International Journal of Bifurcation and Chaos, Vol. 14, No. 9 (2004) 3167–3177
c World Scientific Publishing Company
CHEN’S ATTRACTOR EXISTS
TIANSHOU ZHOU∗ and YUN TANG
Department of Mathematical Sciences, Tsinghua University,
Beijing 100084, P. R. China
∗ [email protected]
GUANRONG CHEN
Department of Electronic Engineering, City University of Hong Kong, P. R. China
Received May 27, 2003; Revised August 11, 2003
By applying the undetermined coefficient method, this paper finds homoclinic and heteroclinic
orbits in the Chen system. It analytically demonstrates that the Chen system has one heteroclinic orbit of Ši’lnikov type that connects two nontrivial singular points. The Ši’lnikov criterion
guarantees that the Chen system has Smale horseshoes and the horseshoe chaos. In addition,
there also exists one homoclinic orbit joined to the origin. The uniform convergence of the series
expansions of these two types of orbits are proved in this paper. It is shown that the heteroclinic
and homoclinic orbits together determine the geometric structure of Chen’s attractor.
Keywords: Chen’s attractor; homoclinic orbit; heteroclinic orbit; Ši’lnikov criterion.
1. Introduction
In 1999, Chen found an interesting system [Chen
& Ueta, 1999; Ueta & Chen, 2000], lately referred to as the Chen system by others, which is
a dual system to the Lorenz system [Lorenz, 1963;
Sparrow, 1982] in the sense defined by Vaněček and
Čelikovský [1996]: For the linear part of the system, A = [aij ]3×3 , the Lorenz system satisfies the
condition a12 a21 > 0 while the Chen system satisfies a12 a21 < 0. Afterwards, Lü and Chen [2002] furthermore found a chaotic system, which satisfies the
condition a12 a21 = 0 and represents the transition
between the Lorenz and Chen systems. Research
along this line eventually led to the finding of a
large class of relevant chaotic systems — a family
of generalized Lorenz systems [Čelikovský & Chen,
2002] — defined according to the system structures,
which encompasses all the aforementioned chaotic
systems of similar behaviors.
∗
It was numerically verified that the Chen system has a seemingly chaotic attractor, named
Chen’s attractor, and displays very sophisticated
dynamical behaviors [Ueta & Chen, 2000; Zhou
et al., 2003]. It has been noticed that most studies
thus far have heavily depended on numerical simulations with the aid of computer graphics, and the
existence of Chen’s attractor in the sense of mathematics has not yet been confirmed as of today.
Because computer simulations have finite precisions
and experimental measurements have finite ranges
in both time and frequency domains, it is quite
possible that the observed dynamical behavior is
either a numerical or graphical artifact of the finiteprecision machines, or a pseudo-orbit nearby a still
uncovered but truly chaotic one. In fact, there were
occasions where a claimed chaotic orbit turned out
to be a long-periodic orbit, or a quasi-periodic one.
To clarify the claims, a rigorous mathematical analytical approach, if ever possible, is often desirable.
Author for correspondence.
3167
3168
T. Zhou et al.
In this endeavor, Tucker [1999] proved the existence
of the Lorenz attractor, which was based on normalform theory and computer simulations.
It should be noted that one of the commonly
agreeable analytic criteria for proving chaos in
autonomous systems is based on the fundamental work of Ši’lnikov [1965, 1970], and its subsequent embellishments and slight extension [Silva,
1993; Tresser, 1984]. This is known as the Ši’lnikov
method or Ši’likov criterion today, and its role is
in some sense equivalent to that of the Li–Yorke
Lemma in the discrete setting [Li & Yorke, 1975;
Kennedy et al., 2001].
In this paper, a rigorous proof is given to show
the existence of Chen’s attractor. Specifically, it will
be shown that the Chen system has two types of
orbits: one is the heteroclinic orbits of Ši’lnikov type
(see Sec. 3) and the other, the homoclinic orbits
(see Sec. 4). Moreover, their precise algebraic expressions in a series expansion form will be derived
with uniform convergence proved. By applying the
Ši’lnikov criterion, it is convinced that the Chen
