Cellular neural networks: Space-dependent template, mosaic patterns and spatial chaos

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Papers

International Journal of Bifurcation and Chaos, Vol. 12, No. 8 (2002) 1717–1730

c

World Scientific Publishing Company

CELLULAR NEURAL NETWORKS: SPACE-DEPENDENT

TEMPLATE, MOSAIC PATTERNS AND SPATIAL CHAOS

JONQ JUANG∗ and SHIH-FENG SHIEH† Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan

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

LARRY TURYN

Department of Mathematics and Statistics, Wright State University, Dayton, OH, USA

[email protected]

Received September 29, 2000; Revised August 9, 2001

We consider a Cellular Neural Network (CNN) with a bias term in the integer lattice Z2 _on

the planeZ2_{. We impose a space-dependent coupling (template) appropriate for CNN in the}

hexagonal lattice onZ2. Stable mosaic patterns of such CNN are completely characterized. The

spatial entropy of a (p1, p2)-translation invariant set is proved to be well-defined and exists.

Using such a theorem, we are also able to address the complexities of resulting mosaic patterns. Keywords: Cellular Neural Networks; space-dependent template; spatial chaos.

1. Introduction

The Cellular Neural Networks (CNNs) was orig-inally formulated by Chua and Yang in [1988a, 1988b]. The CNNs without input terms are of the form dxi,j dt =−xi,j+ z + X |k|≤d, |`|≤d ak,`;i,jf (xi+k,i+`) , (i, j)∈Z2, (1a) xi,j(0) = x0i,j. (1b)

Here xi,j denote the state of a cell Ci,j, and z is

an independent voltage source. When z = 0, (1) is called unbiased; when z 6= 0 it is called biased. The nonlinearity of f is a piecewise-linear function of the form

f (x) = 1

2(|x + 1| − |x − 1|) . (2) For fixed i, j, the numbers ak,`;i,j|k|, |`| ≤ d, k,

`∈Zand d a positive integer, denote the (local) in-teraction weights between the center cell Ci,jand its

neighboring cells Ck,`. The numbers ai,j;k,`,|k| ≤ d,

|`| ≤ d, can be arranged in a (2d + 1) × (2d + 1) matrix form, which is called a space-dependent

∗_{Visiting Department of Mathematics, Texas A&M University, College Station, TX and Department of Mathematics and} Statistics, Wright State University, Dayton, OH, USA. E-mail: [email protected]

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Int. J. Bifurcation Chaos 2002.12:1717-1730. Downloaded from www.worldscientific.com

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Ai,j-template Ai,j =      

ai,j;−d,d ai,j;−d+1,d . . . ai,j;d,d

ai,j;−d,d−1 ai,j;−d+1,d−1 . . . ai,j;d,d−1

..

. ... ...

ai,j;−d,−d ai,j;−d+1,−d . . . ai,j;d,−d

      (2d+1)×(2d+1). (3)

Ai,j is called space-invariant if Ai,j ≡ A for all i, j ∈ _Z2_{. If, for each i, j, a}

i,j;k,` = ai+k,j+`;−k,−`

for all |k|, |`| ≤ d, then Ai,j are called symmetric. For further physical and mathematical backgrounds of CNNs as well as their applications, we refer to [Chua, 1998; Chua & Roska, 1993; Chua & Yang, 1988a, 1988b; Crounse et al., 1996; Thiran, 1997, Thiran et al., 1997; Special Issue, 1995]. Much of the theoretical work concerning CNNs focus on the space invariant template. Our motivation for study-ing a space-dependent template is two-fold. First, chaotic dynamics is only reported (see e.g. [Yen, 1998; Zou et al., 1993; Zou & Nossek, 1991] in case that the template is space-dependent. It seems to be very unlikely that a space-invariant template would yield chaotic dynamics. Second, suppose one considers the regular hexagonal lattice in R2 and that each cell, lying on the vertices of hexagons, only directly interacts with its nearest neighbors. Such a problem is equivalent to placing each cell on the integer lattice in_R2 _{with a space-dependent}

template. To see this, consider the following hexag-onal lattice in_R2.

We assume that each cell only directly interacts with its nearest neighbors. For instance, the cell F only directly interacts with cells E, F , G and I. Now, if one squeezes each hexagon into a square, we will get the following integer lattice inR2

From Fig. 2, we see that the cell E interacts with the cell A, which is on top of E, while the cell

Fig. 1. The hexagonal lattice inR2.

