SUGISAKAMASANORI

Journal of Advances in Artificial Life Robotics
Vol. 1(3); December (2020), pp. 133–137
ON LINE ISSN 2435-8061; ISSN-L 2435-8061
https://alife-robotics.org/jallr.html

Research Article

Analysis and Circuit Design of a Novel 4D Chaotic System
Yan Sun, Yongchao Zhang, Jiaqi Chen

Department of Automation, Tianjin University of Science and Technology, 1038 Dagunanlu Road, Hexi District, Tianjin 300222, PR China

ARTICLE INFO
Article History
Received 11 November 2019
Accepted 23 August 2020
Keywords
Novel 4D chaotic system
Dynamic characteristics
Simulation analysis
Analog circuit desigABSTRACT

In this paper, a novel 4D chaotic system is proposed. First, the basic dynamic characteristics ofthis system are analyzed theoretically. Second, dynamical properties of the system areinvestigated by phase portraits, Poincaré sections, Lyapunov exponent spectrum and bifurcation
diagram. In addition, an analog circuit of the system is designed and simulated by Multisim, acircuit simulation software. The experimental results show that the circuit simulation results areconsistent with the numerical simulation results. Therefore, the existence of chaotic attractor in
the system is verified from the physical level.

© 2022 The Author. Published by Sugisaka Masanori at ALife Robotics Corporation Ltd.
This is an open access article distributed under the CC BY-NC 4.0 license(http://creativecommons.org/licenses/by-nc/4.0/).

Through chaos theory, we can not only further
investigate the nonlinear characteristics and theirgeneration mechanism in these fields, but also provide atheoretical basis for the construction of new chaoticsystems.
In this paper, a novel 4D chaotic system is proposed.
Firstly, the dissipation and equilibrium of the system aretheoretically analyzed. Secondly, simulation analysis iscarried out through phase portraits, Poincaré section
diagrams, Lyapunov exponent spectrum and bifurcation
diagram to analyze the dynamic characteristics of thesystem in detail. Then, in order to illustrate the existenceof chaotic attractor in the system from the physical level,
the circuit design and simulation are carried out with thecircuit simulation software Multisim. The existence of
chaotic attractor in the system is further verified in theexperiment.

1. Introduction
Since 1963, when Lorenz discovered the butterflyeffect [1], more and more scholars have studied chaos. In
recent decades, with the development of mathematicaltheory, chaos research has also made great progress withemerging a number of newly built systems. For example,in 1993, Chua et al. designed Chua circuits in which themathematical model became a very classical chaotic
system [2]. And Sprott J et al. proposed Sprott system inthe following year [3]. In 1999, Chen system wassuccessfully constructed by Chen et al. [4] In the early
21century, Lü et al. discovered the Lü system [5]. A
three-dimensional continuous autonomous chaoticsystem was constructed by Liu et al. called Liu system[6]. Qi et al. proposed a four-dimensional chaos system,
namely Qi system ]. Except to studying how to constructa system with better dynamic performance, on the basisof existing systems, many scholars also investigate andrealize chaotic system through numerical analysis,
analog circuit implementation, digital circuit
implementation and other methods [7],[8],[9],[10],[11].
In recent years, chaos theory has been applied in many
fields, including information studying encryption andsecure communication [12],[13],[14], astrophysics [15],electromagnetic mechanics [16] and economics [17].

2. 4D System ModelIn this paper, a novel 4D chaotic system is proposed.
The system equation of motion is as follows:−cx + ay + xz + yw
 x = y =−ax + bw
(1)− x 2 + w2
z = w =− xy − by − zw + u

Corresponding author’s E-mail: sunyantjkj@qq.com URL: www.tust.edu.cn133

In formula (1), x, y, z, w are the system state variables,a, b, c are the system adjustable parameters, u is theexternal input of the system, and all four numbers arepositive.
2.1 Dissipation

150

150100100505000-50-50-100-100-150-150

The divergence of formula (1) is:∂x ∂y ∂z ∂w
∇V = ++ +=−c + z − z =−c∂x ∂y ∂z ∂w(2)-50050100150-150-150-100-500(a)(b)
Fig.2 Poincaré sections of system (1):(a) w = 0; (b) y = 0.

50100150

As can be seen from the Poincaré sections, the map ofsystem (1) under these two sections are many points with
dense distribution and fractal structure. This indicates thechaotic behavior occurs in this system.

The parameter c is positive, so the divergence ofsystem (1) is negative. This indicates the system (1) is adissipative system.