system indeed is chaotic, with Smale horseshoes and
the horseshoe type of chaos.
The paper is organized as follows. In Sec. 2,
some basic concepts and terminologies related to
homoclinic and heteroclinic orbits are first reviewed. Section 3 introduces the undetermined
coefficient method, which will be used to find
homoclinic and heteroclinic orbits in the Chen
system. In this section, the algebraic expression
of the heteroclinic orbit will also be derived, and
the uniform convergence of its series expansion is
proved. In particular, the Ši’lnikov criterion will be
verified, ensuring that the Chen system has both
Smale horseshoe and the horseshoe type of chaos.
Similarly, the homoclinic orbit is analyzed with
uniform convergence concluded in Sec. 4. The final
section provides some concluding remarks.
2. Preliminaries
Only three-dimensional continuous autonomous
systems are studied in this paper.
2.1. Basic concepts and
terminologies
γ; σ ± iω ,
σγ < 0 ,
ω 6= 0 ,
where γ, σ and ω are real.
A homoclinic orbit refers to a bounded trajectory of system (1) that is doubly asymptotic
to an equilibrium point of the system. A heteroclinic orbit, on the other hand, is similarly defined
except that there are two distinct saddle foci being connected by the orbit, one corresponding to
the forward asymptotic time, and the other, to the
reverse asymptotic
P time 2limit.
Denote by
∈ R a plane that cuts transversely across the recurrent system orbital flow,
which occurs locally to homoclinicPor heteroclinic
P
orbits. Define a 2-D map P : U ∈
→ , called
the Poincaré map, where the neighborhood
U desP
ignates those points that return to
at least once
along the orbital flow of system (1). Then, P defines
a 2-D discrete dynamical system:
xk+1 = P (xk ) ,
k = 0, 1, . . . ,
(2)
which characterizes the system (1). For the case
of a homoclinic orbit or a heteroclinic orbit, the
corresponding Poincaré map is sometimes called a
Ši’lnikov map.
An important concept in this study is the Smale
horseshoe [Wiggins, 1988; Čelikovský & Vaněčk,
1994]. This is a set of orbits analytically detected by
the Ši’lnikov method for discrete dynamical systems
generated by the Ši’lnikov map (2), which guarantees that the original continuous system (1) is
chaotic in a rigorous mathematical sense.
The heteroclinic Ši’lnikov method, namely, the
Ši’lnikov criterion for the existence of chaos, is summarized in the following theorem [Silva, 1993].
Theorem 2.1. Suppose that two distinct equilib-
rium points, denoted by x1e and x2e , respectively, of
system (1) are saddle foci, whose characteristic values γk and σk ± ωk (k = 1, 2) satisfy the following
Ši’lnikov inequality:
|γk | > |σk | > 0 ,
Consider the third-order autonomous system:
dx
= f (x) ,
dt
where the vector field f (x): R3 → R3 , belongs to
class C r (r ≥ 2).
Let xe ∈ R3 be an equilibrium point of system (1). Then xe is called a hyperbolic saddle focus
(or simply, saddle focus) if the eigenvalues of the
Jacobian A = Df , evaluated at xe , are
k = 1, 2
(3)
γ 1 γ2 > 0 .
(4)
under constraint
t ∈ R,
3
x∈R ,
(1)
σ1 σ2 > 0 ,
or
Chen’s Attractor Exists
Suppose also that there exists a heteroclinic orbit
joining x1e and x2e . Then
30
20
10
y
(i) the Ši’lnikov map, defined in a neighborhood of
the heteroclinic orbit, has a countable number
of Smale horseshoes in its discrete dynamics;
(ii) for any sufficiently small C 1 -perturbation g of
f , the perturbed system
dx
= g(x) , x ∈ R3
(5)
dt
has at least a finite number of Smale horseshoes
in the discrete dynamics of the Ši’lnikov map
defined near the heteroclinic orbit;
(iii) both the original system (1) and the perturbed
system (5) have horseshoe type of chaos.
3169
0
−10
−20
−30
30
20
50
10
40
0
30
−10
20
−20
−30
x
Fig. 1.
10
0
z
The chaotic Chen’s attractor.
Figure 1: The chaotic Chen’s attractor.
For convenience, a heteroclinic orbit satisfying
(3) and (4) is referred to as the Ši’lnikov type. Thus,
Note that the linearized
system
(6) corresponding
to theofequilibrium
point
Note that
theof linearized
system
(6) correthe heteroclinic Ši’lnikov criterion implies that
if
following form:sponding to the equilibrium

