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International Journal of Bifurcation and Chaos, Vol. 9, No. 7 (1999) 1465–1466
c
World Scientific Publishing Company
YET ANOTHER CHAOTIC ATTRACTOR
GUANRONG CHEN∗
Department of Electrical and Computer Engineering,
University of Houston, Houston, TX 77204-4793, USA
TETSUSHI UETA†
Department of Information Science and Intelligent Systems,
Tokushima University, 2-1, Minami-Josanjima, Tokushima, 770-8506 Japan
Received March 30, 1999
This Letter reports the finding of a new chaotic attractor in a simple three-dimensional
autonomous system, which resembles some familiar features from both the Lorenz and R¨ossler
attractors.
In the pursuit of anticontrol of chaos (also called
chaotification), which means making a nonchaotic
system chaotic [Chen, 1997; Chen & Lai, 1998;
Wang & Chen, 1999], we have recently found a new
chaotic attractor from the following system:
˙x=a(y−x)
˙y=(c−a)x−xz +cy
˙z=xy −bz ,
(1)
which has the attractor shown in Fig. 1 when
a=35,b=3,c=28.
To compare, we recall the Lorenz system
˙x=a(y−x)
˙y=cx −xz −y
˙z=xy −bz ,
(2)
which is chaotic when
a=10,b=8/3,c=28;
-30 -20 -10 010 20 30
x
-30 -20 -10 010 20 30
y
0
10
20
30
40
50
z
Fig. 1. The new chaotic attractor. a= 35, b=3,c= 28,
x(0) = −10, y(0) = 0, z(0) = 37.
and the R¨ossler system
˙x=−y−z
˙y=x+ay
˙z=xz −bz +c,
(3)
∗E-mail: gchen@uh.edu
†E-mail: tetsushi@is.tokushima-u.ac.jp
1465
1466 G. Chen &T. Ueta
which is chaotic when
a=0.2,b=5.7,c=0.2.
It is easy to verify that systems (1) and (3)
are not topologically equivalent since the former
has three equilibria but the latter has only two.
Although systems (1) and (2) look alike, it is
straightforward (but somewhat tedious) to verify
that there is no nonsingular (linear or nonlinear)
coordinate transforms (i.e. diffeomorphism) that
can convert system (1) to (2) or vice versa. There-
fore, systems (1) and (2) are not topologically equiv-
alent either. More detailed analysis will be provided
in a forthcoming paper.
References
Chen, G. [1997] “Control and anticontrol of chaos,” Proc.
First Int. Conf. Control of Oscillations and Chaos,
COC’97, St. Petersburg, Russia, August 27–29, 1997,
pp. 181–186.
Chen, G. & Lai, D. [1998] “Feedback anticontrol of
discrete chaos,” Int. J. Bifurcation and Chaos 8,
1585–1590.
Wang, X. F. & Chen, G. [1999] “On feedback anticon-
trol of discrete chaos,” Int. J. Bifurcation and Chaos
9, 1435–1441.