2.3.2 Lyapunov exponent spectrum andbifurcation diagram

2.2 Equilibrium point
Let all the left sides of equation (1) are 0, then equation(1) becomes:−cx + ay + xz + yw
0 =
0 =
−ax + bw
− x 2 + w2
0 =
0 =
− xy − by − zw + u-100The most common way to determine whether a system
is chaotic is to determine the number of positive
Lyapunov exponent of a system. For a 4D system, if there
are two positive Lyapunov exponents, the system ishyperchaotic. If there is one positive Lyapunov exponent,
the system is chaotic. As analyzing this chaotic system,
the fixed parameter is a = 50, b = 25, u = 70, and theinitial values are (1,1,1,1). The Lyapunov exponent
spectrum and bifurcation diagram of system (1) changing
with the parameter c in the range (0.3,1) are made. The
results are as shown in Fig.3(a) and Fig.3(b) respectively.

(3)

Through formula (3), calculate the equilibrium pointof system (1). It is found that there is no equilibrium pointin the system. A chaotic attractor generated by a system
without equilibrium point is a hidden chaotic attractor.
2.3 Basic dynamic analysis

Fig.3.Lyapunov exponent spectrum of system (1)2.3.1Phase portraits and Poincaré mapping
diagram
The parameters and external input are a = 50, b = 25,c = 0.47, u = 70, the initial values are (1,1,1,1), the phase
portraits of system (1) are shown in Fig.1.

(a)

100

15080Fig.3 (a): Lyapunov exponent spectrum and (b):bifurcation diagram of system (1).

5040w

20y(b)

100

60

0-50

0-100

-20-150-40

100
50-60-50-50

z

-100
-100

-50

050

By combining the Lyapunov exponent spectrum andbifurcation diagram, it can be analyzed that whenc ∈ (0.3, 0.74) , system (1) is chaotic; When c ∈ (0.74,0.783) , system (1) is quasi-periodic. When c ∈ (0.783,1) , system (1) is periodic. Select c = 0.47, c = 0.7224, c= 0.76 and c = 0.9 of system (1) in these three ranges
respectively. Make the corresponding phase portraits, theresults are as shown in Fig.4. When c = 0.47, the four
Lyapunov exponents are LE1 = 2.5001, LE2 = 0, LE3 = –0.2192, LE4 = –2.8070, which accord with the feature ofchaotic state. When c = 0.7224, the four Lyapunov

100500-80-100-100-150

0x100x(b)(a)Fig.1 Phase portraits of system (1) with a = 50, b = 25, c =0.47, u = 70: (a) x − y ; (b) x − z − w .

As can be seen from Fig.1, chaos is generated in thesystem with the selected parameters and initial values. In
order to further illustrate that the state is chaotic indeed,
the Poincaré mapping diagrams under the same
parameters and initial values are given. w = 0 and y = 0
are the selected sections, and the results are shown inFig.2(a) and Fig.2(b).

134

Now the circuit is designed with the improved method.
The circuit is standardized in advance, and theparameters are applied into equation (5). Then, equation(5) becomes:

00-50-50-100-100-100-50

050-150-150100x-100-50050

100−47 x − 5000(− y ) − 1000(− x) z − 1000(− y ) w
 x = y =−5000 x − 2500(− w)(7)−1000 x 2 − 1000(− w) w
z = w =−1000 xy − 2500 y − 1000 zw − (−7000)

150x(a)(b)100808060

The circuit model built according to equation (7) isshown in Fig.5. The four parts of the circuit correspondto four first nonlinear differential equations respectively,
and the circuit design is realized by using operational
amplifiers—LF347N chips, multipliers—AD633 chips,
resistors and capacitors [18],[19].

604040200zz

20-200-20-40-40-60-60-80-100-60-40-20020

40x60

80

100120(6)In order to observe the phase portraits of system (1) inthe oscilloscope, time scale transformation is required.
Let t τ==100 and use t to replace T, so:0T ,τ 0-80-50

050

100-yx(c)(d)Fig.4 The phase portraits with different values of parameter c:(a) c = 0.47; (b) c = 0.7224; (c) c = 0.76; (d) c = 0.9.

x

R19R320kΩ

-x

2.13MΩ
C1A1YzX

10nF
IC=1V

0.1V/V 0V-yA4Yw

R110kΩ
R2X

R52
1310kΩ

0.1V/V 0V

R4V115V

1kΩ
U1A
1150

1kΩ

LF347N

U2A-x

2V215V

13

41005011150

100−0.47 x − 50(− y ) − 10(− x) z − 10(− y ) w
 x = y =−50 x − 25(− w)−10 x 2 − 10(− w) w
 z = w =−10 xy − 25 y − 10 zw − (−70)
4150zz

exponents are LE1 = 1.2961, LE2 = 0, LE3 = –0.2706, LE4= –1.7442, which accord with the feature of chaotic state.
When c = 0.76, the four Lyapunov exponents are LE1 = 0,
LE2 = 0, LE3 = –0.3503, LE4 = –0.4908, which accord
with the feature of quasi-periodic state. When c = 0.9, thefour Lyapunov exponents are LE1 = 0, LE2 = –0.2664, LE3= –0.3201, LE4 = –0.3752, which accord with the featureof periodic state. These features indicate that the dynamic
behavior of the system is very divers.

