Int'l Conf. Scientific Computing | CSC'15 |
169
Complex Dynamics of Hybrid Cellular Automata Composed of Two
Period Rules
Bo Chen, Fangyue Chen, Zhongwei Cao and Xia Zhang
Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, China
Abstract— The members of Chua’s period rules, which are
considered to have the simplest dynamic behaviors before,
actually define chaotic subsystems by introducing the hybrid
mechanism. Through exploiting the mathematical definition
of hybrid cellular automata (HCAs), this work presents an
analytical method of symbolic dynamics of the HCA rule 77
and 168 as well as the HCA rule 28 and 33. In particular,
the two HCAs are all topologically mixing and possess the
positive topological entropy on their subsystems. This result
therefore naturally argues that they are chaotic in the sense
of both Li-Yorke and Devaney on the subsystems. Finally, it
is worth mentioning that the method presented in this paper
is also applicable to other HCAs therein.
Keywords: Hybrid cellular automata, symbolic dynamics, chaos,
topologically mixing, topological entropy
1. Introduction
Cellular automata (CAs) are a class of spatially and
temporally discrete, deterministic mathematical systems with
large degrees of freedom characterized by local interactions
and an inherently parallel form of evolution [1-5]. Basing
on previous work, L. O. Chua et al. provided a nonlinear
dynamics perspective to Wolfram’s empirical observations
and grouped elementary cellular automata (ECAs) into six
classes depending on the quantitative analysis of the orbits
[6-10]. These six classes are established as period-1, period2, period-3, Bernoulli στ -shift, complex Bernoulli-shift and
hyper Bernoulli-shift rules. It is worth mentioning that some
of their work is consistent with previous studies of other
authors.
In view of an one-dimensional CA, when the evolution
of all its cells is dependent on the one and only global
function, it is called uniform, otherwise it will be called
hybrid, i.e. hybrid cellular automata (HCA) [11,12]. For
instance, denoted by HCA(N ,M ), HCA rule, composed of
ECA rule N and ECA rule M , is specified to obey the ECA
rule N at odd sites of the cell array and obey the rule M
at even sites of the cell array. There are much research on
HCAs which have been applied in cryptographically secure,
see [13-15] and references therein.
Though HCAs are endowed with simple hybrid rules and
evolve on the same square tile structures, the evolution of
HCAs may exhibit rich dynamical behavior with local interactions. More accurately, it can be asserted that the dynamics
of the CAs might be changed from simple to complex and
vice versa by just introducing the hybrid mechanism. Noting
that the dynamics of Chua’s period rules are extremely
simple, we have opted for two of these rules to compose the
HCAs and discovered that several HCAs ultimately produce
behavior of complexity. Although ECA rule 77 and ECA
rule 168 are belong to Chua’s period-1 rules, ECA rule 28
and ECA rule 33 are belong to Chua’s period-2 rules, it is
found that HCA(77,168) and HCA(28,33) are endowed with
glider phenomena.
2. Preliminaries
First and foremost, several terminology and notations are
the necessary prerequisite to the rigorous consideration in the
following. The set of bi-infinite configurations is denoted by
S Z = · · · S × S × S · · · and a metric d on S Z is defined as
+∞
i ,xi )
d(x
d(x, x) = i=−∞ 21|i| 1+
i ,xi ) , where S = {0, 1, . . . , k −
d(x
·) is the metric on S defined as
1}, x, x ∈ S Z and d(·,
i , xi ) = 1. As for
d(xi , xi ) = 0, if xi = xi ; otherwise, d(x
a finite symbol S, a word over S is finite sequence a =
(α0 , ..., αn ) of elements of S.
In S Z , the cylinder set of a word a ∈ S Z is [a]k = {x ∈
Z
S |x[k,k+n] = a}, where k ∈ Z. It is apparent that such a
set is both open and closed (called clopen) [17]. The cylinder
sets generate a topology on S Z and form a countable basis
for this topology. Therefore, each open set is a countable
union of cylinder sets. In addition, S Z is a Cantor space.