F does not interact with the cell B. Thus, to study the dynamical systems on hexagonal lattice in _R2

we consider an equivalent problem on integer lattice inR2 with the following space-dependent template Ai,j. Ai,j =    0 ai,j;0,1 0

ai,j;−1,0 ai,j;0,0 ai,j;1,0

0 ai,j;0,−1 0    =                              0 a3 0 a1 a a2 0 0 0     if i + j is even     0 0 0 b2 b b1 b3     if i + j is odd. (4)

In this paper, we investigate the mosaic patterns of (1), (2), and (4) and the complexity of mosaic terns. Some simulations of the CNN mosaic pat-terns are given in the appendix. A mosaic solution x = (xi,j) is a stationary solution of (1a) satisfying

|xi,j| > 1 for all (i, j) ∈ Z2 and y = (f (xi,j)) is

called a mosaic pattern.

Mosaic solutions of lattice dynamical systems have been studied by many authors ([Chow & Mallet-Paret, 1995a, 1995b; Chow et al., 1996a, 1996b] and the work cited therein). In the case of CNNs, mosaic patterns of one-dimensional CNNs with symmetric, space-invariant template, [b, a, b], were studied by Thiran [1997] etc. Mosaic patterns

Fig. 2. The integer lattice inR2 squeezed from Fig. 1.

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of two-dimensional CNNs with symmetric coupling between nearest neighbors, and also next-nearest neighbors are completely characterized by Juang and Lin [2000]. In [Shih, 1998], mosaic pat-terns of two-dimensional CNNs with asymmetric template were investigated. Local patterns for general space-invariant templates were studied in [Hsu et al., 2000].

2. Notations and Preliminaries

Given a template Ai,j, as defined in (3), and a bi-ased term z, the stationary equation for (1a) is xi,j = z +

X

|k|≤1,|`|≤1

ak,`;i,jf (xi+k,j+`) , (i, j)∈Z2.

(5) Let x = (xi,j) be a solution of (5). Two types

of stationary solutions are of interest: mosaic and

defect. A defect solution x = (xi,j) satisfies |xi,j| >

1 for (i, j) ∈ Z2\D and |xk,`| < 1 for (k, `) ∈ D,

where D 6= ∅ and D 6= Z2. Its corresponding pat-tern y = (f (xi,j)) is called a defect pattern. Let

x = (xi,j) be a (defect or mosaic) solution of (5),

we denote Γ+, Γ− and Γ× as follows

Γ+={(i, j) ∈Z2 : xi,j> 1} Γ₋={(i, j) ∈Z2 : xi,j<−1} , (6a) and Γ_×={(i, j) ∈Z2 :|xi,j| < 1} , (6b) respectively. Let _Z2 E = {(i, j): i, j ∈ Z and i + j

is even} and Z2_O = {(i, j): i, j ∈ Z and i + j is odd}. If (i, j) ∈Z2_E (resp.Z2_O), then (i, j) is called even (resp. odd point). Stability is then studied us-ing spectral theory. Let ξ = (ξi,j) ∈ ˜`2, a suitable

weighted `2 space. The linearized operatorL(x) of (5) at x is given by

(L(x)ξ)i,j =

(

−ξi,j+ Li,j, if (i, j)∈ Γ+∪ Γ−,

(ai,j;0,0− 1)ξi,j+ Li,j, if (i, j)∈ Γ×.

(7a) Here, Li,j = X (k,`)∈N+_(i,j)_∈Γ × ai,j;k,`ξk,` (7b) and N+(i, j) ={(p, q) ∈_Z2:|p − i| + |q − j| = 1} , (7c)

Definition 2.1. Let x = (xi,j) be a solution of (5)

with |xi,j| 6= 1 for all (i, j) ∈ Z2. x is then called

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

Theorem 2.1. Let x = (xi,j) be a solution of (5)

and let the templates Ai,j be given as in (4). Then the following holds. (i) If Ai,j are symmetric for all i, j, then L(x) is self-adjoint. (ii) If x is a mosaic solution of (5), then x is stable. (iii) If ai,j;0,0 > 1

for all (i, j)∈ _Z2_{, and x is a defect solution, then}

x is unstable. Remark

(1) Theorem 2.1 is a direct generalization of Theo-rem 2.4 of [Juang & Lin, 2000].

(2) If Ai,j are given as in (4), then Ai,j are sym-metric if and only if ai = bi i = 1, 2, 3.

