SUGISAKAMASANORI

Journal of Advances in Artificial Life Robotics
Vol. 4(4); March (2024), pp. 230–235
ON LINE ISSN 2435-8061; ISSN-L 2435-8061
https://alife-robotics.org/jallr.html

International JournalAdvances in Artificial Life Robotics
ICAROB series
Editor-in-Chief
Masanori Sugisaka
Co-Editor-in-Chief
Takao Ito

ISSN 2435-8061
Publisher Masanori Sugisaka(Global Register of Publishes)
regstrated in Jan, 25, 1996
Handadai Higashi 2-8-4, Oita, 870-1108, Japan

Research Article

Study and circuit design of heterostructured chaotic
synchronous control of Chen and Lü systems
Haozhe SunCollege of Electronic Information and Automation, Tianjin University of Science and Technology, 300222, China

ARTICLE INFO
Article History
Received 11 November 2023
Accepted 18 August 2024
Keywords
Chaos
Chaos synchronization of different systems
Nonlinear feedback control
Chaotic circuit

ABSTRACTIn this paper, the Chen system and Lü system are taken as research objects, and the nonlinear
feedback synchronization control method is used to synchronize the Chen system with different
initial values and the Lü system with different initial values by adding synchronization controllers
to achieve heterostructure chaotic synchronization, and the error curves and through thesimulation in Simulink, state synchronization curves of the corresponding states in the responding
Lü system and the driving Chen system are plotted, which proves that the responding Lü systemand the driving Chen system can achieve stable heterostructure chaotic synchronization.. Finally,an analog synchronization control circuit is built in Multisim. By comparing the synchronization
curves obtained by simulation and numerical simulation, it is found that the two curves tend to beconsistent, so it proves that the synchronization circuit built is correct.
© 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/).

The purpose is to provide sufficient theoretical support
for the successful application of Chen system and Lüsystem

1. Introduction
Chaos is motion that obeys deterministic laws but has
randomness. By obeying deterministic laws, it is meantthat the motion of the system can be expressed in termsof deterministic dynamical equations; while by motion
having randomness, it is meant that chaotic motion has
no deterministic trajectory in phase space. The earliest
mathematical definition of Chaos was put forward by Li
Tianyan and James Alan Yorke in a paper entitled"Period Three Implies Chaos" in 1975. The definition
stated that chaos was a special motion state of nonlinear
systems. Is a random state appearing in a deterministic
system [1].
In this paper, the classical Chen system and the Lüsystem are taken as the object of study. Analyze theirchaotic properties, and based on this, use the method ofnonlinear feedback to achieve chaotic synchronization ofthe two systems [2], and then use the MATLAB software
to perform numerical simulation and design the simulated
circuits, and perform the circuit simulation in Multisim.

heterostructure chaotic synchronization to the field ofsecure communication in the future.
2. Study and circuit design of heterostructured
chaotic synchronous control of Chen and Lüsystems
2.1. Modeling of the Drive Chen system

Corresponding author’s E-mail: 2921407938@qq.com, URL: www.tust.edu.cn

230

𝑥𝑥̇1 = 𝑎𝑎1 (𝑦𝑦1 − 𝑥𝑥1 )(1)(𝑐𝑐�𝑦𝑦̇1 = 1 − 𝑎𝑎1 )𝑥𝑥1 − 𝑥𝑥1 𝑧𝑧1 + 𝑐𝑐1 𝑦𝑦1
𝑧𝑧̇1 = 𝑥𝑥1 𝑦𝑦1 − 𝑏𝑏1 𝑧𝑧1
The mathematical model of Chen system is shown in Eq.
(1) . Where, x1 ，y1 ，z1 ∈ R and a1 = 35，b1 = 3，
c1 = 28 is the typical parameter of Chen system. Further
analysis from Eq. (1) shows that the Chen system is only
symmetric about the z-axis and is dissipative in all cases.
Eq. (1) is the mathematical model of the drive Chensystem.

