SUGISAKAMASANORI

Journal of Advances in Artificial Life Robotics
Vol. 4(2); September (2023), pp. 110–115
ON LINE ISSN 2435-8061; ISSN-L 2435-8061
https://alife-robotics.org/jallr.html

Research Article

Creation and Testing of a Conservative Chaotic Multi-Scroll
SystemJiaxin Li, Haozhe Sun, Yong Liu

College of Electronic Information and Automation, Tianjin University of Science and Technology, 300222, China

ARTICLE INFO
Article History
Received 15 November 2022
Accepted 02 November 2023
Keywords
Multi-Scroll
Conservative chaotic system
Balance point analysis
Lyapunov index

ABSTRACTThe multi-scroll conservative chaotic system represents a novel category of chaotic systems
renowned for their complex dynamics. Due to their intricate behavior, they have garneredsignificant interest among researchers in the field. This study focuses on the creation of aconservative multilayer chaotic system. The design involves integrating a polygon-free sinusoidal
function to generate multiple scrolls within this conservative multilayer chaotic system. By
incorporating nonlinear functions, this system generates one-dimensional linear multi-roll andtwo-dimensional multi-roll lattice distributions.

© 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

2. Creation and Testing of a Conservative Chaotic
Multi-scroll Syste

Since the 20th century, nonlinear science has developed
rapidly, and it mainly includes three major aspects: fractal,
chaos and soliton. Among these three aspects, chaos has
attracted the attention of many scholars because of its
particularity. The multi-scroll conservative chaotic systemstands out for its intricately woven dynamical properties,
garnering heightened scholarly attention as a distinctive
breed of chaotic systems.
Researchers are increasingly delving into the creation ofmulti-scroll chaotic systems characterized by unique andcaptivating morphologies. This is driven by the sheer
complexity of their dynamics, coupled with theirpropensity for wider spectral width when compared totheir counterparts. Although somewhat less explored due
to limitations in analytical methods, these multi-scroll
chaotic systems find real-world applications in realms
such as communication encryption and image encryption.
Their distinctive qualities, exhibited by multi-scroll
chaotic attractors [1], assume a central role in thesepractical applications. Investigating these systems not only
enriches our comprehension of the fundamental
mechanisms underpinning chaotic attractors but also
fosters innovation in the design of new chaotic systems.

This paper delves into the exploration of the multi-scroll
conservative chaotic system. The creation of multiple
scrolls within the conservative chaotic system isaccomplished through the application of the non-doubleangle sine function. The analysis and examination of the
Lyapunov exponential map, the bifurcation diagram, andthe attractor phase diagram serve as essential tools forunraveling the dynamic properties of the multi-scroll
attractor.
The Lyapunov exponential map reflects several
Lyapunov indices of a chaotic system through images, andobtains the motion state of the system with the change ofparameters, or the motion state of the system with thechange of initial value through the comparison of zero. If
there is a line on a certain section of the bifurcation
diagram, it indicates that the motion generated by thesystem is periodic; If there are two lines, then the motionof the system is period-double; If the image is a pair ofchaotic points, then the movement of the system is periodic.
The phase diagram can intuitively see the motion of thesystem. The phase diagram mainly reflects whether theCorresponding author’s E-mail: 2921407938@qq.com URL: www.tust.edu.cn

