SUGISAKAMASANORI

Journal of Advances in Artificial Life Robotics
Vol. 3(2); September (2022), pp. 122–127
ON LINE ISSN 2435-8061; ISSN-L 2435-8061
https://alife-robotics.org/jallr.html

Research Article

Synchronization of Novel 5D Hyperchaotic Systems Based onCenter Translation Method
Hong Niu
College of Electronic Information and Automation, Tianjin University of Science & Technology, China

ARTICLE INFO
Article History
Received 23 November 2021
Accepted 23 November 2022
Keywords
Novel 5D hyperchaotic system

ABSTRACTIn this paper, synchronization of novel five-dimensional (5D) autonomous hyperchaotic systems
is studied. The synchronization control law is proposed based on the center translation method.
A structure compensator is formulated to make the mathematical model of the error system thesame as that of the response system, and a linear feedback controller and its simplification are
designed via the Lyapunov stability theory to make the error system globally asymptoticallystable at the origin. Thus, the two 5D hyperchaotic systems with different initial values aresynchronized. Some relevant numerical simulation results, such as the curves of thecorresponding synchronization state variables and the errors, are given to illustrate the feasibility
and effectiveness of the synchronization control law.

Hyperchaos synchronization
Center translation method
Lyapunov stability theory
Global asymptotic stability© 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/).

systems with different initial values is studied based on
the center translation method, so that the mathematicalmodel of the synchronization error system would be thesame as that of the controlled system formulated in [2].
Thus, the control law in [2] can be applied to the designof the linear feedback synchronization controller in thispaper. Corresponding numerical simulation results arepresented to demonstrate the validity of thesynchronization method.

1. Introduction
Hyperchaos was first presented in 1979 by Otto Rössler[1]. The main differences between hyperchaotic systemand chaotic system are as follows. Firstly, the minimal
dimension of the phase space that embeds a hyperchaotic
attractor should be at least four, which requires theminimum number of coupled first-order autonomous
ordinary differential equations to be four. Secondly, thenumber of terms in the coupled equations giving rise toinstability should be at least two, of which at least one
should have a nonlinear function [2]. Hence, hyperchaos
is much more complicated than chaos, and hyperchaos
synchronization has greater application significance andengineering value in secure communication.
Stability control of the novel 5D hyperchaotic system
has been presented in [2]. In this paper, the mathematicalmodel of the novel 5D hyperchaotic system is given as
the drive system. Hyperchaos synchronization of the 5D

2. The Novel 5D Hyperchaotic System
The dynamic equations of the novel 5D hyperchaotic
system areCorresponding author’s E-mail: spots@163.com web: www.tust.edu.cn122ex1 = a ( y1 − x1 ) ,where x1 , y1 , z1 , v1 , w1  R are state variables, and a =y1 = ( c − a ) x1 + cy1 + w1 − x1 z1 ,23, b = 3, c = 18, m = 12 and h = 4 [2].
Let the initial values of the system (1) be (x10, y10, z10,v10, w10) = (1, 1, 1, 1, 1), then the Lyapunov exponents
respectively are λ11 = 0.8732 > 0, λ12 = 0.1282 > 0, λ13 =-0.0013 ≈ 0, λ14 = -0.5770 < 0 and λ15 = -8.4231 < 0. It
indicates that the system (1) is hyperchaotic. The
attractors of the 5D hyperchaotic system (1) are shown in
Figure 1.

z1 = −bz1 + x1 y1 ,v1 = mw1 ,
w1 = − y1 − hv1 ,(1)Figure 1 Attractors of the 5D hyperchaotic system: (a1) z-x-y; (a2) v-x-y; (a3) w-x-y; (a4) x-v-z; (a5) x-w-z; (a6) w-x-v

us = us1

3. Hyperchaos Synchronization Based on CenterTranslation Methodandus 2us 3us 4

us 5 Tuc = uc1 uc 2 uc 3 uc 4 uc 5 T

3.1. Formulation of error system
Take the system (1) as the drive system, then the responsesystem is formulated asx2 = a ( y2 − x2 ) + us1 + uc1 ,y2 = ( c − a ) x2 + cy2 + w2 − x2 z2 + us 2 + uc 2 ,z2 = −bz2 + x2 y2 + us 3 + uc 3 ,v2 = mw2 + us 4 + uc 4 ,w2 = − y2 − hv2 + us 5 + uc 5 ,( 2)

