Cellular neural networks: Mosaic patterns, bifurcation and complexity

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World Scientific Publishing Company

CELLULAR NEURAL NETWORKS: MOSAIC

PATTERNS, BIFURCATION AND COMPLEXITY

JONQ JUANG∗, CHIN-LUNG LI† and MING-HUANG LIU‡

Department of Applied Mathematics,

National Chiao Tung University, Hsinchu, Taiwan, R.O.C. ∗_{[email protected]}

†_{[email protected]} ‡_{[email protected]}

Received December 21, 2004; Revised January 18, 2005

We study a one-dimensional Cellular Neural Network with an output function which is nonflat at infinity. Spatial chaotic regions are completely characterized. Moreover, each of their exact corresponding entropy is obtained via the method of transition matrices. We also study the bifurcation phenomenon of mosaic patterns with bifurcation parameters z and β. Here z is a source (or bias) term and β is the interaction weight between the neighboring cells. In particular, we find that by injecting the source term, i.e. z
= 0, a lot of new chaotic patterns emerge with a smaller interaction weight β. However, as β increases to a certain range, most of previously observed chaotic patterns disappear, while other new chaotic patterns emerge.

Keywords: Cellular neural networks; mosaic patterns; transition matrix; spatial entropy;

bifurcation.

1. Introduction

Of concern is one-dimensional Cellular Neural Networks (CNNs) of the form

dxi

dt =−xi+ z + αf (xi−1) + af (xi)

+ βf (xi+1), i ∈ Z. (1a)

Here xi denote the state of a cell Ci and f (x) is a

piecewise-linear output function defined by

f (x) =      rx + 1 − r, if x ≥ 1, x, if|x| ≤ 1, rx − 1 + r, if x ≤ −1, (1b)

where r is a positive constant. The quantity z is called a source term or a bias term. The numbers α, a and β are arranged in a vector form [α, a, β], which is called a space-invariant A-template

A = [α, a, β]. (2)

A is called symmetric (resp. antisymmetric) if α = β (resp. α = −β).

CNNs were first proposed by Chua and Yang [1988a, 1988b]. Their main applications are in image processing and pattern recognition [Chua, 1998]. For additional background information, applica-tions, and theory, see [Special Issue, 1995; Thiran, 1997; Chua, 1998] among others.

A basic and important class of solutions of (1) is the stable stationary solutions. Specifically, a

stationary solution x = (xi)i∈Z of (1) satisfies the

following equation

f (xi) = _a1{xi− z − αf(xi−1)− βf(xi+1)}, i ∈ Z. (3)

Let x = (xi)i∈Z be a solution of (3). The

asso-ciated output y = (yi)_i∈Z = (f (x))i∈Z is called a

pattern. The following two types of stationary solu-tions are of particular interest.

47

Definition 1.1. A solution x = (xi)i∈Z is called a

mosaic solution if |x_i| > 1 for all i ∈ Z. Its

asso-ciated pattern y = (yi)i∈Z = (f (x))i∈Z is called a

mosaic pattern. If |x_i|
= 1 for all i ∈ Z and there

are i, j ∈ Z such that |xi| < 1 and |xj| > 1, then

x = (xi)i∈Z and y = (f (x))i∈Z are called, respec-tively, a defect solution and a defect pattern.

To define the stability of the stationary solu-tion, we consider the following linearized stability.

Let ξ = (ξi)i∈Z∈ 2, the linearized operator L(x) of

(1) at a stationary solution x = (xi)_i∈Z is given by

(L(x)ξ)i=−ξi+ αf(xi−1)ξi−1+ af(xi)ξi

+ βf(xi+1)ξi+1. (4)

Definition 1.2. Let x = (xi)_i∈Z be a solution of

(3) with |x|
= 1 for all i ∈ Z. The stationary

solu-tion x is called (linearized) stable if all eigenvalues of L(x) have negative real parts. The solution is called unstable if there is an eigenvalue λ of L(x) such that λ has a positive real part.

It is well-known, see e.g. [Juang & Lin, 2000; Hsu, 2000], that for

1

|a| + |α| + |β| > r ≥ 0, (5)

where r, a, α and β are defined as in (1), −L(x) is a self-adjoint and positive operator. Therefore, if r is sufficiently small, all mosaic solutions of (1) are stable. For r = 0, the complexity of stable stationary solutions of (1) with respect to all the parameters has been completely characterized when the template A is symmetric or antisymmetric (see [Thiran et al., 1995; Juang & Lin, 2000]). For r > 0, sufficiently small, a map approach was introduced to study the complexity of stable stationary solu-tions of (1) with limited success (see e.g. [Hsu, 2000; Chang & Juang, 2004]). Specifically, only the parameters region that would yield Smale horse-shoe, hence, the spatial entropy of ln 2, is located in those papers. That is to say, only regions that yield the full shift with two symbols are found. For r = 0 [Juang & Lin, 2000], the parameter regions corre-sponding to the positive entropy less than ln 2 can also be found. Those are the regions that yield the subshift of finite types (see e.g. [Robinson, 1995]). It would be reasonable to expect that for r
= 0, one can find such regions as well.