2.2. Modeling of Response Lü System

According to Lyapunov's second method [3], as longas 𝑉𝑉(𝑒𝑒̃ ) is positively definite and 𝑉𝑉̇ (𝑒𝑒̃ ) is negatively
definite, the error system is uniformly asymptoticallystable at the origin in a large range. This is used todetermine the value of parameter 𝑘𝑘 = [𝑘𝑘1 𝑘𝑘2 𝑘𝑘3 ]𝑇𝑇 of
the synchronization controller.
Firstly, take the positive definite Lyapunov
function 𝑉𝑉(𝑒𝑒̃ ) = 1/2[(𝑎𝑎1 − 𝑐𝑐1 )𝑒𝑒12 /𝑎𝑎2 + 𝑒𝑒22 + 𝑒𝑒32 ] , andthen take its derivative:
a1 -c1V(ė� )=e e ̇ +e e ̇ +e e ̇
a2 1 1 2 2 3 3
k1 (a1 -c1 ) 2e1=(a1 -c1 )e1 e2 -(a1 -c1 )e21 a2(7)+c2 e22 +(c1 -a1 )e1 e2 -k2 e22 -b2 e23 -k3 e23
k1 (a1 -c1 )=- �+(a1 -c1 )� e21 -(k2 -c2 )e22 -(k3 +b2 )e23 ,a2

𝑥𝑥̇ 2 = 𝑎𝑎2 (𝑦𝑦2 − 𝑥𝑥2 )(2)�𝑦𝑦2̇ = −𝑥𝑥2 𝑧𝑧2 + 𝑐𝑐2 𝑦𝑦2
𝑧𝑧̇2 = 𝑥𝑥2 𝑦𝑦2 − 𝑏𝑏2 𝑧𝑧2
The mathematical model of Lü system is shown in Eq.
(2) . Where, x2 ，y2 ，z2 ∈ R and a 2 = 36，b2 = 3，c2 = 20 is the typical parameter of Lü system. Further
analysis from Eq. (2) shows that the Lü system is only
symmetric about the z-axis and is dissipative in all cases.
𝑥𝑥̇ 2 = 𝑎𝑎2 (𝑦𝑦2 − 𝑥𝑥2 ) + 𝑢𝑢𝑐𝑐1(3)�𝑦𝑦2̇ = −𝑥𝑥2 𝑧𝑧2 + 𝑐𝑐2 𝑦𝑦2 + 𝑢𝑢𝑐𝑐2
𝑧𝑧̇2 = 𝑥𝑥2 𝑦𝑦2 − 𝑏𝑏2 𝑧𝑧2 + 𝑢𝑢𝑐𝑐3
Eq. (3) is the mathematical model of the response Lüsystem.
𝑢𝑢𝑐𝑐 = [𝑢𝑢𝑐𝑐1 𝑢𝑢𝑐𝑐2 𝑢𝑢𝑐𝑐3 ]𝑇𝑇istherequired
synchronization controller. This synchronization
controller allows for the synchronization of Chen system
and Lü system with completely different structures.