LF347Ny-w
C240kΩ
R63. Analog circuit design and implementation

20kΩ

R9R10
11kΩ

LF347NU4A-y2V415V134

4V315V

1kΩ
U3A

23

Multisim software is used to design and realize ananalog circuit of a system. The selected parameters are a= 50, b = 25, c = 0.47, u = 70, and the initial values are(1,1,1,1). The rated voltage of the power supply in thecircuit is ±15V , so the linear dynamic range of theoperational amplifier is ±13.5V . But the chaotic attractor
range of the system is much larger than ±13.5V ,therefore, the spatial proportional compression
transformation should be carried out for the system (1).
Let:x = 10X, y = 10Y, z = 10Z, w = 10W

10nF
IC=1V
11x

11R20

LF347N

zA2Yx

C3X

R7A510kΩ
R8X

R141310kΩ

V515V1kΩ

LF347N

U6A-z

2

V615V

1
340.1V/V 0V

R131kΩ
U5A

24Y-w

10nF
IC=1V
110.1V/V 0Vw

11xLF347Nw

yA3C4Xw

R1210kΩyY

R15A640kΩ
R21X

R1813V815V

LF347N

1kΩ

U8A2-w

V915V

13LF347N

10kΩ

0.1V/V 0V
V71V(4)R171kΩ
U7A

24z

1110nF
IC=1V

0.1V/V 0V

11Y

4x

R11147kΩ

Fig. 5 Circuit diagram of system (1).

The equation of the four states realized by the circuit
can be obtained from Fig.5. The result is such as equation(8):

Put equation (4) into equation (1) and replace X, Y, Zand W with x, y, z and w again, then:−cx + ay + 10 xz + 10 yw
 x = y =−ax + bw
(5)
−10 x 2 + 10 w2
 z =
 w =−10 xy − by − 10 zw + u


135

111

x−
(− y ) −(− x) z−
 x =
R3C1
R13C1
10 R1C11(− y ) w
 −R
102 C1
11x−(− w)− y =
RCR

6 2
20 C2

11
 z =x2 −w2−
10 R7 C3
10 R8C3

1
11 w =xy −y−zw−
10 R12 C410 R21C4R15C4
 − 1

R11C4


(8)

By equating equation (7) and equation (8), it can beobtained that:

 1
 R C = 47
 3 1
 11
 = = 5000
 R13C1 R6 C2
 1
1
 = = 2500
 R20 C1 R15C2
 111
 = = = 1000
RCR
CR
1010101 1
2 17 C3 1
11
= = = 1000
10 R8C3 10 R12 C4 10 R21C4

(a)

(b)

(c)

(d)

Fig.6 Results of circuit simulation: (a) c = 0.47;(b) c = 0.7224; (c) c = 0.76; (d) c = 0.9.

4. Conclusion
In this paper, an innovative four-dimensional chaotic
system is proposed. Theoretical analysis shows that thissystem is a dissipative system without any equilibrium
point. The basic dynamic characteristics of the system arestudied by using Lyapunov exponent spectrum andbifurcation diagram and so on. In addition, in order toillustrate the existence of chaotic attractor in the systemfrom the physical level, Multisim software is used todesign and implement the analog circuit of the chaoticsystem. The circuit simulation results are consistent with
the numerical simulation results, so the physicalrealizability of the system is verified. There is noequilibrium point in this system, therefore the chaoticattractor generated by the system is hidden chaotic
attractor, which is worthy of further study.

(9)

If C=
C=
C=
C=
10nF , then values of these123
4
resistances can be calculated:=
R3 2.13M Ω , R13= R=
20 K Ω , R=R=
40 K Ω ,
620
15
R=
R=
R=
R=
R12= R=
127821 10 K Ω
By the above circuit model design and built, perform
circuit simulation experiment with Multisim. When c =0.47, the phase portrait of system (1) observed on thevirtual oscilloscope is shown in Fig.6(a). By the same
method, when c = 0.7224, c = 0.76, c = 0.9, the phase
portraits of system (1) observed on the virtual
oscilloscope are shown in Fig.6(b), Fig.6(c), Fig.6(d)respectively. Compared with Fig.4, it can be seen that thecircuit simulation results can consistent with thenumerical simulation results.

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255

Authors Introduction
Ms. Yan Sun

She received her Bachelor's degree fromthe College of Intelligence andInformation Engineering, Tangshan
University, China in 2018. She iscurrently a graduate student in Tianjin
University of Science and Technology,
China.

Mr. Yongchao Zhang
He received his Master's degree fromthe Department of Automation, Tianjin
University of Science and Technology,
China in 2020.

Mr. Jiaqi Chen

137

He received his Bachelor's degree fromthe Department of Automation, Tianjin
University of Science and Technology,China in 2017. He is currently agraduate student in Tianjin Universityof Science and Technology, China.