The purpose of this paper is to find parameter regions yielding the subshift of finite types when the template A is symmetric. Our approach here

makes use of the techniques originated in [Juang & Lin, 2000] and, later, generalized by [Chu & Shih, 2004]. The paper is organized as follows. In Sec. 2, we introduce the notion of (local) basic mosaic terns. We then identify all these basic mosaic pat-terns. Moreover, the solvability conditions for the existence of such patterns are also given. Section 3 is devoted to the global mosaic patterns for the sym-metric template A and z = 0. Specifically, we find parameters regions whose corresponding positive spatial entropy is less than ln 2. The exact entropy of those regions are obtained via the method of the transition matrix. The effect on the pattern for-mation with the presence of the bias term z and with the intensity of the interaction weight β is recorded in Sec. 4. In particular, with the injec-tion of a source term z (
= 0), a lot of new pat-terns, which correspond to certain subshifts of finite types, emerge with a smaller interaction weight β. However, as β increases to a certain range, most of the previously observed chaotic patterns disap-pear, while other new patterns with positive entropy emerge.

2. Basic Mosaic Solutions and Patterns As in the map approach case, we seek to find the set of solutions of (3) that is uniformly bounded. This is also the essence of the paper in [Chu & Shih, 2004].

Specifically, we consider the set of solutions (xi)_i∈Z

for which

|xi| < 1 + δ for all i ∈ Z, (6a) or equivalently,

|f(xi)| < 1 + rδ for all i ∈ Z, (6b) where δ > 0 is a constant.

To study (3), we first define the following concepts.

Definition 2.1. Given any i ∈ Z, let xi−1 and xi+1

be any real numbers for which|x_j| < 1 + δ, j = i−1,

i + 1. If there is a unique xi satisfying (3), then

[xi−1, xi, xi+1] is called a basic solution of (3).

Its corresponding output [f (xi−1), f (xi), f (xi+1)]

is called a basic pattern of (3). If, in addition, |xj| > 1, j = i − 1, i, i + 1, then [xi−1, xi, xi+1]

(resp. [f (xi−1), f (xi), f (xi+1)]) is called a basic

mosaic solution (resp. pattern) of (3). Note that the template A is space-invariant. Therefore, a basic solution pattern is independent of the spatial variable i.

Notation 2.1. For any mosaic pattern {y_i}_i∈Z, we

shall denote by + (resp. −) if y_i = f (xi) > 1 (resp.

yi = f (xi) < −1). There are only eight types of

basic mosaic patterns. We list as below.

[+ + +]_δ, [− − −]δ, [+ + −]δ, [− + +]δ,

[+− −]_δ, [− − +]δ, [− + −]δ and [+− +]δ. (7)

Notation 2.2. The parameters regions that would yield the eight basic mosaic patterns are,

respec-tively, denoted by Γ+₂, Γ−₋₂, Γ+₀, Γ+₀, Γ−₀, Γ−₀, Γ+₋₂

and Γ−₂.

Remark 2.1. _{Since the template A under} consider-ation is symmetric, the parameter regions

generat-ing [+ +−]_δ and [− + +]_δ are exactly the same (see

Propositions 2.3 and 4.1). Thus, we make no dis-tinction for the region that would yield those two types of mosaic patterns. Likewise, the same is true

for [+− −]_δ and [− − +]_δ.

We next study the range of parameters a, α, β, z and r for which the existence of each of eight basic mosaic patterns is guaranteed. For simplification, we first consider 0 < r < 1/2, z = 0 and α = β. We need the following useful proposition.

Proposition 2.1. Let A = (a1, 0), B = (b1, 0),

C = (1, 1), D = (1 + δ, 1 + rδ), C = (−1, −1) and D = (−1 − δ, −1 − rδ). Suppose −1 < a₁ < b1 < 1

and 0 < r < 1/2. Let E ∈ AB be arbitrarily given. The necessary and sufficient condition for any straight line l passing through E with the slope mE and intercepting the open line segment CD

(resp. CD) is that the slope mE satisfies the

fol-lowing inequalities.

m_BD< mE < m_AC (8a)

(resp. m_AD
< mE < m_BC
). (8b)

Here m_EF means the slope of the line through E and F .

Proof. _{From Fig. 1, we see clearly that l ∩ open}

segmentCD
= ∅ if and only if

m_ED< mE < m_EC. (9)

Note that we need 0 < r < 1/2 to ensure (9)

holds. The slopes m_ED and m_EC are increasing in

E as long as E is in between A and B. Thus if mE satisfies (8a), then the intersection of l and open

A C B D X Y C D ` ` E O Fig. 1.

segment CD is nonempty. On the other hand, if

mE ≥ mEC, we see immediately that the line

pass-ing through E with such slope mE either intersects

CD at C or does not intersect CD at all, a

con-tradiction. Similarly, if mE ≤ m_ED, we draw the

same conclusion. The proof for second assertion of

the proposition is similar.

