International Journal of Analysis and Applications
ISSN 2291-8639
Volume 5, Number 1 (2014), 27-32
http://www.etamaths.com

A NEW CHAOTIC ATTRACTOR WITH QUADRATIC

EXPONENTIAL NONLINEAR TERM FROM CHEN’S

ATTRACTOR

IFTIKHAR AHMED∗, CHUNLAI MU, AND FUCHEN ZHANG

Abstract. In this paper a new three-dimensional chaotic system is proposed,
which relies on a nonlinear exponential term and a nonlinear quadratic cross

term necessary for folding trajectories. Basic dynamical characteristics of the

new system are analyzed. Compared with the Chen system, the equilibrium
points of the new system does not contain the origin, and has a greater positive

Lyapunov index, can produce more complex shaped chaotic attractor.

1. Introduction

Since Lorenz found the first chaotic attractor in a three first order autonomous
ordinary differential equations (ODEs) when he studied the atmospheric convection
in 1963 [1], many new three dimension (3D) chaotic attractors have been proposed
in the last three decades, such as the Rossler system [2], the Chen system [3], the Lü
system [4], the Liu system [5], and the generalized Lorenz system family [6]. New
chaotic system can also be achieved by adding or changing the linear/nonlinear ter-
m of existing chaotic system. The nonlinear term of system is normally the product
of variables at different state. when the system contains nonlinear terms of the ex-
ponential function Whether there will be chaos phenomenon, yet need research.

Wei and Yang [7] revealed a 3D autonomous chaotic attractor with a nonlin-
ear term in the form of exponential function at the right-hand side in ODEs as
ẋ = ay−ax, ẏ = −by + mxz, ż = n−exy, where the existence of singularly degen-
erate heteroclinic cycles for a suitable choice of the parameters was investigated.
Recently, Liang and Zhonglin [8] discussed the basic dynamic characteristics of the
new chaotic system containing a nonlinear term of exponential function instead of
the nonlinear term in Lü system.
In this paper, a new chaotic system containing a nonlinear term of exponential
function instead of the nonlinear term in Chen’s system is proposed. The dynamic
characteristics and simulation show clearly that proposed system is chaotic same as
Lorenz chaotic attractor and others, but its topological structure is different from
all existing chaotic attractors.

2000 Mathematics Subject Classification. 34C28.
Key words and phrases. New chaotic attractor, Chen’s attractor; Quadratic exponential non-

linear term.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

27



28 IFTIKHAR AHMED, CHUNLAI MU, AND FUCHEN ZHANG

This paper is organized as follows. Section 2 presents the design of a new chaotic
system. Section 3 outlines the basic properties of the new system. Finally, conclu-
sions are given in section 4.

2. Design of a new chaotic system

Chen’s chaotic system is given by


ẋ = a(y −x)
ẏ = (c−a)x + cy −xz
ż = xy − bz

(1)

When a = 35, b = 3, c = 28 ,the Lyapunov exponents of system (1) are found to
be LE1=1.997, LE2=0.0002 , LE3=-11.9943. The fractal dimension of system (1)
is D1=2.1665 . The chaotic attractor is shown in figure 1. System (I) is still in a
state of chaos. In system (I), we use exy instead of xy in the third equation, and
get a new system (2) : 


ẋ = a(y −x),
ẏ = (c−a)x + cy −xz,
ż = exy − bz,

(2)

When a = 35, b = 3, c = 28 the Lyapunov exponents of system (2) are found to be
LEI =3.8065, LE2=0.010904 , LE3=-13.817. There is a positive Lyapunov expo-
nent, which is much larger than that of the system (1). The Lyapunov dimension
of system (2) is:

DL = j +
1

|LEj+1|

j∑
i=1

LEi = 2 +
LE1 + LE2
|LE3|

= 2 +
3.806 + 0.0109

|−13.817|
= 2.2762

0 50 100 150 200 250 300
−20

−15

−10

−5

0

5

10
Dynamics of Lyapunov Exponents

L
1
 =3.8065

L
2
 =0.010904

L
3
 =−13.817

Time

L
ya

p
u

n
o

v 
E

xp
o

n
e

n
ts

Figure 1. Dynamics of Lyapunov exponents of system (2)

The Lyapunov dimensions of the system are fractional. The chaotic attractor is
shown in Fig. 3. System (2) is also at chaos status.



