A Novel Hyperchaotic System and Its Control

Abstract

In this paper, a new hyperchaotic system is constructed via state feedback control. Some basic dynamical properties are studied, such as continuous spectrum, Lyapunov exponents, fractal dimensions, strange attractor and bifurcation diagram of the new hyperchaotic system. In addition, adaptive controllers are designed for stabilizing hyperchaos to unstable equilibrium with the unknown parameters. Numerical simulations are given to illustrate and verify the results. © 2009 World Academic Press, UK. All rights reserved.

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1371371371371378
A Novel Hyperchaotic System and Its Control
Jiang Xu ∗, Gouliang Cai, Song Zheng
School of Mathematics and Physics, Jiangsu university of Science and Technology
Zhenjiang, Jiangsu 212003, PR China
Faculty of science, Jiangsu University, Zhenjiang, Jiangsu 212013, PR China
Received 24 February 2008; Accepted 29 May 2008
Abstract
In this paper, a new hyperchaotic system is constructed via state feedback control. Some basic dynamical properties
are studied, such as continuous spectrum, Lyapunov exponents, fractal dimensions, strange attractor and bifurcation
diagram of the new hyperchaotic system. In addition, adaptive controllers are designed for stabilizing hyperchaos to
unstable equilibrium with the unknown parameters. Numerical simulations are given to illustrate and verify the results.
© 2009 World Academic Press, UK. All rights reserved.
Keywords: hyperchaos, dynamical behaviors, Lyapunov exponents, bifurcation, adaptive control, Lyapunov stability
theory
1 Introduction
In 1963, Lorenz discovered the first chaotic system when he studied atmospheric convection [1]. Since then, the
Lorenz system has been extensively studied in the field of chaos theory and dynamical systems. In 1999, Chen and
Ueta found the Chen system, which is the dual system of Lorenz system via chaotification approach [2]. In 2002, Lü
and Chen found a new chaotic system [3], bearing the name of the Lü system. In the same year, Lü et al. unified
above the three chaotic systems into one chaotic system which is called unified chaotic system [4].
In recent years, hyperchaos generation and control have been extensively studied due to its theoretical and
practical applications in the fields of communications, laser, neural work, nonlinear circuit, mathematics, and so on,
[5-19]. Hyperchaotic system, possessing more than one positive Lyapunov exponent, has more complex behavior and
abundant dynamics than chaotic system. Historically, hyperchaos was firstly reported by Rössler. That is, the noted
four-dimensional (4D) hyperchaotic Rössler system [5], which exhibits complex and abundant hyperchaotic dynamics
behaviors according to the detailed numerical and theoretical analyses. Very recently, hyperchaos was found
numerically and experimentally by adding a simple state feedback controller [12-19].
In this paper, a new hyperchaotic system is constructed, and stabilization of the hyperchaotic system is achieved.
We have briefly studied and analyzed its some basic dynamical properties and behaviors. Further, the new
hyperchaotic system is suppressed via adaptive control theory.
This paper is organized as follows: In Section 2, a new hyperchaotic system is constructed based on a three-
dimensional system by introducing a nonlinear state feedback controller. In Section 3, basic properties and behaviors
are investigated numerically and analytically. In Section 4, simple but effective controllers are designed for stabilizing
the hyperchaotic system to unstable equilibrium. The final section summarizes this work.
2 Construction of the New Hyperchaotic System
Recently, Liu et al. constructed a three-dimensional autonomous chaotic system [20], which has a new reversed
butterfly-shaped attractor. The system is described by
∗ Corresponding author. Email: xjiang2008@yahoo.com.cn (J. Xu).
Journal of Uncertain Systems
Vol.3, No.2, pp.137-144, 2009
Online at: www.jus.org.uk