In the following, we describe the parameters

regions, Γ+₂ and Γ−₋₂, which are the same if 0 <

r < 1/2, α = β and z = 0. Proposition 2.2. Let 0 < r < 1 2, z = 0 and − 1 2(1 + rδ) < α = β < 1 2(1 + rδ). (10)

Then the basic mosaic patterns [+ + +]_δ and [− − −]_δexist provided that (11) or (12) holds. Here (11) and (12) are given in the following.

a + 2β > 1, (11a) (1 + rδ)a + 2(1 + rδ)β < 1 + δ, (11b) β > 0, (11c) and a + 2β(1 + rδ) > 1, (12a) (1 + rδ)a + 2β < 1 + δ, (12b) β < 0. (12c)

Proof. _{We illustrate only the case that β > 0, let}

xi+1 and xi−1 be any numbers in between 1 and

1 + δ, then 2β < β(f (xi−1) + f (xi+1)) =: p < 2β

(1 + rδ) and Eq. (3) reduces to f (xi) = 1

a[xi− p]. (13)

Set A = (2β, 0), B = (2β(1 + rδ), 0) and E = (p, 0). It then follows from Proposition 2.1 that

if (11) holds, then (13) has a unique solution

xi with 1 < xi < 1 + δ. (14)

Similarly, if xi−1 and xi+1 are any numbers in

between−1 − δ and −1, then 2β(1 + rδ) < p < 2β.

Set A = (2β(1 + rδ), 0) and B = (2β, 0), we also conclude that if (11) holds, than (13) has a unique

solution xi with−1− δ < x_i< −1. Since (14) holds

for any 1 < xi−1, xi+1 < 1 + δ or −1 − δ < xi−1,

xi+1 < −1, we conclude that [xi−1, xi, xi+1] is

indeed a local solution.

From Proposition 2.2, we see that for fixed r, 0 < r < 1/2, and δ > 0,

Γ+₂ = Γ−₋₂=



(a, β) : (11) or (12) holds and |β|

< 1

2(1 + rδ) 

=: Γ₂.

We next study the parameters regions Γ+₀

and Γ−₀.

Proposition 2.3. Suppose (10) holds, then the basic mosaic patterns [+ +−]_δ, [− + +]δ, [+ − −]δ and [−−+]_δ exist provided that (15) or (16) holds. Here (15) and (16) are given as the following:

(1 + rδ)a + rδβ < 1 + δ, (15a) a − rδβ > 1, (15b) β > 0, (15c) and (1 + rδ)a − rδβ < 1 + δ, (16a) a + rδβ > 1, (16b) β < 0. (16c)

The parameter regions Γ−₂ and Γ+₋₂are given in

the following.

Proposition 2.4. Suppose (10) holds, then the basic mosaic patterns [+ − +]_δ and [− + −]_δ exist pro-vided that (17 ) or (18 ) holds. Here (17 ) and (18)

are given in the following:

a − 2(1 + rδ)β > 1, (17a) (1 + rδ)a − 2β < 1 + δ, (17b) β > 0, (17c) and a − 2β > 1, (18a) (1 + rδ)a − 2(1 + rδ)β < 1 + δ, (18b) β < 0. (18c)

The proof of Propositions 2.3 and 2.4 are similar to that of Proposition 2.2, and is thus omitted.

Clearly, we have that for fixed r, 0 < r < 1/2, and δ > 0,

Γ+₀ = Γ−₀ =



(a, β) : (15) or (16) holds and |β| < 1 2(1 + rδ)  =: Γ₀, and Γ−₂ = Γ+₋₂ = 

(a, β) : (17) or (18) holds and |β| < 1

2(1 + rδ) 

=: Γ₋₂.

3. Global Patterns and Their Entropy

To construct the global solutions/patterns from the local solutions/patterns, we need the following notation and proposition.

Notation 3.1. Set Γ
_i = R2− Γi, i = 2, 0, −2. Let

Γ_(i1,i2,i3) = R1 ∩ R2 ∩ R3, where ij = 0 or 1, j =

1, 2, 3, and Rj =

_Γ

−2j+4, if ij= 1 ,

Γ
−2j+4, if ij= 0 . For instance, Γ_(1,0,1) = Γ₂ ∩ Γ
₀ ∩ Γ₋₂. The set of basic mosaic patterns whose corresponding parameters are in

Γ_(i1,i2,i3) is denoted by B(i1,i2,i3).

For fixed r and δ, we put Γi, i = 2, 0, −2, on

the a − β plane as in Fig. 2. Let  =  (a, β) : |β| < 1 2(1 + rδ)  . Int. J. Bifurcation Chaos 2006.16:47-57. Downloaded from www.worldscientific.com by NATIONAL CHIAO TUNG UNIVERSITY on 04/26/14. For personal use only.

