SUGISAKAMASANORI

Journal of Advances in Artificial Life Robotics
Vol. 1(4); March (2021), pp. 193–198
ON LINE ISSN 2435-8061; ISSN-L 2435-8061
https://alife-robotics.org/jallr.html

Research Article

Research on characteristics and synchronization control ofa system without equilibrium point
Lianchen Zhao, Yizhun Peng, Yuheng Zhang
Tianjin University of Science and Technology, China

ARTICLE INFO
Article History
Received 09 November 2019
Accepted 15 November 2020
Keywords
Chaotic characteristics
Synchronous control method based on stateobserver

ABSTRACTIn this paper, the dynamic characteristics of a chaotic system without equilibrium point are studied.
Through numerical simulation and theoretical analysis, the chaos characteristics of the system
without equilibrium point are studied. In this paper, the synchronous control method based onstate observer is used. First, the state observer system is designed according to the original system.
And then the simulation results are obtained. At the end of this paper, we give the advantages anddisadvantages of this synchronization method.

© 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/).

1. Introduction

Xin equal based on neural network, studied thesynchronization problem of two chaotic systems in thepresence of disturbance, Lu Xiang and so on designed adelay synchronization controller to realize thesynchronization control between spatiotemporal chaotic
Systems [3]. At present, scholars are keen to combine
electronic circuit, laser system, neural network andcomputer system with chaos synchronization, so as torealize some application schemes of chaos control andsynchronization. Therefore, chaos synchronization is oneof the hottest research directions in the current chaos, andalso one of the most widely used topics. Its future
prospects are very broad.

Chaos is a kind of motion which is more complex andspecial than other forms in nonlinear dynamic system. It
can be found everywhere in nature. Known as the threegreat revolutions of physics in the 20th century, relativity
and quantum mechanics are the other. With thedevelopment of the research on chaos system, theemergence of the system without equilibrium point and theanalysis of its characteristics are becoming more and more
popular [1].
Chaos synchronization usually means that there are at least
two vibration systems and their phases can be coordinated.
Although chaos synchronization and chaos control are
independent in definition, it is generally considered as a
special form of chaos control in academia. Before thechaos synchronization is proposed, the traditional chaos
control is to stabilize the system on the track of unstable
period, and the chaos synchronization is to reconstruct thetwo systems. Many researchers put forward many conceptsof synchronization, such as complete synchronization,samestep,
generalizedsynchronization,lagsynchronization and so on [2]. In addition to the richresearch results in theory, there are also many
synchronization schemes in experiments, such as Guan

2. Theoretical basis of chaotic system
Chaos can be seen everywhere in our life, such as thereproduction of biology, the fluctuation of stock market,
the sudden change of geomagnetic field and so on.
Chaos is a kind of seemingly random but actually
deterministic motion. Compared with the mechanicalmotion in classical mechanics, it can predict the motion
state of the system at a certain time in the future throughCorresponding author’s E-mail: pengyizhun@tust.edu.cn193

calculation and analysis according to the system state at
the initial time and the deterministic law. In chaotic system,
the state of the system will not be accurately predicted, andonly the future motion trend can be determined.
Li Tianyan, an American Chinese scholar, and his thesis
tutor York first proposed the concept of chaos, and gave
the mathematical definition of chaos.
Li Yorke definition [4].
The mapping 𝑓𝑓 contains all periodic points.
Thereis an uncountable subset 𝑆𝑆∈I without periodic points, andthe following conditions are satisfied:①∀𝑥𝑥, 𝑦𝑦 ∈ 𝑆𝑆, 𝑥𝑥 ≠ 𝑦𝑦 𝑙𝑙𝑙𝑙𝑙𝑙 𝑠𝑠𝑠𝑠𝑠𝑠�𝑓𝑓 (𝑛𝑛) (𝑥𝑥) −𝑓𝑓 (𝑛𝑛) (𝑦𝑦)� > 0②③

