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Discrete Dynamics in Nature and Society
Volume 2011, Article ID 454636, 11 pages
doi:10.1155/2011/454636
Research Article
Chaotic Attractor Generation via
Space Function Controls
Hong Shi,1 Guangming Xie,2, 3 and Desheng Liu2
1
Department of Mathematics and Physics, Beijing Institute of Petrochemical Technology,
Beijing 102617, China
2
Intelligent Control Laboratory, College of Engineering, Peking University, Beijing 100871, China
3
School of Electrical and Electronics Engineering, East China Jiaotong University,
Nanchang 330013, China
Correspondence should be addressed to Guangming Xie, xiegming@pku.edu.cn
Received 28 March 2011; Accepted 17 July 2011
Academic Editor: Elmetwally Elabbasy
Copyright q 2011 Hong Shi et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The analysis of chaotic attractor generation is given, and the generation of novel chaotic attractor is
introduced in this paper. The underlying mechanism involves two simple linear systems with onedimensional, two-dimensional, or three-dimensional space functions. Moreover, it is demonstrated
by simulation that various attractor patterns are generated conveniently by adjusting suitable
space functions’ parameters and the statistic behavior is also discussed.
1. Introduction
Owing to theoretical development in mathematics and technological advances in engineering, complex phenomena are rapidly becoming possible to be studied systematically. As one
of the essences of natural complexity, chaos has been found to be very useful in a variety of applications such as science, mathematics, and engineering communities 1–4 and various techniques such as identification and synchronization 3, 5, 6. In recent years, people
tend to introduce the chaos to many applications and for their purpose, chaotic attractors
in different shapes may be needed for desired dynamical behaviors. As a result, effective
generation of different chaotic attractors with simple techniques becomes an interesting
problem in the past decade.
Many chaotic attractors have been found numerically, and experimentally and it is
relatively easy to generate chaotic systems numerically, but it is usually very hard to analyze or verify the dynamical characteristics of nonsmooth systems, even for the switched
2
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systems with low dimensions 7, 8. To deal with the stability of the equilibrium of switched
linear systems, many efforts have been made and strict analysis has been carried out
9, 10. However, in the studies of complex nonsmooth phenomena, there has not been any
effective mathematical method, though differential inclusions provide a strict way to describe
discontinuous dynamics 11.
Some new chaotic systems have been developed in 12–16, but there does not seem
to be a general methodology for generating chaos. In 17, a chaotic attractor in a new funnel
shape is introduced, simply by designing a switched system with hysteresis switching signal.
It also could be regarded as a method of chaotic attractor generation with one-dimensional
space function. In this paper, we propose a new method of chaotic attractor generation for two
linear systems. It is shown that chaos can be generated by applying an appropriate rule of the
space functions. This new rule can generate different types of chaos or chaos-like behaviors
from different pairs of linear systems.
The rest of this paper is organized as follows. Section 2 presents the structure of two
simple linear systems to generate a new chaotic attractor with one-dimensional space function. Section 3 introduces two-dimensional and three-dimensional space functions to
generate new chaotic attractors. Then, Section 4 concentrates on the pattern changes of the
generated attractors with parameters variation. A brief conclusion is given in Section 5.
2. Specification of the Chaotic Attractor Generation by
One-Dimensional Space Function
In this section, we first introduce two simple linear systems for the generation of chaotic
attractors. Consider the following system:
Ẋt A1 Xt,
2.1
Ẋt A2 Xt,
2.2
where the state X x, y, zT ∈ R3 and
⎛
a
b1 0
⎞
⎜
⎟
⎟
A1 ⎜
⎝−b1 0 0⎠,
0 0 c
⎛
0
b2
0
⎞
⎜
⎟
⎟
A2 ⎜
⎝−b2 −a 0 ⎠,
0 0 −c
2.3
Furthermore, the parameters a, b1 , b2 , and c are chosen to satisfy
a > 0,
c > 0,
a2 − 4bi2 < 0,
i 1, 2.