In this case, the error system variable coefficientsshould satisfy the inequality group: Eq. (8)𝑘𝑘1 (𝑎𝑎1 − 𝑐𝑐1 )⎧−[+ (𝑎𝑎1 − 𝑐𝑐1 )] < 0𝑎𝑎2(8)−(𝑘𝑘2 − 𝑐𝑐2 ) < 0⎨⎩−(𝑘𝑘3 + 𝑏𝑏2 ) < 0
Solve the set of inequalities to obtain the following
result:𝑘𝑘1 > −𝑎𝑎2 = −36(9)� 𝑘𝑘2 > 𝑐𝑐2 = 20
𝑘𝑘3 > −𝑏𝑏2 = −3
The value of k satisfying the condition can bearbitrarily chosen. In this paper, (𝑘𝑘1 , 𝑘𝑘2 , 𝑘𝑘3 ) = (0,25,0),
then the synchronization controller 𝑢𝑢𝑐𝑐 is as follows:
𝑢𝑢cl = −(𝑦𝑦1 − 𝑥𝑥1 )⎧𝑢𝑢𝑐𝑐2 = 𝑥𝑥2 𝑧𝑧2 − 𝑥𝑥1 𝑧𝑧1(10)+33𝑦𝑦1 − 7𝑥𝑥2 − 25𝑦𝑦2⎨⎩𝑢𝑢𝑐𝑐3 = −𝑥𝑥2 𝑦𝑦2 + 𝑥𝑥1 𝑦𝑦1
So far, the mathematical model of synchronous
controller has been designed.
The numerical simulation
model of thesynchronization controller satisfying the condition (Eq.
(10)) is built by Simulink in Matlab, as shown in Fig.1.
The corresponding state error curve obtained by runningis shown in Fig.2, and the corresponding statesynchronization curve is shown in Fig.3.
Through observation, it is found that the state error
curve gradually tends to 0 with the evolution of time,
while the synchronization curve of the corresponding
variable continues to run according to the orbit of theChen drive system after the coincidence. It can be shownthat the above formula derivation is correct. Chen system
and Lü system can realize different structure chaos
synchronization, and Chen system and Lü system canrealize the synchronization of different structure chaos.
Therefore, the following circuit can be designed and2.3. Design of Synchronization Controller

The mathematical model of the error system can beobtained by making a difference between Eq. (1) and Eq.
(3), as shown in Eq. (4).
𝑒𝑒̇1 = 𝑥𝑥̇ 2 − 𝑥𝑥̇1⎧
= 𝑎𝑎2 (𝑦𝑦2 − 𝑥𝑥2 ) − a1 (𝑦𝑦1 − 𝑥𝑥1 ) + 𝑢𝑢c1 ,⎪⎪𝑒𝑒̇2 = 𝑦𝑦̇ 2 − 𝑦𝑦̇1= −𝑥𝑥2 𝑧𝑧2 + c2 𝑦𝑦2(4)⎨ −[(𝑐𝑐 − a )𝑥𝑥 − 𝑥𝑥 𝑧𝑧 + 𝑐𝑐 𝑦𝑦 ] + 𝑢𝑢 ,
1
1 1
1 1
1 1c2
⎪
𝑒𝑒̇3 = 𝑧𝑧̇2 − 𝑧𝑧̇1⎪⎩= 𝑥𝑥2 𝑦𝑦2 − b2 𝑧𝑧2 − (𝑥𝑥1 𝑦𝑦1 − b1 𝑧𝑧1 ) + 𝑢𝑢c3 ,
InEq.
(4),
𝑒𝑒 = [𝑒𝑒1 𝑒𝑒2 𝑒𝑒3 ]𝑇𝑇 =𝑇𝑇[𝑥𝑥2 − 𝑥𝑥1 𝑦𝑦2 − 𝑦𝑦1 𝑧𝑧2 − 𝑧𝑧1 ] , 𝑒𝑒1 ，𝑒𝑒2 ，𝑒𝑒3 is the statevariable of the error system.
Therefore the synchronization controller is designed
as:𝑢𝑢𝑐𝑐1 = (𝑎𝑎1 − 𝑎𝑎2 )(𝑦𝑦1 − 𝑥𝑥1 ) − 𝑘𝑘1 𝑒𝑒1⎧
𝑢𝑢𝑐𝑐2 = 𝑥𝑥2 𝑧𝑧2 − 𝑥𝑥1 𝑧𝑧1 + (𝑐𝑐1 − 𝑐𝑐2 )𝑦𝑦1⎪
(5)+(𝑐𝑐1 − 𝑎𝑎1 )𝑥𝑥2 − 𝑘𝑘2 𝑒𝑒2⎨
𝑢𝑢𝑐𝑐3 = −𝑥𝑥2 𝑦𝑦2 + 𝑥𝑥1 𝑦𝑦1 + (𝑏𝑏2 − 𝑏𝑏1 )𝑧𝑧1 − 𝑘𝑘3 𝑒𝑒3⎪⎩
In Eq. (5), 𝑘𝑘1 ，𝑘𝑘2 ，𝑘𝑘3 ≥ 0. By substituting Eq. (5)into Eq. (4), the mathematical model of the ultimate errorsystem can be achieved as demonstrated in Eq. (6):𝑒𝑒̇1 = 𝑥𝑥̇ 2 − 𝑥𝑥̇1 = 𝑎𝑎2 (𝑒𝑒2 − 𝑒𝑒1 ) − 𝑘𝑘1 𝑒𝑒1 ,𝑒𝑒̇2 = 𝑦𝑦̇ 2 − 𝑦𝑦̇1 = 𝑐𝑐2 𝑒𝑒2 + (𝑐𝑐1 − 𝑎𝑎1 )𝑒𝑒1 − 𝑘𝑘2 𝑒𝑒2 , (6)�𝑒𝑒̇3 = 𝑧𝑧̇2 − 𝑧𝑧̇1 = −𝑏𝑏2 𝑒𝑒3 − 𝑘𝑘3 𝑒𝑒3 ,The error system is then written in the form of 𝑒𝑒̇ =𝑓𝑓(𝑒𝑒, 𝑡𝑡) . It is merely necessary to guarantee that theequilibrium state of the error system at the origin is
uniformly asymptotically stable within a wide range,
which indicates the achievement of synchronization
between the Chen system and the Lü system with distinct
structures.
231