In a general nonlinear system, we assume that there is aparameter variable, when the parameter variable changes
very little near a specific value, then the small change willcause a huge change in the topological properties of thephase space. This phenomenon is called bifurcation, andthe critical point is defined as the bifurcation value, whenthe parameter is on the coordinate axis In the abscissa of,
we call the point of bifurcation and define the point that
does not cause bifurcation as the constant point.
As a common phenomenon, bifurcation is very importantin flight linear system. In order to get the bifurcation
diagram, we need to get all the values of the parameters ina certain range, so that the topological properties of thetrajectory lines in the phase space will be shown, and theproperties shown in the diagram are the bifurcation
diagram we want. When the parameters are taken todifferent values, the system will also show different states.
After analysis, we can see that the occurrence ofbifurcation points means that the system will be unstable,
so the bifurcation points are sometimes called unstable
points.
Lyapunov index is a very important reference. It is anindex law to measure the attraction or divergence ofadjacent orbits in phase space, and judge whether theadjacent orbits in phase space attract or diverge according
to its positive and negative energy. When one of the
Lyapunov exponents of the system is less than zero, itmeans that the motion of the system in this direction will
be a stable state [6], otherwise, when one of the Lyapunov
exponents of the system is greater than zero, it means thatthe motion of the system in this direction is unstable, or thephase space of the system will be an expanded state, thatis, the adjacent trajectory will change more and more
Divergence, because of this nature, we can see that thelong-term behavior of the system in this state isunpredictable. Therefore, it can be realized to judge thestate of the system according to its Lyapunov index.
Here we take a three-dimensional chaotic system as an
example to illustrate the relationship between the
Lyapunov exponent and the state of the system [7].
Because the system is a three-dimensional system, it is notdifficult to see that the system has three Lyapunov
exponents. The three of them together are called the
Lyapunov exponent spectrum. After calculation, when allof them are negative, the attractor is called fixed point andthe system is in steady state. When one of them is zero andthe other two are negative, the attractor is periodic and thesystem is in periodic state. The difference between
stationary state and periodic state is not obvious, so they
can also be regarded as the same state. If one of them iszero and the other two are not zero, and the attractor is achaotic attractor, then the system is in a chaotic state [8].

𝑛𝑛→∞∀𝑥𝑥, 𝑦𝑦 ∈ 𝑆𝑆, 𝑙𝑙𝑙𝑙𝑙𝑙 𝑖𝑖𝑖𝑖𝑖𝑖�𝑓𝑓 (𝑛𝑛) (𝑥𝑥) − 𝑓𝑓 (𝑛𝑛) (𝑦𝑦)� = 0
𝑛𝑛→0∀𝑥𝑥 ∈ 𝑆𝑆, 𝑙𝑙𝑙𝑙𝑙𝑙 𝑠𝑠𝑠𝑠𝑝𝑝�𝑓𝑓 (𝑛𝑛) (𝑥𝑥) − 𝑓𝑓 (𝑛𝑛) (𝑦𝑦)� > 0
𝑛𝑛→0

𝑓𝑓 which satisfies the above conditions, is said to be chaotic
on set s. From the mathematical definition put forward by
two scholars, we can get the essential characteristics of achaotic system, which are boundedness, aperiodicity andinitial value sensitivity.
In 1989, there was a new definition of Devaney [5].
Let 𝑋𝑋 be a metric space. A continuous mapping: 𝑓𝑓: 𝑋𝑋 →𝑋𝑋 is called chaos on 𝑋𝑋 , and 𝑓𝑓 satisfies the following
conditions:(1) 𝑓𝑓 is topologically transitive;(2) The periodic points of 𝑓𝑓 are dense in 𝑋𝑋(3) 𝑓𝑓 is highly sensitive to the initial conditions.
The most essential characteristic of chaotic dynamic
system is the sensitivity of initial value, which is thefamous butterfly effect. This concept refers to two
identical systems, and their initial values have extremely
small differences. After a period of change, the output
values of the two systems will be greatly different, just liketwo completely different systems.
2.1. Research methods of chaos
In order to understand the chaotic nature of the system, weusually simulate the bifurcation diagram, time sequence
diagram, phase trajectory diagram and Lyapunov exponent
spectrum of the system to analyze the system.
Select the time parameter as the ordinate and take aseparate state quantity as the abscissa. The graph drawn inthis way is called the sequence diagram. In this way, wecan draw all the state variables into one graph, so we canget the phase trace graph of the system. From the time
sequence diagram, we can clearly know whether thecurrent state of the system is in chaos or periodic statethrough numerical changes. From the phase trajectory, wecan clearly and accurately judge the state of the systemfrom the shape of the attractor.

194

2.2.

hematical model of three-dimensional chaotic
system without equilibrium point

2.3. Characteristic analysis of system withoutbalance point

The state equations of the three-dimensional chaotic
system studied in this paper are as follows：(1) Symmetry
We define a new set of coordinates𝑧𝑧1 = −𝑥𝑥𝑧𝑧(a)� 2 = −𝑦𝑦𝑧𝑧3 = 𝑧𝑧
We foundx
ay + xz=−bx + yz
y =
1 − x2 − y 2z =
az2 + z1 z3−ay − xz =
 z1 =
bx − yz =−bz1 + z2 z3
z2 =
22
 z 3 =1 − x − y =1 − z12 − z2 2
Among them, x, y, z ∈ R is the state variable of thesystem, and the parameters a = 0.05, b = 1 are selected.
The phase trajectory of the system and the projection of
chaotic attractor on each phase plane are simulated by
MATLAB software. Phase trace diagram is shown in theFig.1.