2.4
We introduce one-dimensional space function:
fz z ∈ z1 , z2 2.5
to make the system trajectory switch between z1 and z2 , where z1 and z2 are positive constants satisfying z1 ≤ z2 . The switching rule is constructed as follows. When system 2.1 is
Discrete Dynamics in Nature and Society
x
3
4
4
10
3
3
9
2
2
8
1
1
7
y
0
z
0
−1
5
−2
−2
4
−3
−3
3
−4
0
200
400
600
−4
800 1000
0
200
400
t
600
10
3
9
2
8
200
z
0
−1
8
z
6
6
4
2
4
4
−3
3
−4
−4
2
−4
0
−2
x
2
2
y
4
y
d x-y projection
800 1000
10
−2
4
600
c t-z projection
5
2
400
t
7
1
0
0
b t-y projection
4
−2
2
800 1000
t
a t-x projection
y
6
−1
e y-z projection
0
5
−2
0
−4 −5
x
f chaotic attractor
Figure 1: Chaotic attractor generated by one-dimensional space function 2.5.
active, it will switch to system 2.2 at time t1 if zt1 z2 . Similarly, when system 2.2 is
active, it will switch to system 2.1 at time t2 if zt2 z1 .
With this switching rule, the switched system will generate chaos or chaos-like
behavior if the system parameters are properly chosen. As shown in Figure 1, the switched
system has a chaotic attractor, where
a 0.7,
b1 0.8,
b2 −2,
c 0.4
2.6
and we assume that z1 2, z2 10. The maximum Lyapunov exponent is −6.7206e − 004,
which indicates the chaotic behavior of the switched system.
Solving ẋ ẏ ż 0 yields the two linear systems equilibrium 0 0 0T . The equilibrium is unstable since all the real parts of the three eigenvalues of Jacobian J, at the origin
for system 2.1,
⎛
a
⎜
J⎜
⎝−b1
0
⎞
b1 0
⎟
0 0⎟
⎠,
0 c
2.7
4
x
Discrete Dynamics in Nature and Society
15
15
10
10
5
5
0
y
22
20
18
16
z 14
0
−5
−5
−10
−10
12
10
−15
0
200
400
600
800 1000
−15
8
0
200
400
t
6
0
800 1000
a t-x projection
400
600
800 1000
t
b t-y projection
c t-z projection
22
20
10
5
z
0
18
25
16
20
14
z
12
−5
15
10
10
−10
−15
−20
200
t
15
y
600
5
10
8
−10
0
10
20
6
−20
−10
x
0
10
20
y
d x-y projection
e y-z projection
5
y
0
10
−5
−10 −10
0
x
f chaotic attractor
Figure 2: Chaotic attractor generated by two-dimensional space function 3.1.
are positive due to a > 0 and c > 0. Obviously, the system 2.1 does not have a stable equilibrium.
Similarly, the system 2.2 has a stable equilibrium since all the real parts of the three
eigenvalues of Jacobian J, at the origin for system 2.2,
⎛
0
b2
0
⎞
⎜
⎟
⎟
J⎜
⎝−b2 −a 0 ⎠,
0
2.8
0 −c
are negative due to a > 0 and c > 0.
In addition, it is easy to see that the state trajectory of the systems 2.1 and 2.2 stays
in the region of
D
T
x, y, z ∈ R3 | 0 < z1 ≤ z ≤ z2 .
2.9
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8
22
4
4
20
18
2
2
0
y
−2
z
0
0
200
400
600
−6
800 1000
8
0
t
a t-x projection
0
200
400
600
800 1000
t
c t-z projection
24
22
4
2
0
z
−2
20
25
18
20
16
z
14
5
10
10
8
−5
0
x
5
6
−10
10
d x-y projection
15
10
12
−4
−6
−10
6
200 400 600 800 1000
t
b t-y projection
6
y
14
10
−4
−6
16
12
−2
−4
−8
24
6
6
x
5
−5
0
y
5
5
y
10
e y-z projection
0
10
−5
−10 −10
0
x
f chaotic attractor
Figure 3: Chaotic attractor generated by three-dimensional space function 3.2.