The classical right-shift map σ is defined by [σ(x)]i = xi−1
for any x ∈ S Z ,i ∈ Z. A map F : S Z → S Z is a CA if and
only if it is continuous and commutes with σ, i.e., σ ◦ F =
F ◦ σ. For any CA, there exists a radius r ≥ 0 and a local
rule N : S 2r+1 → S such that [F (x)]i = N (x[i−r,i+r] ).
Moreover, (S Z , F ) is a compact dynamical system.
A set X ⊆ S Z is F invariant if F (X) ⊆ X and strongly
F invariant if F (X) = X. If X is closed and F invariant,
then (X, F ) or simply X is called a subsystem of F . A
set X ⊆ S Z is an attractor if there
exists a nonempty
clopen F -invariant set Y such that n≥O F n (Y ) = X.
Thus, there always exists a global attractor, denoted by
Λ = n≥O F n (S Z ), which is also called the limit set of
F . For instance, let A denote a set of some finite words
over S, and Λ = ΛA is the set which consists of the
bi-infinite configurations made up of all the words in A.
Then, ΛA is a subsystem of (S Z , σ), where A is said to be
170
Int'l Conf. Scientific Computing | CSC'15 |
the determinative block system of Λ. For a closed invariant
subset Λ ⊆ S Z , the subsystem (Λ, σ) or simply Λ is called
a subshift of σ.
3. Dynamics of HCA(77,168)
Among 256 ECA rules, 67 belong to period-1 rules
since most random initial bit strings converge to a period-1
attractor. The 25 dynamically-independent period-1 rules are
N ={0, 4, 8, 12, 13, 28, 32, 40, 44, 72, 76, 77, 78, 104, 128,
132, 136, 140, 160, 164, 168, 172, 200, 204, 232}. And the
dynamics of the remaining rules can be trivially predicted
and determined from one of the corresponding equivalent
rules in N [7]. Here, we construct the HCAs composed of
two period-1 rules in N . Thus, an empirical glimpse on all
spatio-temporal patterns of above HCAs reveals that glider
phenomena are discovered in HCA(13,40), HCA(13,104),
HCA(13,160), HCA(13,168), HCA(13,232), HCA(77,160),
HCA(77,168) and HCA(77,232).
Figure 1 provides an emblematic example of spatiotemporal pattern of HCA(77,168). By exploiting the mathematical definition of HCAs, this section is devoted to an
in-depth study of the symbolic dynamics of HCA(77,168)
in the bi-infinite symbolic sequences space [18-20]. For biinfinite, one-dimensional ECAs, r = 1 and S is denoted
by {0, 1}. Each local rule of ECAs can be endowed with a
boolean function [16]. Then, the boolean function of rule
77 is expressed as [F77 (x)]i = xi xi+1 ⊕ xi−1 xi xi+1 ⊕
xi−1 xi xi+1 , ∀i ∈ Z, where xi ∈ S, “·”, “⊕” and “−”
denote “AN D”, “XOR” and “N OT ” logical operations,
respectively. The Boolean function of rule 168 is expressed
as [F168 (x)]i = xi−1 xi+1 ⊕ xi−1 xi xi+1 , ∀i ∈ Z. Consequently, theBoolean function of HCA(77,168) is induced as
[F77 (x)]i if i is odd,
[F (x)]i =
[F168 (x)]i if i is even.
0,0,0),(1,0,0,0,0,1),(1,0,0,1,0,1),(1,0,0,1,1,0),(1,0,0,1,1,1),(1,
0,1,0,0,0),(1,0,1,0,0,1),(1,0,1,0,1,0),(1,1,1,0,0,0),(1,1,1,0,0,1),
(1,1,1,0,1,0),(1,1,1,1,1,0),(1,1,1,1,1,1)}.
Proof : The proof of above proposition can be obtained
through computer-aided method according to the definitions
of F and σL , the details are omitted here.
Proposition 2: ΛA in Proposition 1 is a subshift of finite
type of (S Z , σ).