are structure compensator and synchronization controller tobe designed. Let us = 0, uc = 0, and the initial values of theresponse system (2) be (x20, y20, z20, v20, w20) = (5, 0, 4, 3,8), then the Lyapunov exponents respectively are λ21 =0.9121 > 0, λ22 = 0.1175 > 0, λ23 = -0.0008 ≈ 0, λ24 = -0.5533< 0 and λ25 = -8.4755 < 0. It shows that the response system(2) is also hyperchaotic.
Lete =  e1 e2e3=  x2 − x1y2 − y1

where123e4e5 

Tz2 − z1 v2 − v1

w2 − w1 The4e4 + e5e5m
= − ( k1 + a ) e12 + ce1e2 − ( k2 − c ) e22

be the synchronization error andus = us1

us 2

us 3

us 4

V = e1e1 + e2e2 + e3e3 +us 5 T0 x z + x z − 2x z 1 1  2 1 1 2=  − x2 y1 − x1 y2 + 2 x1 y1  ,0
0h 2e4 − k5e52mc
3  −  k1 + a −  e12 −  k2 − c  e2222 h− ( k3 + b ) e32 − k4 e42 − k5e52 .
( 5)
mFor V  0 , the parameters k1 , k 2 , k 3 , k 4 and k 5− ( k3 + b ) e32 − k4then the error system is simplified as

e1 = a ( e2 − e1 ) + uc1 ,e2 = ( c − a ) e1 + ce2 + e5 − e1e3 + uc 2 ,e3 = −be3 + e1e2 + uc 3 ,should satisfy thate4 = me5 + uc 4 ,cc
 0,k1  − a,2233k2 − c  0,
k 2  c,22
k3 + b  0,
k3  −b,hk4  0,k4  0,mk5  0,k5  0,

k1 + a −( 3)

e5 = −e2 − he4 + uc 5 .

Comparing the mathematical model of the error system(3) with that of the controlled system (2) in [2], it can befound that the two models are similar. Hence, thesynchronization controller uc is designed as

uc = uc1 uc 2

uc3 uc 4=  −k1e1 −k2e2−k3e3

uc5 T−k4e4−k5e5  ,T

where k1 , k2 , k3 , k4 , k5  0 .

Thus, the linear feedback synchronization controller uc isdesigned as

3.2. Design of linear feedback synchronization
controller
Theorem 1. Let x = 0 be an equilibrium point for
x = f ( x ) , where f : D → R n is a locally Lipschitz
map from a domain D  R n into R n . Let V : Rn → R
be a continuously differentiable function such thatV ( 0) = 0 and V ( x )  0,  x  0
V ( x )  0, x  0then x = 0 is globally asymptotically stable [2].
Take a continuously differentiable function( 4)

as a Lyapunov function candidate for the error system (3).
Then, the derivative V is derived asuc = uc1 uc 2uc 3 uc 4

= 0 −30e2

0 −e4

uc 5 T−e5  .
T(6)From Theorem 1, the error system (3) is globally
asymptotically stable at the origin. It indicates that theresponse system (2) is synchronized with the drive system(1).
3.3. Numerical simulation under the synchronization
controller ucx →  V ( x) → 1h
V =  e12 + e22 + e32 + e42 + e52 2m

k1 = 0,k2 = 30,
k3 = 0,k4 = 1,k5 = 1.

Remark 1. The initial values of the drive system (1) and theresponse system (2) are (x10, y10, z10, v10, w10) = (1, 1, 1, 1,1) and (x20, y20, z20, v20, w20) = (5, 0, 4, 3, 8) respectively inthis paper.
Definition 1. After adding the structure compensator us
and the linear feedback synchronization controller uc to theresponse system (2), the Lyapunov exponents of theresponse system (2) are called sub-Lyapunov exponents [3].
Theorem 2. The response system (2) and the drive system(1) will synchronize only if the sub-Lyapunov exponents are
all negative [3].
124The curves of the errors and the corresponding statevariables before and after adding the structurecompensator us and the linear feedback synchronization
controller uc to the response system (2) are shown in
Figure 2 and Figure 3 respectively. Comparing Figure 3
with Figure 2, it can be found that the errors e1, e2, e3, e4
and e5 converge to zero asymptotically and rapidly and thecorresponding state variables are synchronized well after
adding us and uc to the response system (2). Moreover, thesub-Lyapunov exponents of the response system (2) are
λ21c = -1.0292, λ22c = -1.0355, λ23c = -3.0000, λ24c =-17.4669 and λ25c = -17.4690, which are all negative. From
Theorem 2, the response system (2) and the drive system(1) have synchronized.
(a) Errors(a) Errors(b) State variables
Figure 3 After adding us and uc: (a) Errors; (b) State variables