β a P O R S _T R S T ` ` ` U U` 1 2 1+δ rδ 1+δ rδ -1 2 2(1+rδ) -(1+δ) -(1+δ) -1 1 1 2r r 2r 2(1+rδ) Q Fig. 2. P = (1, 0), Q = ((1 + δ)/(1 + rδ), 0), U = ((2 + δ)/(2 + rδ), (1 − r)/r(2 + rδ)). Note that Γ_(1,1,1) = Γ₂∩ Γ₀∩ Γ₋₂ = Quadrilateral P RQR ∩ 
= φ, Γ_(1,1,0) = Γ₂∩ Γ₀∩ Γ₋₂ = Triangular P SR ∪ Triangular QT_ R ∩
= φ, Γ_(0,1,1) = Γ₂∩ Γ₀∩ Γ₋₂ = Triangular QT R ∪ Triangular P S_ R ∩
= φ, and Γ_(1,0,1) = Γ₂∩ Γ₀∩ Γ₋₂ = φ.

Proposition 3.1. Suppose (10) holds, and that

(a, β) ∈ Γ(i1,i2,i3), ij = 0 or 1, j = 1, 2, 3. If

[xi−1, xi, xi+1] := xL is a local mosaic solution

of (3) for some i, then xL can be extended to be a global solution xG= (xj)j∈Z, where xk= xk, k = i − 1, i, i + 1, and for all i
= j, [xj−1, xj, xj+1] are any local solutions of (3) in B(i1,i2,i3).

Proof. _{We only illustrate the case that (a, β) ∈} Γ_(1,0,0), since the others are similar. In this case

either 1 < xk < 1 + δ or −1 − δ < xk < −1

for all k = i − 1, i, i + 1. Now suppose the

for-mer holds, then we assign xi+3to be any number in

between 1 and 1 + δ. Since (a, β) ∈ Γ_(1,0,0), xi+2can

be uniquely determined and its value lies between 1 and 1 + δ. By proceeding similarly, we get to a

global solution xG as claimed.

From here on, by a mosaic pattern, we mean

that the pattern consists of only + or− sign. That

is to say we make distinction on only the signs of

f (xi). Using Proposition 3.1, we see immediately

that if (a, β) ∈ Γ_(1,0,0), then the only mosaic

pat-terns are of the following two types

... + + + + + + + + + + + +... ... − − − − − − − − − − − −...

Similarly, if (a, β) ∈ Γ(0,1,0)(resp. Γ(0,0,1)), then

the mosaic pattern produced is unique up to the translation.

... + + − − + + − − + + − −... (resp., ... + − + − + − + − + − + −...)

Theorem 3.1. Suppose (10) holds. Then the follow-ing are true.

(i) If (a, β) ∈ Γ_(1,1,1), then any mosaic pattern

(∗_i)_i∈Z, ∗i = + or −, is a pattern for (3).

(ii) If (a, β) ∈ Γ_(1,1,0), then any mosaic pattern

(∗_i)_i∈Z, ∗i = + or −, satisfying the rules that

any + is adjacent to at least one +, any − is adjacent to at least one −, is a pattern for (3).

(iii) If (a, β) ∈ Γ(0,1,1), then any mosaic pattern

(∗)_i∈Z, ∗ = + or −, satisfying the rules that any + is adjacent to at least one −, any − is adjacent to at least one +, is a pattern for (3).

Proof. We illustrate only (ii). The other cases are

similar. If (a, β) ∈ Γ(1,1,0), then its corresponding

basic mosaic patterns are

[+ + +]_δ, [− − −]δ, [+ + −]δ, [− + +]δ, [+ − −]δ,

[− − +]_δ=: B_(1,1,0). (19)

In view of (19) and Proposition 3.1, we see, immediately, that the assertion in (ii) holds

true.

We next study the complexity of the patterns for given choices of parameters.

Definition 3.1. Let µΓ = {(∗i)i∈Z : ∗i = + or −} be a set of stable mosaic patterns of (3) for given choices of parameters in Γ. The spatial entropy

h(µΓ) is defined as the limit

h(µΓ) = lim

n→∞

ln (µn_Γ)

n . (20)

Here (µn_Γ) = the cardinality of the set µn_Γ =

{(∗i)ni=1:∗i= + or −, (∗i)i∈Z∈ µΓ}

Note that µΓ is a translation invariant set

and the limit in (20) is well-defined (see e.g. [Chow et al., 1996]).

Definition 3.2. We say the system (1) or (3) exhibits spatial chaos for given choices of

param-eters in Γ, in case that spatial entropy h(µΓ) is

pos-itive. We say that the system (1) or (3) exhibits pattern formation for given choices of parameters

in Γ in case the spatial entropy h(µΓ) is zero.

We next recall a well-known result (see e.g. [Robinson, 1995]).