To study the complexity of mosaic patterns, we review some definitions and results concerning spa-tial entropy. Let A be a finite set of d elements and D ≥ 1 be an integer representing the lattice dimension. Denote by AZD the set of all map-pings y:ZD → A. In our case, D = 2, d = 2 and A = {−1, 1} for the mosaic patterns. Consider any nonempty subset U ⊆ AZ2. Here U will represent the mosaic patterns. The setU is said to be transla-tion invariant if Sk(U) = U, k = 1, 2, . . . , D, where

Sk:AZ D

→ AZD

is a shift operator. To save our no-tation later, we write Sk(U) = U as U + ek = U.

Here ek is the unit basis vector in the kth direction

ofRD. Let ΓN(U) count the number of distinct

pat-terns among the elements ofU restricted in a given rectangle of size N1× N2.

Definition 2.2. The set U is said to be (p1, p2

)-translation invariant if there exist non-negative in-tegers p1, p2 such that P2i=1pi ≥ 1 and U +

(p1, 0) =U +(0, p2) =U. The spatial entropy h(U)

of the setU is defined as the limit

h(U) = lim

N1,N2→∞

log ΓN(U)

N1N2

, (8)

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where the limit is taken over all possible choices of N1× N2 rectangles.

In CNNs, if the template is space-invariant, then any stationary solution of (1a) is translation invariant. Note that if the template Ai,j is given as in (4), then any stationary solution of (1a) is (1, 1)-translation invariant. One of our main results, The-orem 4.1, will show that the spatial entropy h(U) is well-defined and exists provided that U is (p1, p2

)-translation invariant.

Definition 2.3. Let U be a translation invariant

subset of AZD; U exhibits spatial chaos if the spa-tial entropy h(U) is greater than zero. Otherwise, U exhibits pattern formation.

3. Local Mosaic Patterns

To simplify our representation, we consider the templates given in (4) with a = b, and ai = bi,

i = 1, 2, 3. We first set the following notations

Ai,j =                              0 a1 0 a1 a a1 0 0 0    := AE if (i, j)∈Z2E     0 0 0 a1 a a1 0 a1 0    := AO if (i, j)∈Z2O, (9) N_E+(i, j) ={(p, q) ∈Z2:|p − i| + |q − j| = 1, q− j > 0} , i + j is even , N_O−(i, j) ={(p, q) ∈Z2:|p − i| + |q − j| = 1, q− j < 0} , i + j is odd .

Definition 3.1. Given any proper subset T ⊆Z2,

x_|T(≡ xT) is called a local mosaic solution if xT is a restriction of some mosaic solution x of (5) on T . The restriction (f (xi,j))_|T on T is called a local

pat-tern, and will be denoted by (f (xi,j))_|T = (yTi,j) =

yT_{; when T =}_Z2_{, y is called a global mosaic pattern.}

Definition 3.2. A set T ⊆_Z2 _{is called basic with}

respect to the template Ai,j, as given in (4), if T = {(i, j)} ∪ N_E+(i, j), where i + j is even or T ={(i, j)} ∪ N_O+(i, j), where i + j is odd. A basic

mosaic pattern is a local pattern defined on some basic set T .

Notation 3.1. Let yT be a local mosaic pattern on

a subset T ⊆Z2. For any (i, j) ∈ T , if f(xi,j) = 1

(resp. f (xi,j) = −1), then to draw a figure for yT

the output f (xi,j) of the state of the cell Ci,j will

be denoted by + (resp. −).

For any (i, j)∈Z2_E, a basic mosaic pattern yT must have one of the following forms.

(i) ++• + (ii) −+• + (iii) +−• + (iv) ++• − (v) −−• + (vi) −+• − (vii) +−• − (viii) −−• −.

(10) Here • is either + or −, the output of the state of the (center) cell Ci,j. For any (i, j) ∈ Z2O, a basic

pattern yT also has eight possible forms which are obtained by rotating the forms in (10) by 180◦.

Notation 3.2. Let V_• = {v_• ∈ _R3, v_• =

(v1, v2, v3)•, |vi| = 1, i = 1, 2, 3}. Here • = + or

−. Clearly, V•can be used to represent all eight

pos-sible forms in (10). For instance, we may identify the output of the state of the cells Ci−1,j, Ci,j+1(or

Ci,j−1), and Ci+1,j as v1, v2, and v3, respectively.

With such identification, _{+ + +}+ = (1, 1, 1)+ and +

+− − = (1, 1, −1)−. We shall use V E

• or VO• to

distinguish the position of the center cell provided the distinction is needed.

Definition 3.3. For any (i, j)∈_Z2, the total

out-put of any basic mosaic pattern whose center is at (i, j) is defined to be v1+ v2+ v3.