 O 2 has the fol point

system (1) has one heteroclinic orbit of the Ši’lnikov
x1
ẋ1
lowing form:




  
 
type, which connects two distinct saddle foci of the

ẋ2  = A  x2 
ẋ
x1 

1
system, then it has both Smale horseshoes and the
 x3 ,
 
ẋ3
(7)
 ẋ2  = A  x2  ,
horseshoe type of chaos, which is rigorous in the
where


above-defined mathematical sense.
x
ẋa3
a 3
0
2.2. Chen’s attractor
The Chen system is [Chen & Ueta, 1999]


 ẋ = a(y − x)
ẏ = (c − a)x + cy − xz


ż = −bz + xy ,



where A = 

b(2c a) 
c
c


p
p


b 0
−a b(2c a) b(2c
a a)
.
p

The characteristic
polynomial of the Jacobian A is
−c
c
− b(2c − a)  .
A=


p λ3 + (a + b c)λ
p2 + bcλ + 2ab(2c a) = 0.
b(2c − a)
b(2c − a)
−b
(6)
Letting λ = µ
where a > 0, b > 0 and c are parameters (2c > a).
With the typical parameter values of a = 35,where
b=3
and c = 28, the corresponding numerical Chen’s
attractor can be found and is depicted in Fig. 1.
Note that system
(6) has
p three equilibp
2c−a)
ria: O1 (0, p
0, 0), O2 ( b(2c
p− a), b(2c − a), Denote
and O3 (− b(2c − a), − b(2c − a), 2c − a). From
Fig. 1, one can see that in this attractor, outside trajectories are typically attracted into the vicinity of
its steady state, and they are alternatively swirling
between the two equilibria O2 and O3 . Intuitively,
there should be a heteroclinic orbit joining O 2 and
O3 , and this will be confirmed in the next section,
where one heteroclinic orbit is found with a precise
algebraic expression.
a+b
3
c
p
yields
(7)
(8)
(9)
(8)
µ3 + pµ + q = 0,
The characteristic polynomial of the Jacobian
(10)
A is
(a + b c)2
,
p = bc
λ3 + (a + b − c)λ23 + bcλ +32ab(2c − a) = 0 .
bc(a + b c)
2(a + b c)
+ 2ab(2c a).
q=
3 yields
Letting λ = µ −27(a + b − c)/3
p0,3
2 =
µ3 + pµ+
q q
∆=
where
p = bc −
q=
2
+
3
(9)
(10)
.
(a +5 b − c)2
,
3
(11)
2(a + b − c)3 bc(a + b − c)
−
+ 2ab(2c − a) .
27
3
(12)
3. The Heteroclinic Orbit
∆=
When
q 2
2
+
∆ > 0,
p 3
3
(11)
(12)
Denote
In the section, the undetermined coefficient method
is applied to find the heteroclinic orbit in the Chen
system.
O2 has the
.
(13)
3170
T. Zhou et al.
the algebraic equation (10) has a unique negative
real root, α1 , and a conjugate pair of complex roots,
β1 ± γ1 i, with
r
r
q √
q √
3
α1 = − + ∆ + 3 − − ∆ ,
(14)
2
2
r
r
1 3 q √
q √
3
β1 = −
− + ∆+ − − ∆ ,
2
2
2
(15)
√ r
r
3 3 q √
q √
3
− + ∆− − − ∆ .
γ1 =
2
2
2
(16)
Therefore, when ∆ > 0, three roots of Eq. (9)
are, respectively,
λ1 = −
a+b−c
+ α1 ,
3
(17)
λ2 = −
a+b−c
+ β1 + γ1 i ,
3
(18)
a+b−c
λ3 = −
+ β1 − γ1 i,
(19)
3
√
where λ1 < 0 and i = −1.
Note that the characteristic polynomial of the
Jacobian of the linearized system of (6), evaluated at the equilibrium point O3 , is exactly the
same as (9). So, it also has three roots, identical to
(17)–(19).
3.1. Series expansion of the
heteroclinic orbit
In this subsection, the heteroclinic orbit of (6) that
links equilibria O2 and O3 is determined.
First, it follows from (7) that
y =x+
ẋ
,
a
(20)
ẍ + (a − c)ẋ
+ 2c − a ,
ax
d ẍ + (a − c)ẋ
ẍ + (a − c)ẋ
+b
dt
x
x
z=−
(21)
+ ax2 + xẋ − ab(2c − a) = 0 .
(22)
If x(t) is found, then y(t) and z(t) will also
be determined. Therefore, finding the heteroclinic
orbit of (6) is now changed to seeking a function
ϕ(t) such that x(t) = ϕ(t) satisfying (22) and
p
ϕ(t) → − b(2c − a) as t → +∞ ,
p
ϕ(t) → b(2c − a) as t → −∞ ,
or
p
ϕ(t) → − b(2c − a) as t → −∞ ,
p
ϕ(t) → b(2c − a) as t → +∞ .
Without loss of generality, one may stipulate
a definite direction as follows: from O 2 to O3 corresponds to t → +∞, while from O3 to O2 , to
t → −∞.
Next, suppose that, for t > 0,
ϕ(t) = −γ +
∞
X
ak ekαt ,
(23)
k=1
p
where γ = b(2c − a), α(< 0) is an undetermined
constant, and ak (k ≥ 1) are also undetermined coefficients. The convergence of this series expansion
will be given in the next subsection.
To this end, substituting (23) into (22) and then
comparing the coefficients of ekαt (k ≥ 1) of the
same power terms, one has: for n = 1,
α
(α + a − c)(α + b) + γ(α + 2a) a1 = 0 . (24)
γ
and, for n > 1,