4
14
3
12
2
10
1
y
0
z
−1
6
−2
4
−3
−4
−5
z
8
0
5
x
a x-y projection
2
−4
−2
0
y
b y-z projection
2
4
14
12
10
8
6
4
2
4
2
y
0
5
−2
−4 −5
0
c chaotic attractor
Figure 4: Chaotic attractor generated by z1 2, z2 14.
x
6
Discrete Dynamics in Nature and Society
10
3
9
2
y
10
9
8
7
6
5
4
3
4
8
1
7
z
0
z
6
−1
5
−2
4
−3
−4
3
−4
0
x
−2
2
4
−2
a x-y projection
0
y
2
2
4
b y-z projection
0
y −2
5
−4 −5
0
x
c chaotic attractor
Figure 5: Chaotic attractor generated by z1 3, z2 10.
8
22
6
20
4
18
16
2
y
14
0
z
−2
12
10
−4
8
−6
6
−8
−10
−5
0
5
10
4
−10
0
−5
5
10
y
x
a x-y projection
b y-z projection
25
20
z 15
10
5
10
5
y
10
0
−5
−10 −10
0
x
c Chaotic attractor
Figure 6: Chaotic attractor generated by F1 6, F2 250.
In fact, once the state of the system 2.1 reaches the plane {z z2 }, according to the switching
rule, it will switch to system 2.2, then the velocity along the direction z is ż −cz2 < −cz1 <
0, which means that z will not be greater than z2 . Similarly, when the state of the system 2.2
reaches the plane {z z1 }, it will switch to system 2.1 and the velocity along the direction
z is ż cz1 > 0, which implies that z will never be less than z1 . Hence, the system trajectory
switches between z1 and z2 .
Discrete Dynamics in Nature and Society
7
6
22
20
4
18
2
y
16
z
0
14
12
−2
10
−4
8
−6
−10
−5
0
5
10
6
−10
0
−5
5
10
y
x
a x-y projection
b y-z projection
25
20
z 15
10
5
10
5
y
10
0
−5
−10 −10
0
x
c Chaotic attractor
Figure 7: Chaotic attractor generated by F1 48, F2 250.
From the foregoing discussion, we observe that as t → ∞, the system switches
between system 2.1 and system 2.2 and the state orbits never go out from the region D.
Hence, the state orbits are folded and stretched repeatedly, leading to the generation of chaos
or chaos-like behaviors.
3. Generating Chaotic Attractors by Two-Dimensional and
Three-Dimensional Space Functions
From the analysis in Section 2, the region of the system orbits is restricted by one-dimensional
space function, leading to the generation of chaos or chaos-like behaviors. Similarly, we can
introduce two-dimensional and three-dimensional space functions to generate new chaotic
attractors. Still considering the system 2.1 and the system 2.2, we introduce a simple twodimensional space function:
f y, z y2 z − 62 ∈ F1 , F2 3.1
to make the system trajectory switch between F1 and F2 , where F1 and F2 are positive
constants satisfying F1 ≤ F2 . The switching rule is constructed as follows. When system 2.1
8
Discrete Dynamics in Nature and Society
15
24
22
10
20
5
y
18
0
z
16
14
−5
12
−10
10
−15
−20
8
0
−10
10
20
6
−20
0
y
−10
x
a x-y projection
10
20
b y-z projection
25
20
z 15
10
5
20
10
y
0
20
−10
−20 −20
0
x
c Chaotic attractor
Figure 8: Chaotic attractor generated by F1 10, F2 300.
is active, it will switch to system 2.2 at time t∗1 if fyt∗1 , zt∗1 F2 . Similarly, when system
2.2 is active, it will switch to system 2.1 at time t∗2 if fyt∗2 , zt∗2 F1 .
With this switching rule, we can generate chaos or chaos-like behavior by the system
parameters chosen as 2.6 and we assume that F1 10, F2 250. As shown in Figure 2, the
maximum Lyapunov exponent is 0.0017.