110system can produce chaotic motion and whether themotion trajectory in the phase space is chaotic.
2.1. The advent of multi-scroll chaotic systems
The four-dimensional structure investigated in this study
represents a conservative chaotic model, whose dynamic
equation x is assigned to sinx. At the same time, given 𝑥𝑥̇ =0, 𝑦𝑦̇ = 0, 𝑧𝑧̇ = 0, 𝑤𝑤̇ = 0 , and the resulting equation isshown in Eq, (1).
0 = 𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏𝑏𝑏
0 = −asin (𝑥𝑥)
�(1)0 = 𝑐𝑐𝑐𝑐
0 = −𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏(𝑥𝑥)𝑦𝑦 − 𝑐𝑐𝑐𝑐
The Jacobian matrix [2] of the equilibrium point isshown in Eq. (2).
0𝑎𝑎 + 𝑏𝑏𝑏𝑏 0 𝑏𝑏𝑏𝑏
−𝑎𝑎 𝑐𝑐𝑐𝑐𝑐𝑐(𝑥𝑥)00 0𝐽𝐽(𝑥𝑥) = �� (2)000 𝑐𝑐−𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏(𝑥𝑥)𝑦𝑦 −𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏(𝑥𝑥) −𝑐𝑐 0
Order 𝐽𝐽 − 𝜆𝜆𝜆𝜆 = 0 ,when 𝜆𝜆1 = −𝑎𝑎𝑎𝑎, 𝜆𝜆2 = 𝑎𝑎𝑎𝑎, 𝜆𝜆3 =−𝑐𝑐𝑐𝑐, 𝜆𝜆4 = +𝑐𝑐𝑐𝑐, 𝑎𝑎 > 0, 𝑏𝑏 > 0, 𝑐𝑐 > 0 , Such equilibrium
points are central points, When 𝜆𝜆1 = 𝑎𝑎, 𝜆𝜆2 = −𝑎𝑎, 𝜆𝜆3 =−𝑐𝑐𝑐𝑐, 𝜆𝜆4 = +𝑐𝑐𝑐𝑐, 𝑎𝑎 > 0, 𝑏𝑏 > 0, 𝑐𝑐 > 0 , Such equilibrium
points are called unstable saddle point [3].
The eigenvalues of the equilibrium point andcorresponding types are shown in Table 1.

Fig.1. Lyapunov exponent diagram when a=9, c=14
Generated bifurcation diagram corresponding to
Lyapunov exponential diagrams. When parameters a=9,c=14 is shown in Fig. 2.

Fig.2. Bifurcation Diagram

Table 1. Characteristic value and type of balance pointbalancepoint(𝑥𝑥0 , 𝑦𝑦0 , 𝑧𝑧0 , 𝑤𝑤0 )(2kπ, 0,0,0)((2k + 1)π, 0,0,0)

characteristic value(𝜆𝜆1 , 𝜆𝜆2 , 𝜆𝜆3 , 𝜆𝜆4 )(−𝑎𝑎𝑎𝑎, 𝑎𝑎𝑎𝑎, −𝑐𝑐𝑐𝑐, +𝑐𝑐𝑐𝑐)(𝑎𝑎, −𝑎𝑎, −𝑐𝑐𝑐𝑐, +𝑐𝑐𝑐𝑐)

typeCenter point
Unstable saddlepoint

When k=0, the initial value is [π/2,1,1,1], and theparameter a=4, c=4, the Lyapunov exponent diagram ofthe system equation is displayed in Fig. 3.

Triangular wave, sawtooth wave, time delay function
and sine function without double Angle are often used togenerate multi-scroll. In this paper, the sine function
without double Angle is used to generate multiple volumes.
2.2. Foundational Investigation of the dynamics of aMulti-Scroll Chaotic System
Fig.3. Lyapunov exponent diagram when a=4, c=4

Taking the initial value as varying, is [𝜋𝜋/2 +2𝑘𝑘𝑘𝑘, 1,1,1], 𝑘𝑘 = (0,1,2) , when k=0, the initial value is[π/2,1,1,1], let the parameters a=9, c=14, the Lyapunov
exponent of the system equation is displayed in Fig. 1.

Generated bifurcation diagram corresponding toLyapunov exponential diagrams. when Parameters a=4,c=4 is shown in Fig. 4.

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Fig.6. Bifurcation Diagram
Fig.4. Bifurcation Diagram

It is clear that the range of chaos produced by the system
is periodic by contrasting the bifurcation diagram in Fig. 6with the Lyapunov exponent diagram in Fig. 5 of thefigures. The shift in Lyapunov exponent inside this rangeof [0, 2𝜋𝜋] resembles the representation of a sine function.
When k = 0, the initial value is [π/2,1,1,1], and the 16scroll chaotic attractors is shown in Fig. 7.