Theorem 3.2. Suppose there is a one-to-one and onto correspondence between the set µΓ and the

sequence space Σ_A. Here A is a matrix of dimension n × n whose elements are 0 and 1, and that ΣA=

{(si) : (A)si,si+1 = 1 f or all i}. Then h(µΓ) =

ln λ, where λ is the maximal eigenvalue of A.

Theorem 3.3. Suppose (10) holds. If (a, β) ∈ Γ_(i1,i2,i3), ij ∈ {0, 1}, j = 1, 2, 3, then system

(1) exhibits spatial chaos if and only if i2 = 1

and i1 + i3 ≥ 1. Moreover, h(µΓ(1,1,1)) = ln 2

and h(µΓ(1,1,0)) = h(µΓ(0,1,1)) = ln(1 +

√ 5)/2. Consequently, Γ(1,1,1), Γ(1,1,0) and Γ(0,1,1) are the

only chaotic parameter regions.

Proof. We first show that the mosaic patterns

pro-duced from Γ_(1,1,1), Γ_(1,1,0)and Γ_(0,1,1)are all stable.

Note that the stability condition (5) reduces to |a| + 2|β| < 1

r. (21)

If 0 < r < 1/2, then Q, see Fig. 2, is to the left of the a-intercept of the line a + 2β = 1/r. Moreover, a direct computation could yield that the point U , see Fig. 2, lies on the line a + 2β = 1/r. Similarly,

the point U lies on the line a − 2β = 1/r. Thus, the

mosaic patterns under consideration are all stable.

We illustrate only the cases that (a, β) ∈ Γ(1,1,0),

and (a, β) ∈ Γ_(0,1,1). We assign four symbols + +,

+−, − + and − − to be 1, 2, 3 and 4, respectively.

We define iland ir, respectively, to be the left (resp.

right) side of the symbol corresponding to i. For

instance, let 2 = + −, then 2_l = + and 2_r = −.

We construct a 4× 4 transition matrix A = (a_i,j)

as follows. Set ai,j =      1, if ir = jl and [il, ir, jr] is a basic mosaic pattern in B_(1,1,0), 0, otherwise. (22)

Thus the transition matrix with the choice of

parameters in Γ_(1,1,0) is      1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1     =: A(1,1,0).

Now, the set of µΓ(1,1,0) has a

one-to-one and onto correspondence with the sequence

space Σ_A(1,1,0). Here Σ_A(1,1,0) = {(s_i) : si ∈

{1, 2, 3, 4}, (A(1,1,0))si,si+1 = 1 for all i}. Clearly,

the characteristic polynomial for A_(1,1,0) is λ4 −

2λ3+ λ2− 1 = 0 or equivalently (λ2− λ + 1)(λ2− λ − 1) = 0. It then follows from Theorem 3.2 that h(µΓ(1,1,0)) = ln(1 +

√

5)/2. If (a, β) ∈ Γ(0,1,1), we

will define the corresponding transition matrix A(1,1,0) as      0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 0     =: A(0,1,1),

the characteristic polynomial of A(0,1,1) is (λ2 +

λ + 1)(λ2− λ − 1) = 0. Thus h(µΓ(0,1,1)) = ln(1 +

√

5)/2.

4. The Effect of the Source Term on Patterns

In this section, we first consider the effect of the source term z on patterns. With the presence of the

source term z
= 0, the regions Γ+₂ and Γ−₋₂ are no

longer identical. Same can be said of the two pairs

of regions Γ+₀ and Γ−₀, and Γ−₂ and Γ+₋₂. Therefore,

some new patterns emerge as z moves away from zero.

Proposition 4.1. Suppose −1 + 2|β|(1 + rδ)

< z < 1 − 2|β|(1 + rδ) and 0 < r < 1 2. (23) Then the Table 1 holds true.

Table 1.

Notation Parameters’ Regions Corresponding Patterns

a + z > 1 − 2β, (1 + rδ)a + z < 1 + δ − 2(1 + rδ)β, β > 0.

Γ+₂ or [+ + +]_δ

a + z > 1 − 2(1 + rδ)β, (1 + rδ)a + z < 1 + δ − 2β, β < 0

Γ−₋₂ replacingz by −z in the equations right above. [− − −]_δ,

a + z > 1 + rδβ, (1 + rδ)a + z < 1 + δ − rδβ, β > 0.

Γ+₀ or [+ +−]_δ, [− + +]_δ

a + z > 1 − rδβ, (1 + rδ)a + z < 1 + δ + rδβ, β < 0

Γ−₀ replacingz by −z in the equations right above. [+− −]_δ, [− − +]_δ

a + z > 1 + 2(1 + rδ)β, (1 + rδ)a + z < 1 + δ + 2β, β > 0.

Γ+₋₂ or [− + −]_δ

a + z > 1 + 2β, (1 + rδ)a + z < 1 + δ + 2(1 + rδ)β, β < 0

Γ−₂ replacingz by −z in the equations right above. [+− +]_δ

The first two inequalities imply that |β| <

1/(2(1 + rδ)) =: β4.