Similarly, we can introduce a three-dimensional space function as following:
g x, y, z x2 y2 z − 62 ∈ G1 , G2 3.2
to make the system trajectory switch between G1 and G2 , where G1 and G2 are positive
constants satisfying G1 ≤ G2 . The switching rule is constructed as follows. When system 2.1
∗∗
∗∗
∗∗
is active, it will switch to system 2.2 at time t∗∗
1 if gxt1 , yt1 , zt1 G2 . Similarly, when
∗∗
∗∗
∗∗
system 2.2 is active, it will switch to system 2.1 at time t2 if gxt∗∗
2 , yt2 , zt2 G1 .
With this switching rule, we can generate chaos or chaos-like behavior by the system
parameters chosen as 2.6 and we assume that G1 50, G2 300. As shown in Figure 3, the
maximum Lyapunov exponent is 5.1735e − 004.
Discrete Dynamics in Nature and Society
9
15
25
10
20
5
y
15
z
0
10
−5
5
−10
−15
−20
−10
0
10
20
0
−20
0
−10
10
20
y
x
a x-y projection
b y-z projection
25
20
z
15
10
5
0
20
10
y
0
20
−10
−20 −20
0
x
c Chaotic attractor
Figure 9: Chaotic attractor generated by G1 40, G2 300.
4. Various Patterns with Parameter Changing
In this section, we pay attention to the dynamical behaviors of the system 2.1 and the system
2.2 with parameter of space functions selected in the “chaotic” regions in order to show the
effective generation of various patterns of attractors based on the parameter selection.
At first, we consider one-dimensional space function: let z1 and z2 change in the
stability intervals. Then, the system displays different patterns for different values of z1 and
z2 , as shown in Figures 4 and 5.
In the two cases, the largest Lyapunov exponents are
LE 1.4720e − 004 z1 2, z2 14,
LE −7.9221e − 004 z1 3, z2 10.
4.1
Then, we consider two-dimensional space function: let F1 and F2 change in the stability
intervals. Then, the system displays different patterns for different values of F1 and F2 , as
shown in Figures 6, 7, and 8.
10
Discrete Dynamics in Nature and Society
15
25
10
20
5
y
15
z
0
10
−5
5
−10
−15
−20
0
x
−10
10
20
0
−20
0
y
−10
a x-y projection
10
20
b y-z projection
25
20
z
15
10
5
0
20
10
y
0
20
−10
−20 −20
0
x
c Chaotic attractor
Figure 10: Chaotic attractor generated by G1 50, G2 330.
The largest Lyapunov exponents are given as following:
LE 7.5329e–004 F1 6, F2 250,
LE 0.0012 F1 48, F2 250,
4.2
LE 0.0016 F1 10, F2 300.
Finally, we consider three-dimensional space function; the system displays different
patterns for different values of G1 and G2 , as shown in Figures 9 and 10.
The largest Lyapunov exponents are given as following:
LE 0.0021 G1 40, G2 300,
LE 0.0016 G1 50, G2 330.
4.3
These numerical simulations verify that space functions dominate rich complex patterns when adjusting parameters. From this, we can see that the proposed space functions are
quite effective in the generation of attractor with obviously quasiperiodic or chaotic behaviors
based on the change of parameters. Moreover, the statistic behavior is also researched by
giving the largest Lyapunov exponents.
Discrete Dynamics in Nature and Society
11
5. Conclusion
This paper has presented a new control method for generating chaos or chaos-like dynamics.
The generation of novel chaotic attractors via two simple three-dimensional linear systems
with various space functions is introduced. The results once again support the long-accepted
belief that properly designed simple systems can perform complex dynamical behaviors.
Moreover, this system can produce various attractor patterns within a wide range of parameter values and the statistic behavior which reveals the regularities in the complex dynamics
is also discussed; other space functions also can be chosen and can generate chaotic attractors
with various system parameters too. In addition, the method which has been developed in
this paper can also be applied to nonlinear dynamical systems and other fields. It is desirable
that one could design more chaos generators by means of the method proposed in this paper.
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