(b)This means that the coordinate value (x, y, z) of thedifferential equation of the system is transformed into (- x,- y, z) without change, that is, the system has symmetry
about the Z axis.
(2) Dissipation∇V =∂f1 ∂f 2 ∂f 3++= 2z∂x ∂y ∂zWhen ∇ V < 0 , the system is always dissipative andconverges with the index DV / dt = e ^ (- t). In this system,if ∇ V < 0 is to be determined first, then with t →∞, eachvolume element including the system trajectory will shrink
to 0. The asymptotic dynamic behavior of all systemtrajectories will be fixed in one attractor and limited to alimit subset of volume 0, which further proves theexistence of attractors.
Based on the simulation of the system without balance
point with different parameter values and the same initialvalue of [- 1-14], Fig.2. is obtained. These two figures are
the sum of the three Lyapunov exponents of the systemfrom - 5 to 5 after fixing one parameter. It can be seen thatno matter which parameter is changed by the system, thesum cannot be stable at zero, so that It can be said that thesystem is not a conservative system.

Fig.1. Phase trace diagram of the system in each plane
When the given initial value is (x (0), y (0), Z (0)) = (- 1, 1, 4), the three Lyapunov exponents of the system are λ 1"= 0.0386" λ 2 "= - 0.1232" λ 3 "= - 0.0968" by software
simulation, and the formula can be used at the same time:
𝐷𝐷𝐿𝐿 = 𝑗𝑗 +1𝑗𝑗� 𝜆𝜆𝐿𝐿𝑖𝑖
�𝜆𝜆𝐿𝐿𝑗𝑗+1 � 𝑖𝑖+1

The dimension of the attractor is calculated as DL = 2.18,
which shows that the system appears chaos.

195

Fig.3. Bifurcation diagrams and Lyapunov exponents withinitial values of [-1 -1 4] and [-2 -2 4]
3. Synchronous control analysis of a system
without balance point
The following formula is defined as the driving system. By
using observer based method, the following formula istransformed into another form, where Ax is the linear part,
Bf ( x ) is the nonlinear part, G is the constant part, andg ( x ) is the output of the drive system [9].
The drive system is:x1 ax2 + x1 x3
=−bx1 + x2 x3
 x2 =
 x3 =−
1 x12 − x2 2
Transform:

Fig.2. The sum of Lyapunov exponents with parameters aand b(2) Equilibrium analysis
0 = 𝑎𝑎𝑎𝑎 + 𝑥𝑥𝑥𝑥（a）
0 = −𝑏𝑏𝑏𝑏 + 𝑦𝑦𝑦𝑦（b）
0 = 1 − 𝑥𝑥 2 − 𝑦𝑦 2 （c）
for（a）and（b）:
𝑥𝑥𝑥𝑥𝑥𝑥 = −𝑎𝑎𝑦𝑦 2 = 𝑏𝑏𝑥𝑥 2 （d）
If 𝑏𝑏𝑥𝑥 2 + 𝑎𝑎𝑦𝑦 2 = 0 （e）
When a>0,b>0, 𝑥𝑥 = 0, 𝑦𝑦 = 0
Contradictory, that is, the system has no equilibrium point.
2.4.

x =Ax + Bf ( x ) + G
g ( x ) = Wx

Namely:  x1 
 x1 
 x1 x3  x  + B  x x  + GA=x2  2
 2 3 22  x3 
 x1 + x2  x3  

Initial value sensitivity

There are two parameters in the system, a = 0.05 isdetermined first, and the characteristic of parameter B isanalyzed. The bifurcation diagram and Lyapunov
exponent spectrum of the system are drawn with
MATLAB software.
The initial values are [-1 -1 4] and [-2 -2 4], a = 0.05.
Through software simulation, the following figures areobtained. From these two figures, it can be seen that thethree-dimensional chaotic system without equilibrium
point is in periodic state at b ∈ (- 2,0) and in chaotic stateat b > 0, so it can be judged that the change of initial values
has a very significant impact on the chaotic state of thesystem. The consequence is shown as Fig.3.

