 1 and 2, respectively.

3. Circuitry implementation of the new chaotic attractor

In this section, an electronic circuit is designed to realize the new chaotic attractor––L€u system.

The designed circuit is shown in Fig. 3. Operational amplifiers, linear resistors and capacitors are used to realize the

function of L€u system as required. The operational amplifiers AD713 are adopted in this design. The four-quadrant,

analog multiplier AD633 is adopted to realize the multiplication. The three state variables, x, y and z, are obtained from

the outputs of U1B, U2B and U2C, respectively.

The adjusting parameter c in L€u system can be implemented by adjusting the resistor R25 in Fig. 3. If R25 ¼ 30:8 kX,

it is corresponding to c ¼ 13, Lorenz attractor can be observed from the L€u system shown in Fig. 3; R25 ¼ 20 kXcorresponding to c ¼ 20, it is transition state between Lorenz and Chen attractors; and R25 ¼ 14:3 kX, corresponding to

c ¼ 28, Chen attractor can be observed from the L€u system. An oscilloscope observed x–y trajectories of L€u system with

different c are shown in Fig. 4, x–z trajectories of L€u system with different c are shown in Fig. 5. The digital storage

oscilloscope is used to record the waveforms. The scale of each connector is shown in the upper-left hand of the

corresponding figures.

The design is simple, only two AD713 (U1 and U2), two AD633 (U3 and U4), some linear resistors and capacitors

are used. By changing the value of adjustable resistor R25 the Lorenz and Chen’s attractors can be observed from the

oscilloscope.

4. Conclusions

In this letter, a novel electronic circuitry design is proposed for the realization of the new chaotic attractor––L€usystem, which connects the Lorenz attractor and the Chen’s attractor, and represents the transition from one to the

other. L€u system is one of the simplest chaotic systems that bridge the gap between the Lorenz and the Chen systems.

Experimental results verify the effectiveness of this electronic circuitry design.

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