When the parameters alter, the distribution of theLyapunov exponent diagram will undergo significant
changes. The system can produce chaos over a wide rangeof parameter values.
2.3. A multi-scroll attractor and its stability analysis
The span within which the system exhibits chaos isevidently affected by a parameter adjustment. In thisscenario, altering the system's initial value unveils diverse
phase diagrams, and by manipulating the phase diagram,one can scrutinize the attractor attributes of the system.
Take the system parameter a = b = c = 4, then only
the initial value of the system is changed.
Let the initial value 𝑦𝑦0 = 𝑧𝑧0 = 𝑤𝑤0 = 1, and the
Lyapunov index diagram changing with the initial value x
is displayed in Fig. 5.

Fig.7. 16-scroll chaotic attractors
When k = 1, the initial value is [5π/2,1,1,1], and the
16-scroll chaotic attractors is shown in Fig. 8.

Fig.5. Lyapunov exponent diagram

6.

The corresponding bifurcation diagram is shown in Fig.
Fig. 8. 8-scroll chaotic attractors
When k = 2, the initial value is [9π/2,1,1,1], and the
16-scroll chaotic attractors is shown in Fig. 9.

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According to this nonlinear function, the Lyapunov
exponent diagram with different initial values can beobtained, as shown in Fig. 10.

Fig.9. 13-scroll chaotic attractors
As can be concluded from Fig. 7, 8 and 9, the amount ofscrolls in the x-y phase diagram varies dramatically withthe modification of the initial value, indicating that thesystem is multi-stable and seems to be conservative. The
multi-volume phase diagram indicates that the multistability is confirmed and the one-dimensional multi-scroll
chaotic attractor is obtained by injecting sin (x). To enable
the system to produce multi rolls, the non-double angle
sine function is introduced for y, z, and w.
When a = b = c = 4 , the eigenvalues of theequilibrium point and their corresponding types are shownin Table 2.

Fig.10. Lyapunov exponent diagram
In Fig. 10, it can be seen that the system before x = 4π.
The Lyapunov exponent of the system that is larger thanzero nonetheless follows a change rule that is roughly
periodic.After x = 4π, the system because of the use of
two multiplicative angle sine function, the law of change
has changed, breaking the rule of change before. The
graphic illustrates the Lyapunov exponent's trend of slow
expansion following 4π.
Here, the initial value is [π/2, π/2,1,1] ,and theparameter is a = b = c = 4 , and the two-dimensional
multi-volume attractor phase diagram can be obtained. andthe amount of attractors is associated with the taken by then. If the amount of scroll attractors produced by the system
be M , then there is Eq. (4).
𝑀𝑀 = (2𝑛𝑛)2 + (2𝑛𝑛 + 1)2(4)Let the order of the two-dimensional multivolume it
generates be R, the order Eq. (5) is available.
𝑅𝑅 = 4𝑛𝑛 + 1(5)
Taking n=1, in other words, N = 2π,the phase diagramof the generated 5 × 5 multi-scroll chaotic attractor phase
diagram is shown in Fig. 11.

Table 2. Characteristic value and type of balance pointbalance point(𝑥𝑥0 , 𝑦𝑦0 , 𝑧𝑧0 , 𝑤𝑤0 )(2kπ, 2kπ, 0,0)(2kπ, (2k+ 1)π, 0,0)((2k + 1)π, 2kπ, 0,0)

�(2k + 1)π, (2k+ 1)π, 0,0�

characteristic valuetype(𝜆𝜆1 , 𝜆𝜆2 , 𝜆𝜆3 , 𝜆𝜆4 )(−4𝑖𝑖, 4𝑖𝑖, −4𝑖𝑖, +4𝑖𝑖)(4, −4, −4𝑖𝑖, +4𝑖𝑖)

Center point
Unstablesaddle point
Unstablesaddle point(4, −4, −4𝑖𝑖, +4𝑖𝑖)(−4𝑖𝑖, 4𝑖𝑖, −4𝑖𝑖, +4𝑖𝑖)