138 J. Xu, G. Cai, and S. Zheng: A Novel Hyperchaotic System and Its Control
()
,
x
ay x
y bx kxz
zczhxy
=−
⎧
⎪=+
⎨
⎪=− −
⎩
&
&
&
(1)
in which a, b, c, h and k are positive real constants. When a=10, b=40, c=2.5, h=1 and k=16, the system (1) has a
reversed butterfly-shaped attractor shown in Fig.1.
To generate hyperchaos from system (1), one needs to extend the dimension of system (1). Moreover, the
improved system has to satisfy some requirements. Here, two basic requirements are listed as follows:
(i) Hyperchaos exists only in higher-dimensional systems, i. e., not less than four-dimensional (4D) autonomous
system for the continuous time cases.
(ii) It was suggested that the number of terms in the coupled equations that give rise to instability should be at
least two, in which one should be a nonlinear function.
On the basis of the system (1), a new hyperchaotic system can be generated by adding an additional stat variable
u to it. Then, we can get the following four-dimensional autonomous system
()
,
x
a
y
xu
y
bx kxz
zczhx
y
uxzdy
=
−+
⎧
⎪=+
⎪
⎨=− −
⎪
⎪=−
⎩
&
&
&
&
(2)
where a, b, c, h, k and d are also positive real parameters.
Fig.1: (a) Three-dimensional view x-y-z space (b) x-z phase plane strange attractor
Let the parameters be a=10, b=40, c=2.5, h=1, k=16 and d=2, thus, the new 4D system (2) is hyperchaotic in this
case. Its attractors are shown in Fig. 2. We will reveal the hyperchaotic dynamical properties and behaviors of this
four-dimensional system.

Journal of Uncertain Systems, Vol.2, No.4, pp.137-144, 2009 139
Fig.2: Phase portraits of hyperchaotic system (2), with a=10, b=40, c=2.5, h=1, k=16 and d=2:
(a) x-y plan; (b) x-z plan; (c) y-z plan; (d) x-u plan; (e) y-u plan; (f) z-u plan.
3 Properties and Dynamical Behaviors Analysis of Hyperchaotic System (2)
In this section, basic properties and complex dynamics of the new system (2) are invested, such as continuous
spectrum, Lyapunov exponents, fractal dimensions, strange attractors and bifurcation diagram. The new hyperchaotic
system (2) has the following basic properties.
1) Symmetry and invariance
Note that the invariance of the system (2) under the transformation (x, y, z, u)→(-x, -y, z, u), i.e. under reflection
in the z-axis. The symmetry persists for all values of the system parameters.
2) Dissipativity and the existence of attractor
For system (2), one has
12.5
xyzu
Vac
xyzu
∂∂∂∂
∇= + + + =−−=−
∂∂∂∂
&& &
&.
So system (2) is dissipative, with an exponential contraction rate: dv/dt=e-12.5. That is a volume element V0 is
contracted by the flow into a volume element V0e-12.5t in time t. This means that each volume containing the system
trajectory shrinks to zero as t→∞ at an exponential rate -12.5. In fact, numerical simulations have shown that system
orbits are ultimately confined into a specific limit set of zero volume, and the system asymptotic motion settles onto
an attractor.
3) Equilibrium and stability
The equilibrium of system (2) can be easily found by solving the four equations 0xyzu====
&& &
&, which lead
to () 0, 0, 0a y x u bx kxz cz hxy−+= + =−− =
, and 0xz dy
−
=. It can be easily verified that there is only one
equilibrium E (0, 0, 0, 0), because the parameters are all positive constants.
For equilibrium E (0, 0, 0, 0), system (2) is linearized, the Jacobian matrix is defined as
0
0 1 10 10 0 1
0040000
0002.50
00200
aa
bkz kx
Jhy hx c
zdx
−−
⎛⎞⎛⎞
⎜⎟⎜⎟
+
⎜⎟⎜⎟
==
⎜⎟⎜⎟
−−− −
⎜⎟⎜⎟
−−
⎝⎠⎝⎠