We are now in a position to study the local ba-sic mosaic patterns. Let (i, j) ∈ Z2, the state xi,j

of the cell Ci,j is greater than one if and only if

a1(yi−1,j+ yi,j+1+ yi+1,j) + a + z− 1 > 0 . (11a)

Similarly, xi,j <−1 if and only if

a1(yi−1,j+ yi,j+1+ yi+1,j)− a + z + 1 > 0 . (11b)

Since (yi−1,j, yi,j+1, yi+1,j) ∈ V•, inequality (3.3)

can be, respectively, simplified as follows

ka1+ a + z− 1 > 0 , (12a)

and

ka1− a + z + 1 > 0 , (12b)

where k = 3, 1,−1, −3.

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[4,3] [4,2] [2,4] [4,4] (-1,0) (1,0) [0,0] [3,3] 3 1 -1 -3 l l l [3,4] -3 l₃ [2,2] [1,4] [4,1] [4,0] [0,4] [0,4] [0,2] [2,0] [3,0] [3,1] [2,3] [1,3] [0,3] [0,1] _[1,0] [3,0] [4,0] [0,3] [1,1] [3,2] -1 1 a E r r r r B C D z

Fig. 3. B = 1/1 + 3|ε|, C = 1/1 + |ε|, D = 1/1 − |ε|, E = (1/1 − 3|ε|) 0 < |ε| < 1/3. Here [m, n]ε= [m, n], `i,|ε| = `i and ri,|ε|= ri, i∈ {3, 1, −1, −3}. The classification of parameter space with respect to the mosaic patterns.

If we treat a1, a and z as three independent

variables, the regions separated by those inequali-ties in (12) are difficult to picture. To better visu-alize the regions, we set a1 = aε. The inequalities

(12) then reduce to

a(1 + εk) + z > 1 (13a) a(−1 + εk) + z > −1, where k = 3, 1, −1, −3 .

(13b) Let rk,ε and `k,ε be straight lines whose equations

are, respectively,

a(1 + εk) + z = 1 (14a)

and

a(−1 + εk) + z = −1 . (14b) Let ` be a straight line that does not pass through the origin in a plane. Denote by `(0) the

open half-plane containing the origin, while `(×) denotes the other open half-plane.

For fixed 0 < |ε| < 1/3, the straight lines in (14) divide z− a plane into the following disjoint regions.

In Fig. 3, we see, for instance, [4, 4]ε= `3,|ε|(×) ∩ r−3,|ε|(×) ,

and

[2, 2]ε= `_−1,|ε|(×) ∩ `1,|ε|(0)∩ r−1,|ε|(0)∩ r1,|ε|(×) .

In general, for m, n∈ {1, 2, 3, 4} and a > 0, [m, n]ε= r_−2m+5,|ε|(×) ∩ r_−2m+3,|ε|(0)

∩ `2n−3,|ε|(0)∩ `2n−5,|ε|(×) . (15a)

Here, if |k| > 3, then r_k,_|ε|(·) and `_k,_|ε|(·) are inter-preted as the z−a plane P2. For m, n∈ {1, 2, 3, 4}

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and a < 0, [m, 0]ε= r2m−3,|ε|(0)∩ r2m−5,|ε|(×) ∩ `3,|ε|(0) , (15b) [0, n]ε= `5−2n,|ε|(×) ∩ `3−2n,|ε|(0) ∩ r_−3,|ε|(0) , (15c) and [0, 0]ε= `3,|ε|(0)∩ r−3,|ε|(0) . (15d)

We omit the subscript ε where the meaning is clear.

Remark. For 1/3 ≤ |ε| < 1, the a-intercept E (see

Fig. 3) of r_−3,|ε|and `3,|ε|lies either below the z-axis

or does not exist. This implies that the region [4, 4] will disappear. For |ε| ≥ 1, the regions [3, 3], [3, 4], and [4, 3] will no longer exist.

Theorem 3.1. Let 0 < |ε| < 1/3, and [m, n]ε be

the disjoint (open) regions described in (15) and shown in Fig. 3. Suppose (z, a) ∈ [m, n]ε and

aε > 0. Then any basic mosaic pattern in V+

(resp. V₋) whose total output is greater or equal to 5−2m (resp. no greater than 2n−5) exists. Suppose (z, a)∈ [m, n]ε and aε < 0. Then any basic mosaic

pattern in V+ (resp. V−) whose total output is no

greater than 2m−5 (resp. no less than 5−2n) exists.