3.4. Simplification of synchronization controller
Corollary 1. Let x = 0 be an equilibrium point for
x = f ( x ) , where f : D → R n is a locally Lipschitz
map from a domain D  R into R . Let V : Rn → R bea continuously differentiable, radially unbounded, positive
definite function such that V ( x )  0 for all x  Rn . LetnnS = x  R n V ( x ) = 0 and suppose that no solution can(b) State variables
Figure 2 Before: (a) Errors; (b) State variables

stay identically in S, other than the trivial solutionx ( t )  0 . Then, the origin is globally asymptoticallystable [2].
Still take Equation (4) as a Lyapunov function candidate
for the error system (3). Now let k4 = 0 and substitute k1 =125

k3 = k4 = 0 into Equation (5). Then, the derivative V is
reduced tohe4e4 + e5e5m= −ae12 + ce1e2 − ( k2 − c ) e22 − be32 − k5e52

V = e1e1 + e2e2 + e3e3 +c
3  −  a −  e12 −  k2 − c  e22 − be32 − k5e52 .
22 (7)For V  0 , the parameters k 2 and k 5 should satisfy that

33k = 30,
k2 − c  0,
k 2  c, 222k5 = 1.
k5  0,k5  0,From Corollary 1, to find S =e  R V ( e ) = 0 ,are shown in Figure 2 and Figure 4 respectively. Comparing
Figure 4 with Figure 2, it can be found that the errors e1, e2,e3, e4 and e5 converge to zero asymptotically and rapidly
and the corresponding state variables are synchronized wellafter adding us and ucs to the response system (2).
Furthermore, the sub-Lyapunov exponents of the responsesystem (2) are λ21cs = -0.5284, λ22cs = -0.5302, λ23cs =-3.0006, λ24cs = -17.4668 and λ25cs = -17.4690, which are all
negative. From Theorem 2, the response system (2) and thedrive system (1) have synchronized. However, comparing
Figure 4 with Figure 3, it can be seen that the speed ofconvergence and synchronization under ucs is a little bit
slower than that under uc. Nevertheless, the simplified
synchronization controller ucs only has two feedback
variables, such that it is easier to implement via circuit thanthe synchronization controller uc. Hence, the simplified
synchronization controller ucs has higher value inengineering application.

5note thatV = 0  e1 = e2 = e3 = e5 = 0

Hence, S = e  R 5 e1 = e2 = e3 = e5 = 0 .

( ) be a solution that belongs identically toLet e t

S = e  R5 e1 = e2 = e3 = e5 = 0 , so thate1 ( t ) = e2 ( t ) = e3 ( t ) = e5 ( t )  0
 e1 ( t ) = e2 ( t ) = e3 ( t ) = e4 ( t ) = e5 ( t )  0
 e4 ( t )  0(a) Errors

Therefore, the only solution that can stay identically in is the trivial solution

S = e  R5 V ( e ) = 0e ( t )  0 . From Corollary 1, the origin is globally

asymptotically stable. Finally, the synchronization
controller uc in Equation (6) is simplified asucs = ucs1 ucs 2 ucs 3 ucs 4 ucs5 T= 0 −30e2

0 0 −e5  .
T(8)3.5. Numerical simulation under the simplified
synchronization controller ucs
The curves of the errors and the corresponding statevariables before and after adding the structurecompensator us and the simplified linear feedback
synchronization controller ucs to the response system (2)(b) State variables
Figure 4 After adding us and ucs: (a) Errors; (b) State variables

126

4. Conclusions
Synchronization of the novel 5D hyperchaotic systems isproposed based on the center translation method in thispaper. A linear feedback synchronization controller and its
simplification are designed via the Lyapunov stability
theory. Numerical simulation results illustrate thefeasibility of the synchronization method. The study hassome engineering significance. Furthermore, the circuit
implementation of the synchronization system is underinvestigation and will be reported elsewhere.
References
1.
2.

3.

O. E. Rössler, An equation for hyperchaos, Physics Letters
A, 1979, 71(2–3): pp. 155–157.
H. Niu, Analysis and stability control of a novel 5D
hyperchaotic system, Journal of Robotics, Networking andArtificial Life, 2022, 8(4): pp. 245-248.
L. M. Pecora, T. L. Carroll, Synchronization in chaoticsystems, Physical Review Letters, 1990, 64(8): pp. 821-824.
Author Introduction
Dr. Hong Niu
She received the Ph.D. degree incontrol science and engineeringfrom Tianjin University, China, in2014. She is currently a lecturer
with the College of Electronic
Information and Automation,Tianjin University of Science &Technology, China.

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