A NEW CHAOTIC ATTRACTOR 29

−25 −20 −15 −10 −5 0 5 10 15 20 25
−30

−20

−10

0

10

20

30

x(t)

y(
t)

−25 −20 −15 −10 −5 0 5 10 15 20 25
5

10

15

20

25

30

35

40

45

50

x(t)

z(
t)

−30 −20 −10 0 10 20 30
5

10

15

20

25

30

35

40

45

50

y(t)

z(
t)

Figure 2. Phase portraits of chaotic attractors of system (I)

−3 −2 −1 0 1 2 3
−4

−3

−2

−1

0

1

2

3

4

x(t)

y(
t)

−3 −2 −1 0 1 2 3
0

10

20

30

40

50

60

70

80

90

x(t)

z(
t)

−4 −3 −2 −1 0 1 2 3 4
0

10

20

30

40

50

60

70

80

90

y(t)

z(
t)

Figure 3. Phase portraits of chaotic attractors of system (2)



30 IFTIKHAR AHMED, CHUNLAI MU, AND FUCHEN ZHANG

3. Basic properties of the new system

A. Equilibria
For equilibrium points we take




a(y −x) = 0,
(c−a)x + cy −xz = 0,
exy − bz = 0,

(3)

When a = 35, b = 3, c = 28, system (2) has three equilibrium points: E1(0, 0, 0.333),
E2(2.035, 2.035, 21), E3(−2.035,−2.035, 21). The equilibrium point of system (2)
does not contain the origin. The Jacobian matrix of system (2) is given by

J =


 −a a 0c−a−z c −x

yexy xexy −b


(4)

For the equilibrium pointE1 = (0, 0, 0.333) , system (2) has three eigenvalues:
λ1 = −3, λ2 = 23.835, λ3 = −30.835. Eigenvalues is not all for the positive or neg-
ative, according to the Routh-Hurwitz , E1 = (0, 0, 0.333) is unstable saddle node.
For the equilibrium points E2 = (2.035, 2.035, 21) and E3 = (−2.035,−2.035, 21)
, system (2) has same eigenvalues: λ1 = −26.053, λ2,3 = 8.026 ± 25.203i. λ1 is
a negative real root and λ2,3 are a pair of conjugate roots with positive real part.
So E2 = (2.035, 2.035, 21) and E3 = (−2.035,−2.035, 21),are unstable saddle-focus
points.

B. Symmetry and invariance
It is easy to see the invariance of system under the coordinate transformation
(x,y,z) → (−x,−y,z) i.e., the system has rotation symmetry around the z -axis.

−4
−2

0
2

4

−4

−2

0

2

4
0

20

40

60

80

100

x(t)

3D view of new attractor

y(t)

z(
t)

Figure 4. 3D view of new chaotic system (2)



A NEW CHAOTIC ATTRACTOR 31

C. Dissipativity
The three Lyapunov exponents and the divergence of the vector field is:

3∑
i=1

LEi = ∆V =
∂ẋ

∂x
+
∂ẏ

∂y
+
∂ż

∂z
= −a + c− b = f,(5)

where
3∑

i=1

LEi denote the three Lyapunov exponents of the system. Note thatf =

−a + c− b = −10 is a negative value, so the system is a dissipative system and an
exponential rate is:

dV

dt
= ef = e−10(6)

From(6), it can be seen that a volume element V0 is contracted by the flow into
a volume element V0e

−10t in time t . This means that each volume containing
the system trajectory shrinks to zero as t → ∞ at an exponential rate of −10 .
Therefore, all system orbits are ultimately confined to a specific subset having zero
volume and the asymptotic motion settles onto an attractor .

D. Sensitivity to initial conditions Figure 4 shows that the evolution of the
chaos trajectories is very sensitive to initial conditions. The initial values of the
system are set to [2, 2, 1]T for the solid line and [2.01, 2, 1]T for the dashed line.

Figure 5. Sensitivity of system(2) to initial conditions

4. Conclusions

A new chaotic system is proposed in this paper, which has exponential term
instead of the nonlinear term of the Chen’s system. Some basic properties of the
system have been investigated. Compare with Chen’s system, the new system has
greater chaos interval and much larger Lyapunov exponent. Its equilibrium point
does not contain the origin. Even though more important analysis of the system
like chaos control, boundedness, and synchronization, will take into account in the
future work.



32 IFTIKHAR AHMED, CHUNLAI MU, AND FUCHEN ZHANG

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College of Mathematics and Statistics,Chongqing University, Chongqing 401331, PR

China

∗Corresponding author