X
1  X
γ
(a + jα)ai aj − α(nα + b)
bi cj  ,
an =
w(nα)
i+j=n
where
(25)
i+j=n
w(nα) = (nα)3 + (a + b − c)(nα)2 + bc(nα) + 2ab(2c − a) ,
(26)
bi = i(iα + a − c)ai ,
(27)
Chen’s Attractor Exists
and
cj =
X
j
X
j Y
a k lk
γ
(l1 ,...,lj )∈Sj k=1
!
lk !
k=1
j
Y
,
(28)
(lk )!
k=1
with
Sj =
(
(l1 , l2 , . . . , lj )|l1 ≥ 0, . . . , lj ≥ 0,
j
X
i=1
i · li = j
)
.
(29)
Assume a1 6= 0 (otherwise, one can inductively
obtain ak = 0 for all k > 1). In this case, it follows
from (24) that
α3 + (a + b − c)α2 + bcα + 2ab(2c − a) = 0 .
(30)
It might be surprising to see that (30) is nothing
but the characteristic polynomial of the Jacobian
of the linearized equation of (6) evaluated at the
equilibrium point O3 .
Since (9) has the unique negative root when
∆ > 0, one has
w(kα) = (kα)3 + (a + b − c)(kα)2 + bc(kα)
+ 2ab(2c − a) 6= 0 ,
k > 1.
(31)
Consequently, an is completely determined by α, a,
b, c, and a1 , which has the following form by (25):
an = ϕn · an1 ,
n > 1,
(32)
where ϕn (n > 1) are some known functions depending on α, a, b and c.
In the next subsection, the expansion (23) is
proved to be uniformly convergent for all t ≥ 0.
Here, it is first to completely determine the heteroclinic orbit of (6).
To this end, the first part of the heteroclinic
orbit corresponding to t > 0 has been determined.
Next, its second part, denoted by ψ(t) corresponding to t < 0 will also be determined.
To find the expression of ψ(t), it is assumed
that for −t < 0,
ψ(t) = γ +
∞
X
bk e−kβt .
(33)
k=1
Completely similar to the case of t > 0, one can
obtain that
β = α,
bk = ψk bk1 ,
k > 1,
(34)
(35)
3171
where b1 is an undetermined constant, and ψk have
similar expressions to ϕk , that is, if b1 = −a1 , then,
ψk = −ϕk (k > 1).
Consequently, under the conditions mentioned
above, the first component ϕ(t) of the heteroclinic
orbit of (6), which connects the equilibria O 2 and
O3 , takes the following form:

∞
X



ak ekαt
−γ +




k=1


ϕ(t) = 0



∞

X



bk e−kαt
γ
+


for t > 0
for t = 0
(36)
for t < 0 .
k=1
In addition, one needs to impose additional
conditions for undetermined constants a 1 and b1 so
that
ϕ(0− ) = ϕ(0+ ) ,
ϕ0 (0− ) = ϕ0 (0+ ) ,
(37)
ϕ00 (0− ) = ϕ00 (0+ ) .
A naturally arising question is whether or
not these a1 and b1 exist. Fortunately, numerical
simulation shows that such a1 and b1 indeed exist. For this reason, one may consider the following
algebraic equation:
F (a1 ) ≡ ϕn an1 + ϕn−1 an−1
+ · · · + ϕ2 a21
1
+ a1 − γ = 0 ,
n > 1,
(38)
which corresponds to the case of b1 = −a1 . Obviously, F (0) = −γ < 0. In addition, one can prove
that w(nα) > 0 and ϕn > 0 when n is sufficiently
large. Therefore, F (a1 ) > 0 for a1 and n sufficiently
large. These imply that Eq. (38) has at least one
real root with respect to a1 . For the requirement,
one must exclude the case of multiple roots, but
this seems to be not easy. However, one has that
F 0 (a1 ) > 0 when n and a1 are large enough. In
particular, for the typical parameter set: a = 35,
b = 3 and c = 28, it has been numerically verified that, when n ≥ 17, (38) has an “almost stable” real root near the value 7.9247, with a relative
error no greater than 1%. Therefore, one may assume without tedious proof that these a 1 and b1
satisfying (37) exist. Consequently, under the conditions mentioned above, the first component ϕ(t)
of the heteroclinic orbit of (6), which connects the
3172
T. Zhou et al.
equilibria O2 and O3 , takes the following form:

∞
X



ak ekαt for t > 0
−γ
+




k=1

0
for t = 0
ϕ(t) =
(39)


∞

X



γ
−
ak e−kαt for t < 0 .