Center point

Where N is the gate that limits the sine function, and its
action controls the width. When different functions areused in different cases, f (u)is linear when u < −N andu > N, and f (u) is a nonlinear function only when −N ≤u ≤ N, the system equation can produce multiple volumes.
When N = 2nπ, the equation of the nonlinear function [4]is shown in Eq. (3).
𝑢𝑢 + 𝑁𝑁, 𝑢𝑢 < −𝑁𝑁𝑓𝑓(𝑢𝑢) = � sin(𝑢𝑢) , −𝑁𝑁 ≤ 𝑢𝑢 ≤ 𝑁𝑁𝑢𝑢 − 𝑁𝑁, 𝑢𝑢 > 𝑁𝑁

Fig.11. 5 × 5 multi-scroll chaotic attractor phase
diagramFig. 11 shows that the number of attractors is M =(2)2 + (2 + 1)2 = 13.

(3)

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the multi-scroll chaotic attractor are conducted through
MATLAB simulations.
The primary objective of this paper is the integration of
the angle-free sine function within the conservative system,
resulting in the derivation of a multi-scroll conservative
chaotic system—this serves as the central research endeavor.
Through the adjustment of nonlinear function introduction,
the paper successfully achieves the generation of onedimensional and two-dimensional chaotic attractors while
also confirming the multistability of the system.
The research in this paper has verified some theories andachieved certain results, but there are still many
shortcomings to be corrected and a lot of work to becompleted in the research process, as follows:
The sinusoidal function without multiple angles used inthis paper is too simple and can only be used as the basic
switching function. Moreover, the chaotic attractor
obtained is too simple, with only one and two-dimensional
mesh-like structures. It is necessary to further deepen theuse of the switching function, so that attractors with more
complex structures can be generated, and threedimensional attractor topologies can be made on the basisof two-dimensional. At the same time, we can study theapplication of sinusoidal function in hyperchaotic systemor higher dimensional chaotic system, so as to constructmore complex and characteristic chaotic attractors.

Taking n=2, in other words, N = 4π, the phase diagramof the generated 9 × 9 multi-scroll chaotic attractor phase
diagram is shown in Fig. 12.

Fig.12. 9 × 9 multi-scroll chaotic attractor phase
diagram 1
Keep n=2 unchanged, alter the initial value to[π/2,5π/2,1,1], the phase diagram of the generated 9 × 9
multi-scroll chaotic attractor phase diagram is shown inFig. 13.

References

Fig.13 9 × 9 multi-scroll chaotic attractor phase
diagram 2
By modifying the initial values, two distinct phase
diagrams of multi-scroll attractors were obtained, asillustrated in Fig. 12 and Fig. 13. A comparative analysisof these two diagrams revealed that changing the initialvalues had a notable impact on the internal motion
trajectory within the phase diagram. However, it's
noteworthy that the number of scrolls in the attractor
remained constant, unaffected by these changes.

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3. Conclusion
This paper focuses on the conservative chaotic system andexplores the application of the sine function without
double angles as the nonlinear function in this context. By
altering the selection and quantity of variables related tothis nonlinear function, the research investigates thegeneration patterns of multi-scroll chaotic attractors. The
validation and characterization of both the multi-scroll
conservative chaotic system and the principles governing

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Authors Introduction
Mr. Jiaxin Li

Mr. Haozhe Sun

Mr. Yong Liu

In 2021, he attained his Bachelor ofEngineering degree from Tianjin
University of Science and Technology,China, majoring in the School ofElectronicInformationandAutomation. Currently, he is enrolled
in a master's program in engineering atthe same university, continuing hisacademic journey.
In 2023, he successfully obtained hisBachelor of Engineering degree fromTianjin University of Science andTechnology, specializing in the Schoolof Electronic Information andAutomation. Presently, he is activelyworking towards a master's degree inengineering at the same institution,
continuing his academic pursuits.
In 2021, he completed his Bachelor ofEngineering program at the School ofElectronic Information and Automation
within Tianjin University of Scienceand Technology. His primary focus interms of research lies within the domainof chaotic systems.

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