Proof. We illustrate only for [3, 3]ε and aε > 0.

The illustration for other regions can be similarly derived. Note that [3, 3]ε is bounded by `1,|ε|, `3,|ε|,

r_−3,|ε| and r_−1,|ε|. If (z, a) ∈ r_−1,|ε|(×) ∩ r_−3,|ε|(0), a > 0 and ε > 0, then any positively saturated cells whose three neighbors are shown in (10) must have at least one positively saturated neighbor. Thus, if (z, a) ∈ r3,|ε|(0) ∩ r1,|ε|(×), a > 0 and

ε > 0, then the total output of any basic mosaic pattern in V+ is no less than −1. Moreover, if

(z, a) ∈ `_3,|ε|(0)∩ `_1,|ε|(×), a > 0 and ε > 0, then the total output of any mosaic pattern in V₋ is no greater than 1.

Remark. Suppose (z, a) ∈ [m, n], where aε > 0.

Then any basic mosaic pattern in V+ (resp. V−)

ex-ists provided that the center of the pattern must be coupled to at least (4− m)+ ’s (resp. (4 − n)− ’s). See (10). Suppose (z, a)∈ [m, n] and aε < 0. Then any basic mosaic pattern in V+(resp. V−) exists

pro-vided that the center of the pattern must be coupled to at least (4− m) −’s (resp. (4 − n) +’s).

Denote byFE([m, n]) (resp.FO([m, n]) the set

of all basic mosaic patterns that have parameter ε in [m, n] and is centered at (i, j), where (i, j)∈Z2_E (resp. Z2_O).

Corollary 3.1. Suppose aε > 0, and let• be either

+ or −. Then (i) FE([4, 4]) = n _• • + •,• − •• o ;FO([4, 4]) = n_{• + •} • ,• − •• o . (ii) FE([3, 3]) = n _• + +•, + • + •,• + +• ,− − •• ,• − •− ,• − −• o . FO([3, 3]) is the rotation of FE([3, 3]) by 180◦. (iii) FE([2, 2]) = n ₊ + +•, • + + +, + • + +,− − •− ,− − −• ,• − −− o . FO([2, 2]) is the rotation of FE([2, 2]) by 180◦. (iv) FE([1, 1]) = n ₊ + + +, − − − − o ; FO([1, 1]) = n_{+ + +} + , − − − − o . (v) F ([0, 0]) =∅.

The other cases can be constructed accordingly.

4. Global Mosaic Patterns and

Their Complexities

To see the complexities of a certain set of global mo-saic patterns, we first show that the limit in (8) is well-defined and exists provided that U is (p1, p2

)-translation invariant. We note that with the tem-plates given as in (4), any stationary solution x of (1a) is (2, 2)-translation invariant. To this end, we need the following notation.

Definition 4.1. Let N = (N1, N2) be a two-tuple

of positive integers, and let Γ(i,j)_N (U) count the num-ber of distinct patterns among the elements of U restricted to a rectangle of size N1×N2whose lower

left corner point is at (i, j). When the reasoning is general, we may omit the (i, j) and write only ΓN(U).

Definition 4.2. Let p1 and p2 be as in

Defini-tion 2.2. A standard window N is a rectangle of size N1× N2, where N1 and N2 are positive integer

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multiples of p1 and p2, respectively, whose lower

left corner point is at (p1k1, p2k2) for some integers

k1, k2.

We note that if N is any rectangle then there are standard windows Ñ and ˆN with Ñ ⊆ N ⊆ ˆN and the areas of N\ Ñ and ˆN\N are less than δ := 4(p1N2+ p1N2). It follows that

Γ_Nˆ(U) ≤ 2δΓN(U) and ΓN(U) ≤ 2δΓ_N˜(U) ,

because the cardinality ofA being two implies that there are at most 2δ ways to put symbols in the regions ˆN\N or N\ ˜N . So,

2−δΓ_Nˆ(U) ≤ ΓN(U) ≤ 2δΓ_N˜(U) . (16)

We are now ready to state one of our main re-sults in this section.

Theorem 4.1. Suppose U is (p1, p2)-translation

invariant. Then the limit in (8) is well-defined and exists. Proof. First, lim N1,N2→∞ log ΓN(U) N1N2

exists when the limit is restricted to standard win-dows N , using the methods of [Chow et al., 1996, Sec. 5]. Next, let N be any window and let ˜N and

ˆ

N be standard windows chosen to get (16). Let the sizes of Ñ and ˆN be Ñ1 × Ñ2 and ˆN1 × ˆN2,

respectively. We have ˆ N1Nˆ2 N1N2 −δ log 2 + log Γ_Nˆ(U) ˆ N1Nˆ2 ≤ log ΓN(U) N1N2 ≤ N˜1N˜2 N1N2 δ log 2 + log Γ_N˜(U) ˜ N1N˜2 . Because δ/N1N2 = 4(p1N2 + p1N2)/N1N2, the existence of lim N1,N2→∞ log ΓN(U) N1N2 follows.