3.2. Convergence of the heteroclinic
orbit series expansion
In this subsection, the convergence of the series
expansion (23) of the heteroclinic orbit is proved.
Only the case where the Chen system with the typical parameter set that generates Chen’s attractor
is considered. For some other parameter sets, if the
heteroclinic orbit exists, the proof is similar.
First, it follows from (25) that
k=1
(a + α)γ 2 α(a − c + α)(2α + b) 2
a −
a1 ,
w(2α) 1
γw(2α)
α(3α + b) 5α + 3(a − c)
α+a−c 3
(2a + 3α)γ
a3 =
a1 a2 −
a1 a2 +
a1 .
w(3α)
w(3α)
γ
γ2
a2 =
For the typical parameter values of a = 35, b = 3 and c = 28, one has
a2 ≈ −0.0293a21 ,
a3 ≈ 5.8812 · 10−4 · a31 .
Therefore,
|a3 | ≈ 5.8812 · 10
−4
3
· |a1 | < 6.6661 · 10
−4
Next, for any n > 3, it can be proved by induction that
1 a1 n
|an | ≤ .
n γ
1 a1 3
· |a1 | ≈ .
3 γ
3
(40)
(41)
In the following, note that a + jα < 0, α(nα + b) > 0 for the typical parameter values, where j, n ∈ N. It
follows from (25) that
"
#
X
X
1
|an | ≤
−γ
(a + jα)|ai ||aj | + α(nα + b)
|bi ||cj | .
|w(nα)|
i+j=n
i+j=n
Note that
γ
X
(a + jα)|ai ||aj | = −γ(2a + nα)(|a1 ||an−1 | + |a2 ||an−2 |) − γ
i+j=n
≤ −γ
≤ −γ
2 2a
+ nα
n−2
a1
γ
n "
a1
γ
n a1
γ
n
−γ
a1
γ
n
i>2,
j>2
X
i+j=n
i>2,
j>2
X
i+j=n
|ai ||aj |(a + jα)
a + jα
ij
n−3
n−3
i=3
i=3
X
X1
1
γ(2a + nα)
+a
+α
n−2
i(n − i)
i
γ(2a + nα)
≤ −γ
+ a + α(ln n + 1 − ξ)
n−2
n γ(2a + nα)
a1
+ a + α ln n ,
≤ −γ
γ
n−2
#
Chen’s Attractor Exists
where ξ is the Euler constant. Also, note that
X
i+j=n
|bi ||cj | = |b1 ||cn−1 | + |b2 ||cn−2 | + |bn−1 ||c1 | + |bn−2 ||c2 | +
i>2,
j>2
X
i+j=n
|bi ||cj |
i>2,
j>2 X
a1 i
(n − 1)α + a − c a1 n
(iα + a − c)|cj | .
≤ −(α + a − c)|a1 ||cn−1 | −
γ
γ −
γ
i+j=n
Furthermore, for j ≥ 3,
j
X
!
lk !
j Y
ak lk k=1
|cj | =
γ
j
Y
(l1 ,...,lj )∈Sj k=1
(lk )!
X
k=1
j
a1 ≤ γ
j
a1 ≤ γ
j
a1 ≤ γ
j
X
lk !
j Y
1 lk k=1
j
kγ
Y
(l1 ,...,lj )∈Sj k=2
(lk )!
X
k=1
X
γ−
X
i+j=n
j
X
lk
j lk
Y
1
k=1
k=2
(l1 ,...,lj )∈ Sj
k
j
X
lk !
k=1
j
Y
!
(lk )!
k=1
X
γ−
(l1 ,...,lj )∈Sj
j
X
k=2
lk
j
X
!
lk !
k=1
j
Y
k=1
1 a1 j
≤ 2 .
j γ
Therefore,
!
[(lk ) · k lk ]!