140 J. Xu, G. Cai, and S. Zheng: A Novel Hyperchaotic System and Its Control
To gain its eigenvalues, let 00IJ
λ
−=
.
These eigenvalues corresponding to equilibrium E (0, 0, 0, 0) are respectively obtained as follows:
λ1= -25.6909, λ2= 15.4899, λ3= -2.5 and λ4= 0.201.
Here λ2 and λ4 are two positive real number, λ1 and λ3 are two negative real numbers. Therefore, the equilibrium
E (0, 0, 0, 0) is a saddle point, which is unstable.
4) Lyapunov exponents and Lyapunov dimension
In order to discover the effective of the parameters on the dynamics of the new 4D system (2), we fix the
parameters a=10, b=40, c=2.5, h=1 and k=16, let the parameter d vary in the interval (0,12]. Given the initial
condition (0.1, 0.1, 0.1, 0.1), according to the detailed numerical as well as theoretical analysis [21-22], the Lyapunov
exponent spectrum are proposed to show how the system (2) changes with in creasing value of parameter d shown in
Fig. 3. And the simulation results are obtained by using 4-order Runge-Kutta method with the step length taken as
0.001.
Suppose that LEi (i=1, 2, 3, 4) are the Lyapunov exponents of the dynamical system, satisfying the condition
4321
LE LE LE LE≤≤≤
. From Fig. 3, the abundant dynamics that the system (2) undergoes versus the parameter d are
summarized as follows.
When (0,0.79]d∈, the largest Lyapunov exponent is positive, implying the system shows chaotic behavior. Fig.5.
displays chaotic attractors for different values of parameter d respectively.
When [0.80,12]d∈, the system (3) has two positive Lyapunov exponents except a very narrow periodic window
near 6.81 (shown in Fig.6), representing that the hyperchaos occurs. See Fig.2.
Now, we consider properties of the system (2) when a=10, b=40, c=2.5, h=1, k=16 and d=2. From Fig.4, we
know the two largest values of positive Lyapunov exponents of nonlinear system are obtained as λL1=1.0088 and
λL2=0.1063. It is related to the expanding nature of different direction in phase space. Another one Lyapunov
exponent λL3=0.0038. It is related to the critical nature between the expanding and the contracting nature of different
direction in phase space.
While negative Lyapunov exponent λL4=-13.6191. It is related to the contracting nature of different direction in
phase space.
So, we can obtain the Lyapunov dimension of the new hyperchaos attractors of system (2), it is described as
123
L14
1
1 1.0088 0.1063 0.0038
D =j+ 3 3 3.0822.
13.6191
jLLL
Li
iL
Lj
λλλ
λλ
λ
=
+
++ ++
=+ =+ =
−
∑
The Lyapunov dimension of the new system (2) is also fractional dimension, therefore, there is really hyperchaos
in this system. The hyperchaotic strange attractors are shown in Fig.2.
Fig.3: Lyapunov exponents spectrum of the Fig.4: Lyapunov exponents of ystem (2) when d=2
system (2) versus parameter d.

Journal of Uncertain Systems, Vol.2, No.4, pp.137-144, 2009 141
Fig.5: Chaos phase portraits of the system (2) in the y-z plane for different d: (a) d=0.05; (b) d=0.5
Fig.6: Periodic phase portraits of the system (2) in the x-z plane for different d:(a) d=6.81; (b) d=6.8105.
5) Waveform, Spectrum, Poincaré mapping and Bifurcation diagram
According to the above analyses, obviously, they are also new reverse butterfly-shape chaotic attractors, while
the sensitive dependence on the initial conditions is a prominent characteristic of chaotic behavior: when the initial
values are changed, the chaotic dynamical behavior of this system disappears immediately.
The waveforms of x(t) in time domain are shown in Fig.7, apparently, they are non-periodic in system (2), which
is one of basic chaotic dynamical properties.
The spectrum of system (2) is also studied, and its spectrum is continuous as shown in Fig.8.
The Poincaré mapping of system (2) is also analyzed. It is clear that the Poincaré mappings are these points in
confusion as shown in Fig.9.
The bifurcation diagram of x with increasing a is given in Fig. 10, and it shows abundant and complex
dynamical behaviors.
Fig.7: x(t) waveform of system (2) Fig. 8: Spectrum of |x| in system (2)