It can be calculated:

 0 0.05 0 
A = -1 00 00 
 0

1 0 0 
B = 0 1 0 
0 0 -1

196

0
G = 0 
1 
The state observer is constructed as:y=
Ay + Bf ( x ) +G − ( g ( y ) − g ( x ) )Where g ( y ) is the output of the state observer, and theerror value e= y − x is set:

e = y − x = Ae − ( Wy − Wx ) = ( A − W ) e
When ( A − W) is a time invariant matrix, select theappropriate gain matrix W so that the eigenvalue of( A − W ) is less than zero, then the system gradually
approaches stability.
The state observer system is:  x2 
 y1 
 x1 x3  y1 − x1  
A  y2  + B  x2 x3  + G − W  y2 − x2 2 y=  y3  x12 + x2 2  y3 − x3  z2 
 

Fig.4. Synchronous curve
4. Conclusion

The feedback gain matrix can be obtained by poleplacement:
 1 0.05 0 
W =  −1 10  001 

The full text is summarized as follows:
1. Through Lyapunov exponent spectrum, bifurcation
diagram, phase trajectory diagram and sequence diagram,the hiding and coexistence characteristics of the system
without equilibrium point are studied.
2. When the parameters are determined, a Lyapunov
function is designed by using synchronization control
method based on state observer. Finally, the simulation
results are obtained through simulation, and the conclusionthat the system can be synchronized by this method isdrawn after analysis.
In the work of this paper, the basic dynamic characteristics
and synchronous control of a system without balance point
are studied preliminarily, but the research on the hidden
characteristics and coexistence characteristics of thesystem without balance point is not deep enough, whether
there are other hidden characteristics needs further study,
and in the synchronous control, there are not enoughmethods,whether there are
bettermethods
Synchronization, which needs to be done in the next phase.

The error system is:  e1  =e2   e3  

 −1 0 0   e1 
 0 −1 0  e 
 2
 0 0 −1  e3 Because the eigenvalue of matrix ( A − W ) is less thanzero, so the two systems are synchronized.
Next, we use MATLAB software to verify whether thestate observer based synchronization method used in thispaper can synchronize the system. Firstly, the Simulinkmodule of MATLAB software is used to build the drive
system and state observer system, and then the simulation
is carried out.
Through simulation, the synchronization curve as shownin Fig. 4 is obtained. It can be seen from the diagram thatthe final output of the drive system and the state observer
system are the same after a period of synchronization
control under the condition of different initial values,
which indicates that the applied state observer
synchronization control method plays the role ofsynchronous control.

References
1. Zhang Wenxiao, Jiang Guoping. Control method ofunstable equilibrium point of chaotic system with
unpredictable state [J]. Computer technology anddevelopment, 2015,25 (01): 79-81 + 86

197

2. Luo Dongmei. Analysis of chaotic sequencecharacteristics and Research on chaotic synchronization
technology [D]. Chongqing University, 2011
3. Wang Yu. Chaos synchronization control of nonlinear
systems [D]. Northeast Petroleum University, 2018
4. Shi Qingmei. Analysis and synchronization of a new
chaotic system [D]. North China Electric Power University,
2012
5. Anirban Ray,D. C. Saha,Papri Saha,A. Roy Chowdhury.
Generation of amplitude death and rhythmogenesis incoupled hidden attractor system with experimental
demonstration[J]. Nonlinear Dynamics,2017,87(2).
6. Hamid Reza Abdolmohammadi,Abdul Jalil M
Khalaf,Shirin Panahi,Karthikeyan Rajagopal,Viet-ThanhPham,Sajad Jafari. A new 4D chaotic system with hidden
attractor and its engineering applications: Analog circuitdesignandfield
programmable
gate
array
implementation[J]. Pramana,2018,90(6).
7. Huitao Zhao, Yiping Lin,Yunxian Dai. Hopf bifurcationand hidden attractor of a modified Chua’s equation[J].
Nonlinear Dynamics,2017,90(3).
8. J. Timmer,H. Rust,W. Horbelt,H.U. Voss. Parametric,
nonparametric and parametric modelling of a chaotic circuit
time series[J]. Physics Letters A,2000,274(3).
9. Yu yeqiang. Research on synchronous control strategy offractional chaotic system [D]. Northeast Petroleum
University, 2017

Mr.Yuheng Zhang============================================================
Authors Introduction

Mr. Lianchen Zhao

Dr. Yizhun Peng

He is a second-year mastercandidate in Tianjin University ofScience and Technology, majoringin control engineering. His researchfirld is intelligent robot. He won thethird prize in the RobotChallenge
2019 humanoid sumo item.

He is currently the master course
student in Tianjin University ofScience & Technology. His
research field is artificial
intelligent. His research direction
is natural language processing
and machine learning.

He is an Associate Professor inTianjin University of Science &Technology. He received a doctor's
degree in control theory andcontrol engineering from theInstitute of Automation，Chinese
Academy of Sciences, in 2006.His
research field is intelligent robot
and intelligent control.

198