SUGISAKAMASANORI

Journal of Advances in Artificial Life Robotics
Vol. 4(2); September (2023), pp. 67–72
ON LINE ISSN 2435-8061; ISSN-L 2435-8061
https://alife-robotics.org/jallr.html

Research Article

Study of Synchronization of Two Heterogeneous Chaotic
Systems
Hanyuan Wang, Yiting Gao, Peng Lu, Xun Yin

Tianjin University of Science and Technology, 300222, China

ARTIC LE INFO
Article History
Received 15 November 2022
Accepted 18 September 2023

Keywords
Chaos synchronization
Chaotic systems with different structures
Direct method
MATLAB

ABSTRACTThis paper describes a method for synchronizing two chaotic systems with three-dimensional
heterogeneous structures. We introduced a mathematical model and designed a new synchronouscontroller that can synchronize the system from different initial values. This paper analyzes theadvantages and disadvantages of the synchronous controller, and also uses MATLAB software togenerate the error curve of the synchronous system when the synchronous controller is activatedon the response system, and shows its effectiveness.

© 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/).

phenomenon of chaotic synchronization for the first timein circuits by aligning the chaotic trajectories of the system
under various initial conditions [2]. In physics, chaotic
systems have been extensively studied and various
methods have been developed. These include adaptivesynchronization,
delayedsynchronization,pulsesynchronization,etc.
Wheninvestigating
thesynchronization of two chaotic systems, many researchers
start by analyzing the equilibrium point of the controlsystem. Then, both numerical simulations and functionalmethods are used to examine the accuracy of thesimulation results based on the Lyapunov law stability
theory [3], [4], [5]. We design an adaptive synchronous
controller and propose an adaptive method to synchronize
chaotic systems.

1. Introduction
Research on chaos theory is an important subject innonlinear phenomena.One of his early pioneers was theAmerican scientist Lorentz, who demonstrated theexistence of chaotic systems in 1963 [1]. Equations
governing chaotic systems are significant, especially since
there are chaotic solutions to nonlinear equations. We
design a mathematical model suitable for a particularsystem, investigate whether chaotic behavior occurs underthe model, and then verify the fundamental characteristics
of chaotic systems.
The discovery of synchrony was first made by the famous
physicist Huygens. Huygens happened to observe two
adjacent pendulums swinging in perfect harmony. This
breakthrough opened up disaster oscillator theory. The
synchronous phenomenon in the natural world and theunderlying mechanism have been brought to light. In 1990,
Pecora and Carroll developed a response-driven
synchronization method and demonstrated the2. Mathematical models of chaotic systems
Chaos theory’s Qi system canmathematically as shown below:

Corresponding author’s E-mail: 1463213510@qq.com URL: www.tust.edu.cn

67

berepresented x = a ( y − x ) + yz
 y = cx − xz − y= z xy − bz

the dimensional value of the chaotic system is fractional innature.
The graphs depicting the state variables of a chaoticsystem can be simulated using Matlab in the following
manner. The Simulink simulation generates the phasetrajectory diagram of the chaotic system as illustrated inFig. 1. The phase trajectory curves for the Qi chaos system
in varied coordinate systems are displayed in Fig. 2.

(1)

State variables of an unidentifiable nature, including
variables like x, y, and z, are an integral part of the system.
The state variable of the system is an unidentified quantity
is x , y , z ∈ R . a = 35 , b = 8 3 , c = 80 are the system’s
typical parameters.
The x, y, and z axes are the transformations for the state() ()variables in the mathematical model x, y , z → x, − y , − z ,( x, y , z ) → ( − x, y , − z ) , ( x, y , z ) → ( − x, − y , z ) .