The following objective is to investigate the complexity
and chaotic dynamics of F 4 on its subsystem ΛA . It follows
from proposition 2 that dynamical behaviors of F 4 on S Z
can be characterized via a subshift ΛA , which is a subshift of
finite type. As the topological dynamics of a subshift of finite
type is largely determined by the properties of its transition
matrix, it is helpful to briefly review some definitions.
A matrix D is positive if all of its entries are nonn
negative, irreducible if ∀i, j, there exist n such that Dij
> 0,
n
and aperiodic if there exists N , such that Dij > 0, n >
N, ∀i, j. If Λ is a two-order subshift of finite type, then it
is topologically mixing if and only if D is irreducible and
aperiodic, where D is its associated transition matrix with
Dij = 1, if (i, j) ≺ Λ; otherwise Dij = 0.
Then, the topologically conjugate relationship between
(ΛA , σ) and a two-order subshift of finite type is established.
The dynamical behaviors of F 4 on ΛA are discussed based
on existing results.
Let S = {r0 , r1 , · · · , r24 , r25 } be a new symbolic
set, where ri , i = 0, . . . , 25, stand for elements of A
respectively. Then, one can construct a new symbolic space
Denote by B = {(rr )|r = (b0 b1 b2 b3 b4 b5 ), r =
SZ on S.
∀2 ≤ j ≤ 5 such that bj = b }.
(b0 b1 b2 b3 b4 b5 ) ∈ S,
j−1
Further, the two-order subshift ΛB of σ is defined
by ΛB = {r = (· · · , r−1 , r0∗ , r1 , · · · ) ∈ SZ |ri ∈
(ri , ri+1 ) ≺ B, ∀i ∈ Z}. Therefore, it is easy to calculate
S,
the transition matrix D of the subshift ΛB as follow:
⎛
Fig. 1: The spatio-temporal pattern of HCA(77,168).
Proposition 1: For HCA(77,168), there exists a subset
4
ΛA of S Z , such that F 4 (x)|ΛA = σL
(x)|ΛA , where
ΛA = {x = (· · · , x−2 , x−1 , x0 , x1 , x2 , x3 , · · · ) ∈
S Z , (xi−2 , xi−1 , xi , xi+1 , xi+2 , xi+3 ) ∈ A, ∀i ∈ Z} and
A = {(0,0,0,0,0,0),(0,0,0,0,0,1),(0,0,0,1,0,1),(0,0,0,1,1,0),(
0,0,0,1,1,1),(0,1,0,1,0,1),(0,1,0,1,1,0),(0,1,0,1,1,1),(0,1,1,0,0,
0),(0,1,1,0,0,1),(0,1,1,0,1,0),(0,1,1,1,1,0),(0,1,1,1,1,1),(1,0,0,
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Int'l Conf. Scientific Computing | CSC'15 |
4
4
Proposition 3: (ΛA , σL
) and (ΛB , σL
) are topologically
4
conjugate; namely, (ΛA , F 4 )and (ΛB , σL
) are topologically
conjugate.
Proof : Define a map from ΛA to ΛB as follows: π : ΛA →
ΛB , x = (..., x−1 , x∗0 , x1 , ...) → (..., r−1 , r0∗ , r1 , ...), Where
ri = (xi , xi+1 , xi+2 , xi+3 , xi+4 , xi+5 ), ∀i ∈ Z. Then, it
follows from the definition of ΛB that for any x ∈ ΛA , one
has π(x) ∈ ΛB ; namely,π(ΛA ) ⊆ ΛB . One can easily check
4
4
that π is a homeomorphism and π ◦ σL
= σL
◦ π. Therefore,
4
4
(ΛA , σL
) and (ΛB , σL
) are topologically conjugate.
Proposition 4: F 4 is topologically mixing on ΛA .
Proof : It follows from [17] that a two-order subshift
of finite type is topologically mixing if and only if its
transition matrix is irreducible and aperiodic. Meanwhile, it
is easy to verify that Dn is positive for n ≥ 4, where D
is the transition matrix of the two-order subshift ΛB . This
implies that D is irreducible and aperiodic.