We note that with the templates given as in (4), even more is true: Γ(0,0)_(2n+1,2n)(U) = Γ(1,0)_(2n+1,2n)(U) for

all positive integers n. We illustrate this by con-sidering the following two rectangles, ABCD and A0B0C0D0, of size 2× 3 whose lower left corners are A(0, 0) and A0(1, 0).

D D0 C C0

A A0 B B0

Clearly, A, D, B0 and C0 ∈_Z2

E and A0, D0, B and

C∈Z2_O. By rotating by 180◦a pattern restricted on the rectangle ABCD, we would get a pattern that can be fit in the rectangle A0B0C0D0. The converse is also true. Hence,

Γ(0,0)_(3,2)(U) = Γ(1,0)_(3,2)(U) .

Notation 4.1. We denote by M[m, n] the set of

all global mosaic patterns that have parameters (z, a)∈ [m, n].

Theorem 4.2. Let aε > 0, m, n ∈ [0, 1, 2, 3, 4],

and let α = max{m, n} and β = min{m, n}. Then (1) exhibits spatial chaos if and only if α ≥ 2 and β ≥ 2.

Proof. From Theorem 3.1, it is clear thatM[m, n]

is increasing with respect to m and n. That is if m1 ≤ m2 and n1 ≤ n2, then

M[m1, n1]⊆ M[m2, n2] .

To prove the theorem, it suffices to show that h(M[4, 1]) = h(M[1, 4]) = 0 (17) and

h(M[2, 2]) > 0 . (18)

We first prove (17). For aε > 0, (z, a) ∈ [4, 1] or [1, 4], the only two global mosaic patterns are either all +’s or all−’s. Hence h(M[4, 1]) = h(M[1, 4]) = 0. To prove (18), we consider a global mosaic pat-tern that is an alpat-ternative array of vertical stripes of width 3 that are alternating in signs. See Fig. 4. We shall assume the lower left corners of the 3× 2 rectangles that are boxed and are even. Re-stricting our observation to the 9×12 rectangle con-taining those boxed rectangles, we conclude that each of the boxed rectangles can either remain the

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-+

+

-

+

Fig. 4. Using Theorem 3.1 to find a lower bound for the spatial entropy ofM[2, 2] with aε < 0.

same or be replaced by a rectangle of the same size that is filled with opposite signs. Thus, we see in general that for (i, j)∈Z2_E

Γ(i,j)_N (M[2, 2]) ≥ 2n2, where N = (3n, 4n). Hence,

h(i,j)_N (M[2, 2]) ≥ log 2 12 .

Using Theorem 4.1, we conclude that h(M[2, 2]) ≥ log 2/12.

Theorem 4.3. Let aε < 0, m, n ∈ {1, 2, 3, 4}.

Then (1) exhibits spatial chaos if and only if m + n≥ 4.

Proof. We only illustrate that

h(M[2, 2]) > 0 , (19)

h(M[1, 3]) = h(M[3, 1]) > 0 , (20)

and

h(M[2, 1]) = h(M[1, 2]) = 0 . (21) The other possibilities can be similarly treated as in Theorem 4.2.

Let aε < 0, (z, a) ∈ M[2, 2], so any + must have among its three interacting neighbors at least two −’s and any − must have among its three in-teracting neighbors at least two +’s. Consider the boxed 3× 2 rectangles + − + − + − and − + − + − + .

As offset in the figure below, any arrangement con-sisting solely of these 3× 2 rectangles whose lower left corners are even, i.e. are in Z2_E, makes a global mosaic pattern in M[2, 2]. So, in every window of

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Fig. 5. Using Theorem 3.1 to find a lower bound for the spatial entropy ofM[2, 2] with aε < 0.

size 3n×2(n+1), there are at least 2n2 distinct pat-terns. It follows that h(M[2, 2]) ≥ log 2/6, giving (19).

Let aε < 0, (z, a) ∈ M[1, 3], so any + must have all three of its interacting neighbors as−’s and that any − must have among its three interacting neighbors at least one +. Define a 3× 3 rectangle

3 :=

− − + − + −

+ − −

.