For fixed 0 < r < 1/2 and δ > 0 to draw param-eter regions in z − a space, we need the following notations.

Notation 4.1. Denote by z = 1 − 2β(1 + rδ), a + z = 1 − 2β, (1 + rδ)a + z = 1 + δ − 2(1 + rδ)β, a + z = 1 + rδβ, (1 + rδ)a + z = 1 + δ − rδβ, a + z = 1 + 2(1 + rδ)β and (1 + rδ)a + z = 1 + δ + 2β

by l0, l1, l4, l2, l5, l3 and l6, respectively. Replacing

z and −z in those equations above, we shall denote

the corresponding equations by r0, r1, r4, r2, r5, r3

and r6, respectively.

Notation 4.2. (i) We shall denote the

intersec-tion of the lines li and rj, i, j = 1, 2, . . . , 6 by

Ai,j. (ii) We shall denote by the quadrilateral

Ai,jAi,kAl,kAl,j = (li, rk, ll, rj) = (i, k, l, j). Here

the a-coordinate of Ai,j is greater than those of

Ai,k, Al,k and Al,j. Note that such tuple is

well-defined.

Let 0 < r < 1/2 and δ > 0 be fixed and

0 < β < (1 − r)δ/2(1 + rδ)(2 + rδ) =: β1. Putting a z 0 ₁ 2 4 5 6 6 1 5 4 3 3 r r r r r _{l 1} l l l l r 2 l r₀ 0

Fig. 3. Orange region: (5, 4, 4, 5), green region: (4, 3, 3, 4) and yellow region: (3, 2, 2, 3).

ri and li, i = 0, 1, 2, . . . , 6, on z − a plane, we have

the Fig. 3. Notation 4.3. Set Λ₁ = Γ+₂, Λ₂ = Γ−₋₂, Λ₃ = Γ+₀, Λ₄ = Γ−₀, Λ₅ = Γ₋₂+ , Λ₆ = Γ−₂, and for ij, j = 1, 2, . . . , 6, ∈ {0, 1}, we define Γ(i1,i2,i3,i4,i5,i6) = ₆ j=1Rj∩z, where Rj =  Λ_j, if ij = 1, R2− Λj, if ij = 0, and Λz={(α, β) : |z| < 1 − 2β(1 + rδ)}, Int. J. Bifurcation Chaos 2006.16:47-57. Downloaded from www.worldscientific.com by NATIONAL CHIAO TUNG UNIVERSITY on 04/26/14. For personal use only.

With z
= 0 and a small β > 0, we see, in the fol-lowing, that a lot more chaotic parameters regions emerge. The case for β < 0 is similar and is, thus, omitted.

Theorem 4.1. Assume that (23) holds and r is suf-ficiently small. Then the following hold:

(i) Suppose 0 < β < min{2(1 − r)δ/(2 + rδ)

(4 + 5rδ), 2/(6 + 5rδ)} =: min{β0, ˆβ0} and

0 < δ < 2/(1−2r). Then all parameter regions in Table 2 are nonempty and all assertions in Table 2 hold true.

(ii) Suppose min{β₀, ˆβ0} < β < β1 and 0 <

δ < 2/1 − 2r. Then the last two parameter regions Γ_{(0,1,0,1,1,1)} and Γ_{(1,0,1,0,1,1)} in Table 2 are empty, and all other regions are nonempty.

(iii) Suppose 0 < β < min{β0, ˆβ0} and 2/(1 −

2r) < δ. Then the last two parameter regions Γ_{(0,1,1,1,1,0)} and Γ_{(1,0,1,1,0,1)} in Table 2 are empty, and all other regions are nonempty.

(iv) Suppose min{β₀, ˆβ0} < β < β4 and 2/(1 −

2r) < δ. Then the last four parameter regions Γ_{(0,1,0,1,1,1)}, Γ(1,0,1,0,1,1), Γ(0,1,1,1,1,0)

and Γ_{(1,0,1,1,0,1)} in Table 2 are empty, and all other regions are nonempty.