simulated according to the synchronous controller
designed above.
Scope1
b1b2
a53z1x1y1

1s3
z2x2y2251sa47
c1x1z1

c2

a3

28

33

y2x2z21s

c1-a1

Scope2

20y1

1
s-7x1

a21s

36232

a1Fig. 2 State error curve: (a) e1; (b) e2; (c) e3

35Fig. 1 Numerical simulation modelx21sFig. 3 State synchronization curve: （a）x1-x2；（b）y1-y2；（c）z1-z22.4. Analog circuit model design and construction
Substitute 𝜏𝜏0 = 100 into Eq. (11) and take 𝐶𝐶4 = 𝐶𝐶5 =𝐶𝐶6 = 10nF, and the calculation is as follows:

In the following, the analog circuit model of thesynchronous controller designed above is established. To
make the designed circuit explicit and manageable, it isdetermined to adopt the method of improved modular
circuit design for the synchronous controller circuit.
Simultaneously, to ensure that the operational amplifier
operates within the normal working range as much aspossible, it is necessary to implement proportional
compression
transformationandtimescaletransformation of the variables.
The state expression of this circuit and its derivation
are shown in Eq. (11).
τ0 uc1 =τ0 [-(y1 -x1 )]=-τ0 y1 -τ0 (-x1 )11(-x )=y R 14 C4 1 R 15 C4 1
τ0 uc2 =τ0 [10x2 z2 -10x1 z1+33y1 -7x2 -25y2 ]=-10τ0 (-x2 z2 )-10τ0 x1 z1-33τ0 (-y1 ) -7τ0 x2 -25τ0 y2
1(-x z )=10R 16 C5 2 2
11(-y )
x z 10R 17 C5 1 1 R 13 C5 1
11x y
R 19 C5 2 R 20 C5 2
τ0 uc3 =τ0 (-10x2 y2 +10x1 y1 )=-10τ0 x2 y2 -10τ0 (-x1 y1 )1=x y
10R 21 C6 2 21(-x y )10R 22 C6 1 1