Consider the 12× 12 rectangles

1 := 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 and 2 := − − + − − + − − + − − + − + − − + − − + − − + − + − − + − − + − − + − − − − + − + − − − + − − + − + − + − + − + − − + − + − − + − + − + − + − − − − + − + − + − + − − + − + − + − + − + − − + − + − − − + − + − − + − − − − + − − + − − + − − + − + − − + − − + − − + − + − − + − − + − − + − − .

Noting that both rectangles 1 and 2 are of the form

3 3 3 3

3 3

3 3 3 3

for some interior choices of 6× 6 rectangles, one can check that any arrangement consisting solely of rectangles 1 and 2 whose lower left corners are even, i.e. are inZ2_E, makes a global mosaic pattern inM[1, 3]. So, in some window of size 12n × 12n, there are 2n2 distinct patterns. It follows that h(M[1, 3]) ≥ log 2/144, giving (20).

As for (21), let aε < 0, (z, a) ∈ M[1, 2], so m = 1 implies that any + must have all three of its interacting neighbors as−’s, and n = 2 implies that any− must have among its three interacting neigh-bors at least two +’s. The latter implies that there

cannot be three horizontally consecutive −’s. So, the only global mosaic patterns which are possible are (a) the single “checkerboard” pattern of alter-nating +’s and −’s, and (b) those patterns which have somewhere two horizontally consecutive −’s. In case (b), one can see by tedious logical impli-cations that the pattern must have two adjacent diagonal “stripes” of all −’s surrounded by alter-nating diagonal stripes of all +’s or all−’s. So, for all windows N which are N1× N2,

ΓN(U) ≤ 2 + 2 max{2 + N1, 2 + N2} .

Inequality (21) follows.

We conclude by mentioning possible future re-lated work. First, the classification of the defect patterns is of interest. It is numerically reported in

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[Yeh, 1998] that CNNs with space-dependent tem-plate such as (4) can generate temporal chaos. The rigorous study of such phenomenon is of consider-able interest. The other dynamics properties, such as stability, traveling waves and scrolling waves of CNNs addressed in this paper have not been ad-dressed yet. Finally, in practice CNNs are imple-mented on a finite lattice. Thus, it is also desirable to study the dynamics of such CNNs on a finite lat-tice. In particular, how do the boundary conditions on finite lattice influence the dynamics properties and pattern formation of CNNs on infinite lattice (see e.g. [Shih, 2000]).

Acknowledgment

We thank Dr. C. J. Yu for providing the simulation work in the paper.

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Appendix

The purpose of the simulations is to support the assertions of Theorem 2.1(ii) and Theorem 3.1. In the following figures, all the lower-left 1× 1 boxes are assumed to be at the integer lattice (0, 0). If the output f (xi, j) of the state of a cell Ci,j is +1

(resp.−1), then we use black (resp. white) to color its associated box. Numerical tables give values of the states xi,j.

In Fig. 6, we pick ε = 1/6 and (z, a) ∈ [3, 2] region. According to Theorem 3.1, in Fig. 6(c), the outputs of the cells at (4, 4) and (4, 1) are initially white but cannot remain white. The basic mosaic patterns whose centers are at (i, j), 0 ≤ i, j ≤ 5, (i, j) 6= (4, 4), (i, j) 6= (4, 1) are initially good.

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1.5 1.5 1.5 1.5 1.5 1.5 1.5 −1.5 −10 −10 −10 1.5 1.5 −7 −5 −10 −1.5 1.5 2.5 −1.5 −8 −1.5 −5 1.5 1.5 −1.5 −2.5 −1.5 −1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 (a) 1.9 1.9 1.5 1.9 1.9 1.9 1.5 −1.3 −1.3 −1.3 1.5 1.9 1.5 −1.3 −1.7 −1.7 −1.3 1.5 1.5 −1.3 −1.7 −1.7 −1.3 1.5 1.5 −1.3 −1.3 −1.3 1.5 1.9 1.9 1.9 1.5 1.9 1.9 1.9 (b) (c) (d) Fig. 6. [3, 2] region: a = 1.2, ε = 1/6, z = 0.1. 1.5 1.5 1.5 −1.5 −1.5 −1.5 1.5 1.5 1.5 −2 −2 −2 −1.5 −1.5 −1.5 1.5 1.5 1.5 −1.5 −1.5 −1.5 1.5 1.5 1.5 1.5 1.5 1.5 −1.5 −1.5 −1.5 1.5 1.5 1.5 −1.5 −1.5 −1.5 (a) 1.167 1.5 1.167 −1.167 −1.5 −1.167 1.167 1.167 1.167 −1.167 −1.5 −1.167 −1.5 −1.167 −1.5 −1.5 −1.5 −1.5 −1.5 −1.167 −1.5 −1.5 −1.5 −1.5 1.167 1.167 1.167 −1.167 −1.5 −1.167 1.167 1.5 1.167 −1.167 −1.5 −1.167 (b) (c) (d) Fig. 7. [2, 2] region: a = 1, ε = 1/6, z = 0.