n
i>2,
j>2
X
a1 2γ(α + a − c)
iα + a − c 
|bi ||cj | ≤ − 
+
γ
(n − 2)2
j2
i+j=n


n
i>2,j>2
n−3
X
X
a1 2γ(α + a − c)
1
i
≤ − 
+ (a − c)
+α
.
2
2
γ
(n − 2)
j
j2
Since
n−3
X
j=3
X
π2
1
<
− 1,
j2
6
i+j=n
α
j=3
i>2,
j>2
X
i+j=n
n−3
X 1
i
< (n − 3)
<n
2
j
j2
j=3
i+j=n
π2
−1 ,
6
n 2
a1 2γ(α + a − c)
π
+
− 1 (a − c + nα)
|bi ||cj | ≤ − γ
(n − 2)2
6
3173
|
3174
and
X
≤
≤
T. Zhou et al.
n 2
+ a c)
π
1 (a c + nα) +
aγ1 n 2γ(α
a1 γ 2(n(2a2)+2 nα) 6
π2 − 6 1
i+j=n
+ aγ + αγ ln n +
+
|an | ≤ − · α(nα + b)(a − c + nα)
|w(nα)|
γ
n−1
6
γ
n h
a γ (2a+nα)
n c + nα)i /|w(nα)|
+2αγ ln n + π 6 + 1n · α(nα+ b)(a
γ
n 1 1 + aγ π
6 a γ
a
1
1
−n6
n
1 < 1 .
1 a 6
π
1
+ γ1 ≤ aγ · < n
γ .+
n ·
6
n
6
γ
γ
n γ
|bi | |cj | ≤
2
2
1
2
1
1
ow, one can easily prove the convergence of the series expansion (23). In fact,
Now, one can easily prove the convergence of the series expansion (23). In fact,
n
∞
∞
a2
X
∞X 1 a1eαt
∞
.
+
an enαt ≤ a1 eαt + 1 e2αtX
a1 etα +
a21 2αt X 1 a1 αt n
αt
tα
2
n nαt
γ
n=3
n=2
.
an e ≤ a 1 e + e +
e
a1 e +
2
n γ
n=2
n=3
0 < a1 ≈ 7.9247 < 7.9373 ≈ γ,
7.93732 ≈
γ, ∞ 0 < a ≈ 7.9247
Since
<
P
a
a1 αt
1
nαt 1
αt
a1 etα +
a
e
+ 21 1 γ12 e2αt ln 1
n
γ e
γ e
≤ a1 1
n=2
∞ 2 1X
a
a11
a1 αt
a21
1
+ 21 1nαt γ12
ln 1
≤ a1 tα
1
2αt
αt
γ
γ
an e ≤ a 1 1 −
e +
1 − 2 e − ln 1 − e
a1 e +
γ
2
γ
γ
n=2
t ≥ 0. Thus, the series expansion (23) is
uniformly convergent with respect all t ≥ 0.
heteroclinic
orbit
r the typical parameter set: a = 35, b = 3 and c = 28, the corresponding
a21
1
1
+
1− 2
≤ a1 1 −
wn in Figure 2.
γ
2
γ
a1
− ln 1 −
γ
one component has the form (39), and the corresponding chaos is of horseshoe type.
20
15
10
y
5
0
4. The Homoclinic Orbit
−5
−10
−15
−20
20
35
10
30
0
25
−10
x
20
−20
15
z
Recall that a homoclinic orbit joining the origin of
the Chen system (6) implies that such an orbit is
doubly asymptotic with respect to time t to the
origin.
As in the previous subsection, let x(t) = ϕ(t),
so that
re 2 The heteroclinic
orbit
in the Chenorbit
system
the parameter
valuesthe
a = 35, b = 3 and
Fig. 2. The
heteroclinic
in with
the Chen
system with
c =3 28.
y(t)
parameter values a = 35, b =
and c = 28.
= ϕ(t) +
nally, to apply the Šilnikov criterion [Šilnikov, 1965], one may impose the following condition:
for all t ≥ 0. Thus, the series expansion (23) is uni2(a + b c)