The mathematical model for the system remains
unchanged when subjected to z-axis transformation.
Therefore, it may be concluded that the system’smathematical model is symmetric around the z-axis.
The mathematical model for the Qi chaotic systemyields a partial derivative:∂x ∂y ∂z
116∇V = ++=− a + b + 1 =−<0∂x ∂y ∂z3()(2)
Fig. 1 The construction of the Qi chaotic system and its
simulation using Simulink

When a negative partial derivative is observed in theequation mentioned before, it indicates that the system’smathematical model is dissipative. And its trajectory iscontrolled within a finite boundary as time elapses.
2.1 Chaotic dynamic properties
Designate the starting value for the Qi chaotic system as( x0 , y0 , z0 ) = (1,1,1) . The MATLAB function toolbox

computes the Lyapunov index for the Qi chaotic system as−0.0027 ≈ 0 . The Lyapunov index=λ1 4.0517 > 0 , λ2 =holds significant importance since =it λ1 4.0517 > 0 , thissignifies that the Qi chaotic system can exhibit chaotic
motion at that specific point in time. Calculating theLyapunov dimension d L of a system involves using theformula for solving by inputting the three Lyapunov
exponents obtained:j∑ λi
4.0517 − 0.0027==2+
2.0948dL =j + i =1
42.7151
λ j +1(3)

The given condition can be satisfied by finding thelargest integer

jj that ∑ λi > 0 . Based on the formulai =1

mentioned above, it can be inferred that, d L =2.0948 , thus,68

chaotic systems, and its observation in the Qi chaoticsystem signifies its susceptibility towards even minutevariations in initial value.
2.2 Characteristic of equilibrium point
The equilibrium state equation can be obtained by setting
the right side of the system's mathematical model equation
to 0:0a ( y − x ) + yz =0cx − xz − y = xy − bz =0
Fig. 2 Phase trajectory curve of Qi chaotic system system
Initiate the system by assigning initial values as follows
x02 , y0 , z0 = 1.0001,1,1andrespectively, and, a mathematical model can beconstructed on Simulink to plot the solution curves of thesystem's three corresponding variables, as illustrated in Fig.
3.

( x01 , y0 , z0 ) = (1,1,1)

() ()

(4)

Solve the system of state, let:2
b  ac + c 2 − 2 a + c ( a + c ) − 4 a x = 
 o2a
22 ab  ac + c 2 − 2 a + c ( a + c ) − 4 a 
 yo =2

a +c + ( a +c ) −4a2ac + c 2 − 2 a + c ( a + c ) − 4 a zo =2

a +c + ( a +c ) −4a(5)

To obtain the equilibrium point of the mathematicalmodel of the system, the given parameters can besubstituted into the formula as follows:S1 = ( 0, 0, 0 )S1 = ( 0, 0, 0 )
=S 2 ( xo , yo , zo ) →=
S 2 ( 26.3899, 8.0531, 79.6948)
S =− ( −26.3899, −8.0531, 79.6948)
 3 ( xo , − yo , zo ) S3 =(6)Linearizing the Qi chaotic system at the equilibrium
point involves writing the coefficient of each variable into
the matrix to determine its Jacobian matrix, which helps
evaluate the equilibrium point by substituting the values
accordingly: −aJ1 = J= c − z
S1
 yFig. 3 Qi Solution curves of chaotic system in different
states with different initial values
Under the same conditions, the solution curves of the Qi
chaotic system will greatly differ in a short period of timeupon modifying only the initial value of state variable x by
0.01%. It is evident that even a minimal change in initialvalue impacts the Qi chaotic system largely, indicating its
susceptibility towards small perturbations. Sensitive
dependence on initial conditions is a significant feature ofa+z y −1 − x x−b 

− a= c
0S1 a 0
−1 0 
0 −b 
(7)

Converting the aforementioned matrix into a
determinant and equating it to zero helps obtain thecharacteristic equation upon its expansion, which is given
as follows:2(8)f ( s) = ( s + b)  s + ( a + 1) s − a ( c − a ) = 0
By performing calculations, the characteristic roots ofthe matrix can be determined as follows:69s1 =−b =−2.6667()− a +1 +s22( a + 1) + 4a ( c − 1)

≈ 37.5788(10)≈ −73.5788(11)2(2) ( a + 1) + 4a ( c − 1)

− a +1 −s3

separated. As a result individual parts of the system may
be unstable, resulting in a complex and difficult structure.
The numerical simulation results demonstrate that theLü chaotic system displays chaotic dynamics equivalent tothat of the Qi chaotic system.

(9)

2(

3. Design of synchronous controller)

The equilibrium S1 = 0, 0, 0 is a saddle node.