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1.5 −1.5 1.5 −1.5 1.5 −1.5 −1.5 1.5 −1.5 2.5 −1.5 1.5 1.5 −2.5 1.5 −1.5 1.5 −1.5 −2.5 1.5 −1.5 1.5 −1.5 1.5 1.5 −1.5 2.5 −1.5 1.5 −1.5 −1.5 1.5 −1.5 1.5 −1.5 1.5 (a) 1.75 2.25 2.25 2.25 2.25 2.25 −1.25 −1.75 1.75 2.25 2.25 1.25 −2.25 −2.25 −1.75 1.25 −1.75 −1.75 −2.25 −1.75 −1.75 1.25 −1.75 −2.25 1.75 1.75 2.25 2.25 1.75 −1.25 2.25 2.25 2.25 2.25 2.25 2.25 (b) (c) (d) Fig. 8. [3, 3] region: a = 1.5, ε = 1/6, z = 0. 1.5 1.5 1.5 1.5 1.5 1.5 1.5 −1.5 −10 −10 −10 1.5 1.5 −7 −5 −10 −1.5 1.5 2.5 −1.5 −8 −1.5 −5 1.5 1.5 −1.5 −2.5 −1.5 −1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 (a) 4 4 3.33 4 4 4 3.33 −1.33 −1.33 −1.33 3.33 4 3.33 −1.33 −2 −2 −1.33 3.33 3.33 −1.33 −2 −2 −1.33 3.33 3.33 −1.33 −1.33 −1.33 3.33 4 4 4 3.33 4 4 4 (b) (c) (d) Fig. 9. [4, 2] region: a = 2, ε = 1/6, z = 1.

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1.5 1.5 1.5 1.5 1.5 1.5 1.5 −1.5 −10 −10 −10 1.5 1.5 −7 −5 −10 −1.5 1.5 2.5 1.5 −8 −1.5 −5 1.5 1.5 −1.5 2.5 −1.5 −1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 (a) 5.5 5.5 4.5 5.5 4.5 5.5 4.5 −2.5 −2.5 −3.5 −1.5 4.5 4.5 −2.5 −3.5 −3.5 −2.5 4.5 5.5 4.5 −2.5 −3.5 −2.5 4.5 5.5 5.5 4.5 −2.5 −1.5 4.5 5.5 5.5 5.5 5.5 4.5 5.5 (b) (c) (d) Fig. 10. [4, 3] region: a = 2, ε = 1/6,= 2, ε = 1/6, z = 1. 1.5 −1.5 1.5 −1.5 1.5 −1.54 −1.5 1.5 −1.5 4.5 −1.5 1.5 1.5 −2.5 1.5 −1.5 1.5 1.5 −2.5 1.5 −1.5 1.5 −1.5 1.5 1.5 −1.5 2.5 −1.5 1.5 −1.5 −1.5 1.5 −1.5 1.5 −1.5 1.5 (a) 1.5 −1.5 1.5 −1.5 1.5 −1.5 −1.5 1.5 −1.5 1.5 −1.5 2.5 2.5 −1.5 1.5 −1.5 2.5 4.5 −1.5 1.5 −1.5 1.5 −1.5 1.5 1.5 −1.5 1.5 −1.5 1.5 −1.5 −1.5 1.5 −1.5 1.5 −1.5 1.5 (b) (c) (d) Fig. 11. [4, 4] region: a = 3, ε = 1/6, z = 1.

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Therefore, in Fig. 6(d) the final outputs seem to be very resonable. Note also that the evolution from the unstable pattern in Fig. 6(c) to the pattern in Fig. 6(d) involves the least number of changes to arrive at a final, stable pattern.

Indeed, we see that all figures of evolution from unstable to stable patterns, for parameters (z, a, ε) corresponding to the [3, 2], [2, 2], [3, 3], [4, 2], [4, 3] and [4, 4] regions, are consistent with the theory proved in Theorem 2.1(ii) and Theorem 3.1.

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