α1 +
β1 <respect all t ≥ 0.
formly convergent
with
3
For the typical parameter set: a = 35, b = 3
and c = 28, the corresponding heteroclinic orbit is
11
shown in Fig. 2.
Finally, to apply the Šilnikov criterion [Šilnikov,
1965], one may impose the following condition:
2(a + b − c)
(42)
α1 + β 1 < −
3
which ensures that the equilibria O 2 and O3 are
saddle foci. Note that for the typical parameter set,
this condition is obviously satisfied.
To summarize, one has obtained the following
result.
Theorem 3.1. If ∆ > 0 and (42) is satisfied, then
the Chen system has one heteroclinic orbit whose
z(t) = −
(42)
ϕ̇(t)
,
a
ϕ̈(t) + (a − c)ϕ̇(t)
+ 2c − a ,
aϕ(t)
(43)
(44)
and ϕ(t) satisfies
d ϕ̈ + (a − c)ϕ̇
ϕ̈ + (a − c)ϕ̇
+b
dt
ϕ
ϕ
+ aϕ2 + ϕϕ̇ − ab(2c − a) = 0 .
(45)
At this point, it is required that ϕ(t) → 0 as
t → ±∞.
Assume that, for t > 0,
ϕ(t) =
∞
X
dk ekβt ,
(46)
k=1
where β < 0 is an undetermined constant, d k (k ≥
1) are also undetermined coefficients with d 1 =
6 0.
Chen’s Attractor Exists
Similarly to the discussion in the case of the heteroclinic orbit, one has
2
β + (a − c)β − a(2c − a) = 0
β
(β + b) [2(2β + a − c)d2 + f2 ] = 0
d1
where
fn =
(47)
which is just the characteristic polynomial of the
linearized equation of (6) evaluated at the equilibrium point O1 . It then follows from (47) that
p
a − c + (a − c)2 + 4a(2c − a)
β=−
.
(48)
2
Given all the similarities described above, it is
important to point out that the exponent component β in the homoclinic orbit is completely different from that in the heteroclinic orbit. In addition,
(49)
and for n ≥ 2,
β
(nβ + b)[(n + 1)((n + 1)β + a − c)dn+1 + fn+1 ]
d1
X
+
(a + iβ)di dj = 0 ,
(50)
i+j=n
3175
X
i+j=n
ej =
i(iβ + a − c)di ej ,
j
X
j Y
−dk+1 lk
X
k=1
j
Y
d1
(l1 ,...,lj )∈Rj k=1
(51)
!
lk !
,
(52)
(lk )!
k=1
with
(
Rj = (l1 , l2 , . . . , lj )|l1 ≥ 0, . . . , lj ≥ 0,
j
X
i=1
)
i · li = j .
(53)
Note that f2 = −(β + a − c)d2 . According to
(49), one has d2 = 0 if β + b 6= 0 and 3β + a − c 6= 0.
A more general result can be inductively obtained
from d2 = 0 and (50)–(53); that is
d2k = 0 ,
k ≥ 2,
(54)
when kβ + b 6= 0 and (k + 2)β + a − c 6= 0. However,
for the coefficients of odd-order terms, one has the
following iterative relations for n ≥ 1:




l
j i>1
X
X
X
Y

−dk+1 k
−1

di dj + β(nβ + b)
i(iβ + a − c)di
d1
d2n+1 =
nβ(nβ + b)[(n + 2)β + a − c] 
d1
 i+j=2n
i+j=2n+1
(l1 ,...,lj )∈Rj k=1


×
j
X
k=1
j
Y
!
lk !
+ β(β + a − c)(nβ + b)d1
(lk )!
X
2n−1
Y (l1 ,...,l2n−1 ,0) ∈ R2n k=1
−dk+1
d1
lk
2n−1
X
if 2nβ + b 6= 0 and 2(n + 1)β + a − c 6= 0.
They can also be expressed in the following simper
form:
,
d2n+1 = ω2n+1 d2n+1
1
n ≥ 1,
where ω2n+1 are some known functions depending
on a, b, c and β.
Similarly to the case of the heteroclinic orbit,
one can prove the convergence of the series expansion (49).
Due to the symmetry of the Chen system, one
component of the homoclinic orbit of (6) has the
lk ! 


,

(lk )! 
k=1
2n−1
Y
k=1
k=1
!
following form:
 ∞
X



d2k+1 ekβt




 k=1
ϕ(t) = 0


∞

X

−

d2k+1 e−kβt


(55)
for t > 0
for t = 0
(56)
for t < 0 .
k=1
Furthermore, due to the continuity of ϕ(t) at
the origin, the following condition
∞
X
+ d1 = 0 .
ω2i+1 d2i+1
1
i=1
3176
T. Zhou et al.
must be satisfied. It has also been numerically confirmed that such an a1 exists, which will be assumed
here without a detailed proof.
Summarizing the above discussion gives the
following conclusion.
Theorem 4.1. If kβ + b 6= 0 and kβ + a − c 6= 0
(k = 1, 2, . . .), then the Chen system has one homoclinic orbit joining the origin, with one component
in the form of (56).
Observe that for the typical parameter set:
a = 35, b = 3 and c = 28, the conditions kβ + b 6= 0
and kβ + a − c 6= 0 (k = 1, 2, . . .) are always
satisfied.
Finally, the structure of Chen’s attractor is
briefly described as follows.
For each suitable set of parameter values, a,
b and c, i.e. they satisfy ∆ > 0 and α1 + β1 <
−(2(a + b − c))/3, by Theorem 4.1, there exists one
heteroclinic orbit of Ši’lnikov type, and as a subsequence, there exist a countable number of Smale
horseshoes. Therefore, there exists an invariant set
constituting the complex Chen’s attractor. If one
takes into account the presence of the singular point
O1 , then it can be verified that such an attractor is
located on a suspended flow (i.e. the unstable manifold of O1 ) and is not structurally stable. In nature, the structure of Chen’s attractor is similar to
that of the Lorenz attractor except that it is more
sophisticated topologically.
5. Conclusive Remarks
By applying the undetermined coefficient method,
two types of orbits in the Chen system, i.e. heteroclinic and homoclinic orbits, have been identified
and reported in this paper, with explicit and convergent algebraic expressions derived. It has been
shown that with the typical parameter values of
a = 35, b = 3 and c = 28, the Chen system has
one heteroclinic orbit of Ši’lnikov type and also one
homoclinic orbit, implying by the Ši’lnikov criterion
that the Chen system has Smale horseshoes and has
horseshoe chaos.
It is interesting to recall that all trajectories of
the Chen attractor pass through transversely two
Poincaré sections, z = 2c − a and z = 2c − a −
((a − c)2 /4a), for infinitely many times [Zhou et al.,
2003]. Now, with the results obtained in this paper, the observation becomes clearer and has a solid
theoretical backing.
Finally, it is worth mentioning that although it
is difficult to numerically determine homoclinic or
heteroclinic orbits of chaotic systems in general, the
undetermined coefficient method used in this paper
provides a powerful tool for doing so.
Acknowledgments
The first author thanks the hospitality of the
Kanazawa University of Japan, where he conducted part of this research under the fellowship
awarded by the Japanese Society for the Promotion of Science. This paper is partly supported
by National Key Basic Research Special Fund
(No. G1998020309), National Nature Science Foundation of China (No. 10272059), and the Hong Kong
Research Grants Council under the CERG grants
(CityU 1018/01E and 1004/02E).
References
Čelikovský, S. & Vaněček, A. [1994] “Bilinear systems
and chaos,” Kybernetika 30, 403–424.
Čelikovský, S. & Chen, G. [2002] “On a generalized
Lorenz canonical form of chaotic systems,” Int. J. Bifurcation and Chaos 12, 1789–1812.
Chen, G. & Ueta, T. [1999] “Yet another chaotic attractor,” Int. J. Bifurcation and Chaos 9, 1465–1466.
Kennedy, J., Kocak, S. & Yorke, J. A. [2001] “A chaos
lemma,” Amer. Math. Monthly 108, 411–423.
Li, T. Y. & Yorke, J. A. [1975] “Period three implies
chaos,” Amer. Math. Monthly 82, 985–992.
Lorenz, E. N. [1963] “Deterministic nonperiodic flow,”
J. Atmos. Sci. 20, 130–141.
Lü, J. & Chen, G. [2002] “A new chaotic attractor
coined,” Int. J. Bifurcation and Chaos 12, 659–661.
Shi’lnikov, L. P. [1965] “A case of the existence of a
countable number of periodic motions,” Sov. Math.
Docklady 6, 163–166 (translated by S. Puckette).
Shi’lnikov, L. P. [1970] “A contribution of the problem of the structure of an extended neighborhood of
rough equilibrium state of saddle-focus type,” Math.
U.S.S.R.–Shornik 10, 91–102 (translated by F. A.
Cezus).
Silva, C. P. [1993] “Shi’lnikov theorem — a tutorial,”
IEEE Trans. Circuits Syst.-I 40, 675–682.
Sparrow, C. [1982] The Lorenz Equations: Bifurcation,
Chaos, and Strange Attractor (Springer-Verlag, NY).
Tucker, W. [1999] “The Lorenz attractor exists,” C. R.
Acad. Paris Ser. I Math. 328, 1197–1202.
Ueta, T. & Chen, G. [2000] “Bifurcation analysis of
Chen’s attractor,” Int. J. Bifurcation and Chaos 10,
1917–1931.
Chen’s Attractor Exists
Vanĕc̆ek, A. & C̆elikovský, S. [1996] Control Systems: From Linear Analysis to Synthesis of Chaos
(Prentice-Hall, London).
Wiggins, S. [1988] Global Bifurcation and Chaos
(Springer-Verlag, NY).
3177
Zhou, T. S., Chen, G. & Tang, Y. [2003] “The complex
dynamical behaviors of the chaotic Chen’s system,”
Int. J. Bifurcation and Chaos 13, 2561–2574.