3.1 Driving system

The approach to analyzing the last two stable points is
identical to that of S1, and S2 and S3 exhibit symmetry
along the z-axis. Therefore, by analyzing one of theequilibria, we can proceed to analyze S2 equilibrium aswell.
Atequilibriumpoint(S 2 = 26.3899, 8.0531, 79.6948

This article adopts the Qi chaotic system as the driver andpresents the mathematical expression for the Qi chaotic
system model:

 x1= a1 ( y1 − x1 ) + y1 z1 y1 = c1 x1 − x1 z1 − y1= z1 x1 y1 − b1 z1)(which is the same as S1), the J matrix of the Qi chaotic
system is linearizeda + 79.6948 8.0531  −ac 79.6948
J 2 =−−1−26.3899
 8.0531
26.3899−b 

Among them, the typical parameters of Qi chaotic
system are: a = 35 , b = 8 3 , c = 80 .

(12)

3.2 Response system

Using the aforementioned method, the matrix isconverted to a determinant and then equated to zero. Next,
the obtained determinant is expanded to obtain thecharacteristic equation, which yields the eigenvalues of J2
through calculation:

s1 = −45.8958=
s2,3 3.6146 ± 32.3465 j(15)This article utilizes the Lü chaotic system as the responsivesystem, and the synchronization controller equation issubstituted with equivalent words to create a synonymous
sentence: x2= a2 ( y2 − x2 ) + uc1
− x2 z 2 + c2 y 2 + u c 2
 y 2 = z = x y − b z + u 2 2 2 2 2 c3

(13)
(14)

While S1 has a negative eigenvalue, the other two
equilibrium points contain a pair of complex roots that are
conjugate to each other with a positive real part. Applying
the Routh stability criterion indicates that S2 and S3 have
similar characteristics as unstable foci within the system.
From the data obtained in the table, we can conclude that
only one of the three stable points shown by the Qi chaoticsystem remains stable, whereas the remaining two havebeen determined to be unstable. This instability leads to adivergence of nearby orbits, which becomes more
pronounced over time. This divergence is a clear
indication of the butterfly effect present within the system.
With its characteristic feature of dissipation, the Qi
chaotic system is capable of producing chaotic motion,
ensuring stability of the system as a whole. The presenceof the dissipative property in the Qi chaotic system aids instabilizing the system, leading to the convergence of outerorbits on the attractor. However, the nearby orbitals
experience a repulsive force and must be exponentially(16)

Typical parameters of Lü chaotic system are: a = 36 ,, c = 20 . The synchronization controller isb =3[u c = u c1uc 2uc 3] .
TTo simulate the error system and the controller’s statesynchronization curve, MATLAB software was employed,
determine if these two chaotic systems with different
architectures can exhibit complete synchronization.
3.3 Direct method
When the system's equilibrium state is highly stable andits output has attained a state of balance. As time
progresses, the energy contained within the system will
gradually diminish until it reaches the stable and minimum
value of the equilibrium state. The Lyapunov direct
method is based on an energy perspective. The system'smotion will result in energy consumption, but the system
capacity will not be depleted to zero.

70Lyapunov's method involves the use of an artificial
function to achieve this goal, this function is called
Lyapunov function, denoted as v ( x , t ) or v ( x ) . Let v ( x )be any scalar function, where x is the unknown variable ofthe system, if v ( x ) satisfies the following properties:(1) v ( x )=dv( x)uc= x= 35( y1 − x1 ) + (x1 − x2 ) + y1 z1
− y2 − x1 z1 + 80 x1
 y =
5 z = x1 y1 − z2 − z13energy change;(2) v ( x ) is positive definite and can reflect themagnitude of energy;(3) When x → ∞ , v ( x ) → ∞ reflects the distribution ofenergy, function v ( x ) is called Lyapunov function.
indicates the control quantity.
(17)

The key to achieving synchronization between two
disparate chaotic systems lies in the discovery of a suitableuc that make lim y (t ) − x (t ) =0.

( x1 , y1 , z1 ) = (1,1,1) , ( x2 , y2 , z2 ) = (30，30，30) .

t →∞For any initial values x (0) and y (0) , the problem ofcontrolling the synchronization of the system can beconverted into a problem of system error throughtransformation.

Fig. 4 and Fig. 5 depict the error and statesynchronization diagrams for both the Qi chaotic systemand the Lü system, which were created using simulation
software.