Table 2. Parameters Exact Location

Region in Fig. 3 Basic Mosaic Patterns Contained Spatial Entropy

Γ_{(1,1,1,1,1,1)} (4, 3, 3, 4) ∩V_z [+ + +]_δ, [− − −]_δ, [+ + −]_δ, ln 2 [− + +]_δ, [+ − −]_δ, [− − +]_δ, [− + −]_δ, [+ − +]_δ. Γ_{(0,1,1,1,1,1)} (5, 3, 4, 4) ∩V_z [− − −]_δ, [+ + −]_δ, [− + +]_δ, lnλ₁ [+− −]_δ, [− − +]_δ, [− + −]_δ, [+ − +]_δ. Γ_{(1,0,1,1,1,1)} (4, 4, 3, 5) ∩V_z [+ + +]_δ, [+ + −]_δ, [− + +]_δ, lnλ₁ [+− −]_δ, [− − +]_δ, [− + −]_δ, [+ − +]_δ. Γ_{(1,1,1,1,0,1)} (3, 3, 2, 4) ∩V_z [+ + +]_δ, [− − −]_δ, [+ + −]_δ, lnλ₂ [− + +]_δ, [+ − −]_δ, [− − +]_δ, [+ − +_δ]. Γ_{(1,1,1,1,1,0)} (4, 2, 3, 3) ∩V_z [+ + +]_δ, [− − −]_δ, [+ + −]_δ, lnλ₂ [− + +]_δ, [+ − −]_δ, [− − +]_δ, [− + −]_δ. Γ_{(0,0,1,1,1,1)} (5, 4, 4, 5) ∩V_z [+ +−]_δ, [− + +]_δ, [+ − −]_δ, ln1 + √ 5 2 [− − +]_δ, [− + −]_δ, [+ − +]_δ. Γ_{(1,1,1,1,0,0)} (3, 2, 2, 3) ∩V_z [+ + +]_δ, [− − −]_δ, [+ + −]_δ, ln1 + √ 5 2 [− + +]_δ, [+ − −]_δ, [− − +]_δ. Γ_{(0,1,1,1,1,0)} (5, 2, 4, 3) ∩V_z [− − −]_δ, [+ + −]_δ, [− + +]_δ, ln1 + √ 5 2 [+− −]_δ, [− − +]_δ, [− + −]_δ. Γ_{(1,0,1,1,0,1)} (3, 4, 2, 5) ∩V_z [+ + +]_δ, [+ + −]_δ, [− + +]_δ, ln1 + √ 5 2 [+− −]_δ, [− − +]_δ, [+ − +]_δ. Γ_{(0,1,0,1,1,1)} (6, 3, 5, 4) ∩V_z [− − −]_δ, [+ − −]_δ, [− − +]_δ, ln1 + √ 5 2 [− + −]_δ, [+ − +]_δ. Γ_{(1,0,1,0,1,1)} (4, 5, 3, 6) ∩V_z [+ + +]_δ, [+ + −]_δ, [− + +]_δ, ln1 + √ 5 2 [− + −]_δ, [+ − +]_δ.

Hereλ₁ andλ₂are the maximal roots of (λ3− λ2− λ − 1) = 0 and (λ3− 2λ2+λ − 1) = 0, respectively.

Proof. We illustrate only (i). To see the nonempti-ness of the parameter regions in Table 2, we first

check that the z-coordinates of both A4,3 and A5,4

are smaller than z = 1 − 2β(1 + rδ). A direction computation would yield so provided that 0 < δ <

2/(1 − 2r) and 0 < β < ˆβ0. We then need to

ver-ify that the intersection A of r3 and r4 lies above

l5. We see, via direct computations, that only if

0 < β < β0, then A lies above l5. Note also that if

r is sufficiently small, the stability condition (5) is satisfied. The verification of the other assertions in the theorem is then similar to the above and those

in Theorem 3.1, and is thus omitted.

Remark 4.1. _{(i) If 0 < δ < 2/(1 − 2r) and 0 < r <}

1/2, then β4 > β1.

(ii) Note that 2 > λ1 > λ2 > (1 +√5)/2. Thus,

Table 2 is arranged in the following way: the higher the row of parameters region the more complex are its corresponding patterns.

(iii) It is clear that the chaotic patterns produced

from the regions Γ_{(1,1,1,1,1,1)} and Γ_(1,1,1) are the

same. Similarly, the pairs Γ_{(0,0,1,1,1,1)}, Γ_(0,1,1) and

Γ_{(1,1,1,1,0,0)}, Γ_(1,1,0) generate the exact patterns. Thus, with the presence of the bias term z
= 0, some new chaotic patterns would emerge. Specifically, the patterns whose parameter regions are from Γ_{(0,1,1,1,1,1)}, Γ_{(1,0,1,1,1,1)}, Γ_{(1,1,1,1,0,1)}, Γ_{(1,1,1,1,1,0)}, a z 0 r l 1 2 3 4 5 6 r r r r r _l l l l l 1 2 4 6 3 5 l₀ r₀

Fig. 4. Orange region: (5, 3, 3, 5) and yellow region: (4, 2, 2, 4).

Γ_{(0,1,1,1,1,0)}, Γ_{(1,0,1,1,0,1)}, Γ_{(0,1,0,1,1,1)}, and Γ_{(1,0,1,0,1,1)} are new and chaotic.

(iv) Note that in Fig. 3, we have 0 < β < β1.

Such condition is to ensure that the β-intercept

of l3 is smaller than that of l4. We also remark

that β1 is the β coordinate of R in Fig. 2.

There-fore when β (< β1) is fixed, we see in Fig. 2 that

the line β = β passes through Γ(1,1,0), Γ(1,1,1) and

Γ_(0,1,1), which corresponds to the line z = 0 in

Fig. 3 going through Γ_{(1,1,1,1,0,0)}, Γ_{(1,1,1,1,1,1)} and

Γ_{(0,0,1,1,1,1)}.