Proposition 5: The topological entropy of F 4 |ΛA is
logλ∗ = log(2.61803) = 0.962424, where λ∗ is the
maximum positive real root of equation λ22 (λ2 − 3λ +
1)(λ2 − λ + 1) = 0.
Proof : Recall that two topologically conjugate systems
have the same topological entropy and the topological
entropy of σ on ΛB equals logρ(D), where ρ(D) is the
spectral radius of the transition matrix D of the subshift ΛB .
Proposition 6: F is topologically mixing on ΛA .
Proof : To prove F |ΛA is topologically mixing, it is necessary to check that for any two open sets U, V ⊂ ΛA ,
∃N such that F (U ) ∩ V = ∅, for n ≥ N . Since F |ΛA
is topologically mixing, it immediately follows that for
any two open sets U, V ⊂ ΛA , there exists N0 such that
(F 4 )k (U )∩V = F 4k (U )∩V = ∅, for k ≥ N0 , we consider
two situations separately.
Case 1: n = 4k. It is obvious that F n (U )∩V = F 4k (U )∩
V = ∅.
Case 2: n = 4k + 1, 4k + 2 or 4k + 3. Firstly, we need
to prove that F : ΛA → ΛA is a homeomorphism. It is
evident that F |ΛA is surjective. Suppose that there exist
x, x ∈ ΛA such that F (x) = F (x ). Thus, F 4 (x) = F 4 (x )
4
4
holds, i.e., σL
(x) = σL
(x ), which implies x = x . Hence,
F |ΛA is injective. Since ΛA is a compact Hausdorff space,
and F |ΛA is one-to-one, onto and continuous, F −1 exists
and is continuous. Consequently, F : ΛA → ΛA is a
homeomorphism. This implies that F i (U ) is also an open
set, thus, one has F n (U ) ∩ V = F 4k ◦ (F i (U )) ∩ V = ∅,
where i = 1, 2, 3.
It follows from [17] that the positive topological entropy
implies chaos in the sense of Li-Yorke. Meanwhile, the
topological mixing is also a very complex property of
dynamical systems. A system with topologically mixing
171
property has many chaotic properties in different senses. For
instance, the chaos in the sense of Li-York can be deduced
from positive topological entropy. More importantly though,
both the chaos in the sense of Devaney and Li-York can be
deduced from topologically mixing.
In conclusion, the mathematical analysis presented above
provides the rigorous foundation for the following theorem.
Theorem 1: F is chaotic in the sense of both Li-Yorke
and Devaney on the subsystem ΛA .
4. Dynamics of HCA(28,33)
Among 256 ECA rules, 25 belong to period-2 rules
since most random initial bit strings converge to a period2 periodic orbit. Moreover, the 13 dynamically-independent
period-2 rules are M ={1, 5, 19, 23, 28, 29, 33, 37,
50, 51, 108, 156, 178} [8]. An empirical glimpse on
all spatio-temporal patterns of HCAs composed of two
period-2 rules reveals that shift phenomena are discovered
in HCA(23,28), HCA(28,33), HCA(28,37), HCA(28,50),
HCA(28,178), HCA(33,156), HCA(37,156) and so on. Remarkably, HCA(29,33) and HCA(29,37) are endowed with
more complicated phenomena.
Figure 2 provides the example of spatio-temporal pattern
of HCA(28,33). Its symbolic dynamics on the space of biinfinite symbolic sequences is also briefly discussed in the
following. The boolean function of rule 28 is expressed
as [F28 (x)]i = xi−1 xi xi+1 ⊕ xi−1 xi , ∀i ∈ Z, where
xi ∈ S. The Boolean function of rule 33 is expressed as
[F33 (x)]i = xi−1 xi xi+1 ⊕ xi−1 xi xi+1 , ∀i ∈ Z. Consequently, theBoolean function of HCA(28,33) is induced as
[F28 (x)]i if i is odd,
[F(x)]i =
[F33 (x)]i if i is even.
Fig. 2: The spatio-temporal pattern of HCA(28,33).