R14 = R15 =11=𝜏𝜏0 𝐶𝐶4 100 × 10 × 10−9= 1MΩ
11=33𝜏𝜏0 𝐶𝐶5 33 × 100 × 10 × 10−9= 30.3kΩ
11R19 =
=7𝜏𝜏0 𝐶𝐶5 7 × 100 × 10 × 10−9(12)= 142.8kΩ
11R 20 ==25𝜏𝜏0 𝐶𝐶5 25 × 100 × 10 × 10−9= 40kΩ
1R16 = 𝑅𝑅17 = 𝑅𝑅21 = 𝑅𝑅22 =100𝜏𝜏0 𝐶𝐶51= 10kΩ=100 × 100 × 10 × 10−9R18 =The circuit model is built according to the parameters
obtained above, are presented in Fig.4. After theconnection is accomplished, activate the oscilloscopes
corresponding to the variables x-y, x-z and y-z
respectively, click the "run" button, await theoscilloscopes to display waveforms, and observe thecorresponding waveforms. Fig.5 and Fig.6 show thecorresponding state synchronization curve and error
curve, respectively

(11)

233

Fig. 4 Circuit simulation modelFig.6. Error curve:（a）e1；（b）e2；（c）e3
Fig.5. Synchronization state curve:（a）x1-x2；（b）y1-y2；（c）z1-z2

3. Conclusion
This paper begins by separately establishingmathematical models for the driven Chen system and theresponsive Lü system. Subsequently, a synchronous
controller is designed to eliminate all nonlinear terms,
using Lyapunov's second method to derive its parameters.
234

References

The designed controller enables numerical simulations in
MATLAB, demonstrating that the state error graduallyconverges to zero over time, while the statesynchronization curve follows the trajectory of the Chendrive system.
Then, with the aid of Multisim software, the analog
circuit model of Chen system and Lü system with distinct
structure chaos synchronization is designed andconstructed by the improved modular approach. It
constitutes a crucial step in the design of this system toinstall independent switches for the compensator and thesynchronization controller in the system to guarantee
separate control, which is largely associated with therealization of the synchronization between theabovementioned two systems. The circuit configured inthis manner is more concise and convenient to use thanother methods. In the simulation of Multisim software,
the oscilloscope indicates that the corresponding statevariable curve will reflect diverse waveforms when theindependent switch is disconnected and closed. When theswitch is disconnected, the observed state curves will
oscillate based on their inherent laws. Nevertheless, after
the switch is closed, this irregularity will graduallyconverge to a curve oscillation that follows the operating
trajectory of Chen's drive system. The chaotic
synchronization analog circuit of the two systems isconstructed by employing the improved modularelectricity. The nonlinear controller circuit and structurecompensator circuit are flexibly connected to the Chensystem and the Lü system, and a single switch is utilized
to control the access and detachment of the two circuits.
When conducting the simulation in Multisim and closing
the switch, it can be clearly observed in the oscilloscopethat the corresponding state variable curve of the original
two does not converge into one but continues to undergo
random oscillation, and the oscillation law is inaccordance with the variable change mode of the driving
Chen's system, thereby verifying the correctness of theimproved modular circuit construction. It is also
demonstrated that the Chen system and the Lü systemachieve chaotic synchronization with different structures.

1.
2.
3.

Li T, Yorke J A, Period three implies chaos[J]. Americanmathematical monthly, 1975, 82(10): 985-992.
W. Kinzel, A. Englert, I. Kanter. On chaos synchronizationand secure Communication[J]. Philosophical Transactions
Of The Royal Society A, 2010, 368, 379–389.
J. W C .An introduction to Lyapunov's secondmethod[J].Transactions of the American Institute ofElectrical Engineers, Part II: Applications andIndustry,1962,80(6):325-332.

Authors Introduction
Mr. Haozhe Sun

235

He is 22 years old, he has obtained abachelor's degree in engineering andis currently studying for a master'sdegree in engineering at Tianjin
UniversityofScienceandTechnology. His major is electronic
information.