Set the error e =[e1 e2 e3 ] =[ x2 − x1 y2 − y1 z2 − z1 ] , choose theΤLyapunov function v ( x )=(21)The construction of a model for both the Qi chaoticsystem and the target Lü system is achieved throughestablishing their system equations and mathematicalmodels, accompanied by the plotting of error and statesynchronization curves. The purpose of this process is todemonstrate the effectiveness of this approach insynchronizing systems with varying structures.
Here, the initial values of the two systems are:

Given two systems: A = ( x1 , y1 , z1 ) , Y = ( x2 , y2 , z 2 ) , uc

 A = f ( A)=B g ( B ) + uc

(20)

The successive generations lead to the refinement of theformulas for the error and the standard parameters of both
systems, resulting in incremental improvements.

is continuous and can reflect the trend ofdt

( a2 − 1)e1 − a2 e2 + ( a2 − a1 )( x1 − y1 ) + y1 z1 
 x2 z2 − (1 + c2 )e2 − (c2 + 1) y1 − x1 z1 + c1 x1 
 x y + (b − 1)e − x y + (b − b ) z  11 232 221 1

Τ

1 n 2
∑ ei , obviously v ( x ) is
2 i =1positive definite, the state error equation will be rendered
asymptotically stable at the origin if v ( x ) is negative
definite,thatis,v ( x )=e1e1 + e2 e2 + e3e3 =0 att →∞Fig. 4 Simulation error curve of synchronous controller
designed by center translation method

identification of a suitable uc is imperative to attainnegative definiteness in v ( x ) .
Set the controller to:[uc = u c1 u c 2uc 3]

T(18)The error system can be formulated through substituting
similar words in the mathematical models of the drive
system and the response system:e1= a2 ( e2 − e1 ) + ( a2 − a1 )( y1 − x1 ) − y1 z1 + uc1e2 =− x2 z 2 + c2 e2 + ( c2 + 1) y1 + x1 z1 − c1 x1 + uc 2(19)e3 = x2 y2 − b2 e3 − x1 y1 + (b1 − b2 ) z1 + uc 3

Choose the Lyapunov function v ( x )=1 n 2∑ ei
2 i =1andcalculate v ( x )=e1e1 + e2 e2 + e3e3 .
71Authors Introduction
Ms. Hanyuan Wang
She received her B.S. degree from Collegeof Tianjin University of Science andTechnology, China in 2021. She iscurrently a Master course student inTianjin University of Science andTechnology. Her research area is about
Inertial Technology.
Ms. Yiting Gao

She is currently a master course student inTianjin University of Science andTechnology, majoring in neural network,
deep learning.

Fig. 5 Simulated state synchronization curve ofsynchronization controller designed with direct method
Mr. Peng Lu

After analyzing the simulation data depicted in thediagrams, it is observed that the system can reach its
equilibrium state in approximately 4.5 seconds.
4. Conclusion
The direct method for designing synchronous controllers
is both theoretically simple and practically efficient,providing significant savings in resources such as time andeffort. With its ability to address synchronization problems
in systems with different structures, it is widely consideredan effective approach.

Ms. Xun Yin

References
1. Lorenz E N, Deterministic nonperiodic flow, Journal ofatmospheric sciences, 1963, 20(2): pp. 130-141.
2. Pecora L M, Carroll T L, Synchronization in chaoticsystems, Physical review letters, 1990, 64(8): pp. 821.
3. Shahi M, Fallah Kazemi M, Adaptive sliding mode control
approach, Transactions of the Institute of Measurement andControl, 2017, 39(1): pp. 86-95.
4. Zhang H, Huang W, Wang Z, et al, Adaptive
synchronization between two different chaotic systems withunknown parameters, Physics Letters A, 2006, 350(5-6): pp.
363-366.
5. Huang J, Chaos synchronization between two novel
different hyperchaotic systems with unknown parameters,
Nonlinear Analysis: Theory, Methods & Applications, 2008,
69(11): pp. 4174-4181.

72

In 2023, he received his master's degree inelectronic Information from Tianjin
University of Science and Technology.
His research interests include artificial
intelligence and multi-sensor informationfusion.

He received his B.S. degree from Collegeof Qufu Normal University, China in
2022. He is currently a Master course
student in Tianjin University of Scienceand Technology. His research area is about
Deep Learning.