Proposition 7: For HCA(28,33), there exists another
4
subset ΛA of S Z , such that F 4 (x)|ΛA = σR
(x)|ΛA , where
ΛA = {x = (· · · , x−3 , x−2 , x−1 , x0 , x1 , x2 , x3 , x4 · · · ) ∈
∀i ∈
S Z , (xi−3 , xi−2 , xi−1 , xi , xi+1 , xi+2 , xi+3 , xi+4 ) ∈ A,
Z} and A2 = {(0,0,0,0,1,0,0,0),(0,0,0,0,1,0,0,1),(0,0,0,0,1,0,
1,0),(0,0,0,0,1,0,1,1),(0,0,0,0,1,1,1,0),(0,0,1,0,0,1,0),(0,0,0,1,
0,0,1,1),(0,0,0,1,0,1,0,1),(0,0,0,1,0,1,1,1),(0,0,1,0,0,0,0,0),(0,
0,1,0,0,0,0,1),(0,0,1,0,0,0,1,0),(0,0,1,0,0,0,1,1),(0,0,1,0,0,1,0,
0),(0,0,1,0,1,0,0,0),(0,0,1,0,1,0,0,1),(0,0,1,0,1,1,1,0),(0,0,1,1,
172
Int'l Conf. Scientific Computing | CSC'15 |
1,0,0,0),(0,0,1,1,1,0,0,1),(0,1,0,0,1,0,0,0),(0,1,0,0,1,0,0,1),(0,
1,0,0,1,0,1,0),(0,1,0,0,1,0,1,1),(0,1,0,0,1,1,1,0),(0,1,0,1,0,1,0,
1),(0,1,0,1,0,1,1,1),(0,1,0,1,1,1,1,0),(0,1,1,1,1,0,0,1),(1,0,0,0,
0,0,1,0),(1,0,0,0,0,0,1,1),(1,0,0,0,0,1,0,0),(1,0,0,0,0,1,0,1),(1,
0,0,0,1,0,0,0),(1,0,0,0,1,0,0,1),(1,0,0,0,1,0,1,0),(1,0,0,0,1,0,1,
1),(1,0,0,0,1,1,1,0),(1,0,0,1,0,0,1,0),(1,0,0,1,0,0,1,1),(1,0,1,0,
0,0,0,0),(1,0,1,0,0,0,0,1),(1,0,1,0,0,0,1,0),(1,0,1,0,0,0,1,1),(1,
0,1,0,0,1,0,0),(1,0,1,1,1,0,0,0),(1,1,1,0,0,0,0,0),(1,1,1,0,0,0,0,
1),(1,1,1,0,0,0,1,0),(1,1,1,0,0,0,1,1),(1,1,1,0,0,1,0,0)}.
ΛA is a subshift of finite type of (S Z , σR ). Further, the
relevant two-order subshift ΛB also can be introduced by
the above similar methods. Therefore, it is easy to calculate
of the subshift Λ as follow:
the transition matrix D
B
⎛
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⎝
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
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0
0
0
0
0
0
1
0
0
0
0
0
0
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0
0
0
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0
0
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0
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1
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1
0
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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0
0
0
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0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
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0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
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0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
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0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
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0
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0
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0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
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0
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0
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1
0
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1
0
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1
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1
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0
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0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
n is positive for n ≥ 10, which implies D
Meanwhile, D
is irreducible and aperiodic. Thus, it can be obtained immediately that F 4 is topologically mixing on ΛA. Therefore,
F is topologically mixing on ΛA. The topological entropy
of F 4 |ΛA is logλ∗ = log(2.139374) = 0.760513 as λ∗ is
the maximum positive real root of λ40 (λ9 − 2λ8 + λ7 −
4λ6 + 2λ5 + λ4 + λ3 − λ − 1)(λ + 1) = 0 which is the
In conclusion, the mathematical
characteristic equation of D.
analysis presented above provides the rigorous foundation
for the following theorem.
Theorem 2: F are chaotic in the sense of both Li-Yorke
and Devaney on the subsystem ΛA.
Acknowledgments
This research was supported by NSFC (Grants no.
11171084 and 60872093).
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