142 J. Xu, G. Cai, and S. Zheng: A Novel Hyperchaotic System and Its Control
Fig. 9: The Poincaré map of x-y Fig.10: Bifurcation diagram for increasing a with
plane of the system (2) b=40, c=2.5, d=2, h=1 and k=16 of system (2)
The above theoretical analysis and numerical simulation both show that system (2) is really a new hyperchaotic
system and has more sophisticated topological structure and abundant hyperchaotic dynamical properties.
4 Stabilizing the Unstable Equilibrium E0 (0, 0, 0, 0)
In this section, by using adaptive control theory the hyperchaos in dynamical system (2) is controlled to unstable
equilibrium in the presence of unknown system parameters. For this purpose, let us assume that the controlled system
is as follows:
1
2
3
4
()
,
xayx u
ybxkxz
zczhxy
uxzdy
µ
µ
µ
µ
=−++
⎧
⎪=+ +
⎪
⎨=− − +
⎪
⎪=−+
⎩
&
&
&
&
(3)
in which a, b, c, k, d, and h are unknown parameters, 1234
,,,
µ
µµµ
are the controllers to be designed.
Choose the controllers 1234
,,,
µ
µµµ
as follows:
1
2
3
4
ˆˆ
(1)
ˆˆ
ˆ
ˆ
(1)
ˆ,
axa
y
u
bx kxz
y
czhx
y
dy xz u
µ
µ
µ
µ
=− −−
⎧
⎪=− − −
⎪
⎨=− +
⎪
⎪=−−
⎩
(4)
and the parameters estimation update law as follows:
2
2
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ.
axyx
bxy
cz
dyu
hxyz
kxyz
⎧=−
⎪
⎪=
⎪=−
⎪
⎪
⎨=−
⎪
⎪=−
⎪
⎪=
⎪
⎩
&
&
&
&
&
&
(5)
We choose Lyapunov function for (2) as follows:
222 2222 222
1(),
2
Vxyzwabcdhk=+++++++++
%%%%
%%

Journal of Uncertain Systems, Vol.2, No.4, pp.137-144, 2009 143
where ˆˆˆ
ˆˆ
,,, ,aaabbbcccdddhhh=− =− =− =− =−
%%%
%% and ˆ,kkk
=
−
%ˆˆˆˆ
ˆˆ
,,,,,abcdhk are the estimate values of these
unknown parameters, respectively.
The time derivate of V along trajectories (3) is
12 3
4
(( ) ) ( ) ( )
()
V x a y x u y bx kxz z cz hxy
wxzdy aabbccddhhkk
µ
µµ
µ
=−++++++−−+
+−+++++++
&
&&&&
%% %% %% %%
&&
%% %% (6)
Substituting Eqs. (4) and (5) into Eq. (6) yields
222 2
0Vxyzw=− − − − <
&.
It is clear that V is positive definite and V
&is a negative definite in the neighborhood of the zero solution for the
system (2); Therefore, based on the Lyapunov stability theory, the controlled system (3) can asymptotically converge
to the unstable equilibrium E0 (0, 0, 0, 0) with the controllers (4) and the parameters estimation update law (5). Fig.10
shows the time responses of the four states of the controlled system (3). The controllers ( 1234
,,,
µµµµ
) action is
shown in Fig. 11. We can see the system is driven to the origin immediately.
Fig.11: The time response of states(x, y, z, w): Fig.12. Control actions 1234
(), (), (), ()tttt
µ
µµµ
stabilizing the equilibrium E0.
5 Conclusion
In this paper, a new hyperchaotic system is introduced. There are abundant and complex dynamical behaviors in new
hyperchaotic system (2). This new hyperchaotic attractor is different from the hyperchaotic Lorenz attractor,
hyperchaotic Chen attractor as well as hyperchaotic system proposed by Liu Congxin and Wang Faqiang, etc. Further,
adaptive controller has been derived for controlling hyperchaos to unstable equilibrium point of the uncontrolled
system with unknown parameters. These new hyperchaotic attractors and their forming mechanism need further study
and exploration. A great deal of achievements will be obtained in the near future.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant Nos 70571030, 90610031) and
the Advanced Talents’ Foundation of Jiangsu University (Grant No 07JDG054).
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