Table 3. Parameters Exact Location

Region in Figure 4 Basic Mosaic Patterns Contained Spatial Entropy

Γ_{(0,0,1,1,1,1)} (5, 3, 3, 5) ∩V_z [+ +−]_δ, [− + +]_δ, [+ − −]_δ, ln1 + √ 5 2 [− − +]_δ, [− + −]_δ, [+ − +]_δ. Γ_{(1,1,1,1,0,0)} (4, 2, 2, 4) ∩V_z [+ + +]_δ, [− − −]_δ, [+ + −]_δ, ln1 + √ 5 2 [− + +]_δ, [+ − −]_δ, [− − +]_δ. Γ_{(0,1,1,1,0,0)} (3, 2, 4, 4) ∩V_z [− − −]_δ, [+ + −]_δ, [− + +]_δ, lnλ₃ [+− −]_δ, [− − +]_δ. Γ_{(1,0,1,1,0,0)} (4, 4, 2, 3) ∩V_z [+ + +]_δ, [+ + −]_δ, [− + +]_δ, lnλ₃ [+− −]_δ, [− − +_δ]. Γ_{(0,0,1,1,0,1)} (3, 3, 4, 5) ∩V_z [+ +−]_δ, [− + +]_δ, [+ − −]_δ, lnλ₄ [− − +]_δ, [+ − +]_δ. Γ_{(0,0,1,1,1,0)} (5, 4, 3, 3) ∩V_z [+ +−]_δ, [− + +]_δ, [+ − −]_δ, lnλ₄ [− − +]_δ, [− + −]_δ.

Hereλ₃andλ₄are the maximal roots of λ4− λ3− 1 = 0 andλ4− λ − 1 = 0, respectively. Clearly, 1 +

√

5

2 > λ3> λ4> 1.

Fig. 5.

(v) In the case that (1 − r)δ/(2 + rδ)(2 +

3rδ) < β < β0, (6, 3, 5, 4) reduces to a

triangu-lar A5,4A5,3A. Here A is the intersection of lines r3

and r4. Likewise, (4, 5, 3, 6) reduces to a triangular

too.

(vi) In the case that β0 < β < β1, (5, 2, 4, 3) and

(3, 4, 2, 5) both reduce to a triangular. Moreover, (6, 3, 5, 4) and (4, 5, 3, 6) disappear.

For β1 < β < min{(1 − r)δ/(1 + rδ)(2 + rδ),

(1− r)δ/(2(1 + rδ)2 + rδ), β4} =: min{β2, β3, β4},

we have Fig. 4 and Table 3.

Theorem 4.2. Let (23) hold, 0 < δ < 2/(1−2r) and r be sufficiently small. In the case that β1 < β < min{β₂, β3, β4}, the parameters regions in Table 3

are nonempty, and all assertions in Table 3 hold true.

Remark 4.2. _{(i) If β1 < β < min{β2, β3, β4}}, then

the a-intercept of l3 is greater than that of l4. We

also note that β2 and β3 are the β-coordinates

of S and T , respectively. So when β1 < β <

min{β₂, β3, β4}, we see from Fig. 2 that Γ(1,1,1)

dis-appears. Thus, not surprisingly, most regions in Fig. 3 are destroyed; however, there are some new chaotic parameter regions as opposed to the case

that 0 < β < β1 appear. Specifically, the parameter

regions with indexes containing three zeros newly emerge.

(ii) For β > min{β2, β3, β4}, most chaotic regions

are destroyed and yield no new chaotic regions. We thus skip the discussion of the case.

We conclude the paper with the following remarks.

(i) The antisymmetric template for (1) can be similarly done. Moreover, the generalization of the work to two-dimensional CNNs with output function (1) and with the symmetric and antisymmetric templates is also straight-forward.

(ii) It is of considerable interest to study the defect patterns for (1).

(iii) Figure 5 is a collection of a computer simula-tion with sets of parameters chosen from the parameter regions in Tables 2 and 3. Specif-ically, we set r = 0.25 and δ = 2 for all cases. The first eleven cases in Fig. 5 corre-spond to the first eleven parameters regions in Table 2. The last four cases in Fig. 5 corre-spond to the last four parameters regions in Table 3. Each collection in Fig. 5 contains two arrays of colors. The first array is the initial outputs. The second array represents the final

outputs. If the state xj of a cell Cj is such

that |x_j| < 1, then we color it green. If the

state xj of a cell Cj is less than −1 (greater

than 1, respectively), then we color it blue (red, respectively). Moreover, the final outputs in each of the collection consist of all basic mosaic patterns allowed in their corresponding

parameter region. For instance, the final out-puts in (1) consist of all eight basic mosaic

patterns. Likewise, in (6) — Γ_{(0,0,1,1,1,1)} and

(12) — Γ_{(0,1,1,1,0,0)}, their corresponding

out-puts contain six and five basic mosaic patterns listed in Tables 2 and 3, respectively.

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