行政院國家科學委員會專題研究計畫 期中進度報告
推廣型類神經網路: 函數和動態(2/3)
計畫類別： 個別型計畫
計畫編號： NSC93-2115-M-009-007-
執行期間： 93 年 08 月 01 日至 95 年 07 月 31 日
執行單位： 國立交通大學應用數學系(所)
計畫主持人： 莊重
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處理方式： 本計畫可公開查詢
中 華 民 國 94 年 5 月 6 日
BOUNDARY INFLUENCE ON THE ENTROPY OF A PROBLEM IN CELLULAR NEURAL NETWORKS
YU-CHUAN CHANG AND JONQ JUANG
Abstract. Let T a two dimensional map induced from a spatial discretization of a Reaction-Difussion system. In [Afraimovich and Hsu, 2003], the following open problems were raised. Is it true that, in general, h(T ) = hD(T ) =
hN(T ) = h`(1),`(2)(T )? Here h(T ) and h`(1),`(2)(T ) (see Definitions 1.3 and
1.4) are, respectively, the spatial entropy of the system T and the spatial entropy of T with respect to the lines `(1)and `(2), and hD(T ) and hN(T ) are
spatial entropy with respect to the Dirichlet and Neuman boundary conditions. If it is not true, then which parameters of the lines `(i), i = 1, 2, are responsible
for the value of h(T ). What kind of bifurcations occurs if the lines `(i)move?
In this paper, we shed some light on this open problem for a Lozi-type map obtained from Cellular Neural Networks.
Key words: Boundary influence, Lozi-type map, dynamics of intersection, en-tropy, cellular neural networks.
1. Introduction
We consider one-dimensional Cellular Neural Networks (CNNs) of the form (e.g., [Ban et al., 2002, 2001; Hsu 2000]).
dxi
dt = −xi+ z + αf (xi−1) + af (xi) + βf (xi+1), i ∈ Z, (1.1a) where f (x) is a piecewise-linear output function defined by
f (x) = rx + 1 − r, if x ≥ 1, x, if |x| 6 1, lx + l − 1, if x ≤ −1. (1.1b)
Here r and l are positive constants. The state of a cell Ci is denoted by xi. The
quantity z is called the threshold or bias term. The constants α, a and β are the interaction weights between neighboring cells. Such triple pair [α, a, β] of the interaction weights is called the template of the system (1.1).
CNNs were first proposed by Chua and Yang [1988a, 1988b]. Their main applica-tions are in image processing and pattern recognition [Chua, 1998]. For additional
background information, applications, and theory, see [Thiran, 1997; Chua, 1998] among others.
A basic and important class of solutions of (1.1) is the steady-state solutions. Specifically, a steady-state solution x = (xi)i∈Zof (1.1) satisfies the following
equa-tion
f (xi+1) =
1
β(xi− z − αf (xi−1) − af (xi)). (1.2) Set ui= f (xi). Then (1.2) becomes
ui+1= 1 β(−αui−1− z + f −1(u i) − aui), (1.3a) or, equivalently, (ui, ui+1) = (ui, 1 β(−αui−1− z + f −1(u i) − aui)) =: T (ui−1, ui). (1.3b)
Clearly, (1.3b) defines a Lozi-type map T of the form
(si+1, ti+1) = T (si, ti) = (ti, F (ti) − bsi). (1.4a)
Here b = α β, (1.4b) and F (y) = a1y + a0− a1+ ¯a0:= a1y + ¯a1, if y ≥ 1, a0y + ¯a0, if |y| 6 1, a−1y + a−1− a0+ ¯a0:= a−1y + ¯a−1, if y ≤ −1. (1.4c) where a1=β1(1r− a) > 0, a0= β1(1 − a) < 0, a−1= 1β(1l − a) > 0, ¯a0= −zβ . (1.4d)
Any bounded trajectory (sj+1, tj+1) = T (sj, tj) corresponds to a bounded
steady-state solution of system (1.1). The following class of steady-steady-state solutions is of particular interest.
Definition 1.1. A solution x = (xi)i∈Z of equations (1.2) is called a mosaic
so-lution if |xi| > 1 for all i ∈ Z. Its associated pattern y = (yi)i∈Z = (f (xi))i∈Z is
called a mosaic pattern.
To define the stability of a steady-state solution, we consider the following lin-earized stability. Let ξ = (ξi)i∈Z ∈ `2, the linearized operator L(x) of (1.1) at a
steady-state solution x = (xi)i∈Z is given by
(L(x)ξ)i= −ξi+ αf0(xi−1)ξi−1+ af0(xi)ξi+ βf0(xi+1)ξi+1. (1.5)
Definition 1.2. Let x = (xi)i∈Z be a solution of (1.2) with |xi| 6= 1 for all i ∈ Z.
The steady-state solution 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 and Lin, 2000; Hsu, 2000], that for 1
|a| + |α| + |β| > max{r, `} ≥ 0, (1.6) where r, `, a, α and β are defined as in (1.1), −L(x) is a self-adjoint and positive operator. Therefore, if r and ` are sufficiently small, all mosaic solutions of (1.1) are stable.
If we have a finite number of n cells, which is usually the case in real applications, we will take the boundary conditions into account. The question then is how well such finite system as its size grows represents the original infinite system (1.1)? One quantity measures such resemblance is “spatial entropy”. We next define the spatial entropy of the infinite system as well as that of the finite system. The following notion of the entropy of the system (1.1) was introduced by [Mallet-Paret and Chow, 1995]. Set Γn,k(T ) (resp., ΓMn,k(T )) to be the number of elements in the
solution set Sn,k(resp., Sn,kM ), where Sn,k(resp., Sn,kM )= {{ui}n+k−1i=k : {ui}∞i=−∞is a
bounded steady-state solution (resp., stable mosaic solution) of (1.1)}. Here k ∈ Z. Since the template of system (1.1) is space invariant, the steady-state solutions of (1.1) are also space invariant. That is to say if {ui}∞i=−∞is a steady state solution of
(1.1), so is {ui+k}∞i=−∞for any k ∈ Z. Hence, Γn,k(T ) and ΓMn,k(T ) are independent
of the choice of k. Thus, we may set Γn,k(T ) = Γn(T ) and ΓMn,k(T ) = Γ M n (T ).
Definition 1.3. (1) The spatial entropy h(T ) of the system (1.1) or the map T is defined to be the limit
h(T ) = lim
n→∞
ln Γn(T )
n .
(2) The spatial mosaic entropy hM(T ) of the system (1.1) or the map T is defined to be the limit
hM(T ) = lim
n→∞
ln ΓM n (T )
n .
Inspired by the open problems raised in [Afraimovich and Hsu 2003], we are led to define the following notation of the spatial entropy for the finite system. Define the line `(m,k) as
`(m,k)= {(x, y) ∈ R2: y = mx + k}. (1.7a)
Here
`(∞,k) is interpreted as {(x, y) ∈ R2: x = k}. (1.7b)
Denote by N (n, `(m1,k1), `(m2,k2), T ) the number of points on the intersection of
Tn`
(m1,k1)∩ `(m2,k2). Should no ambiguity arise, we will write `(mi,ki)as `(i). We
next elaborate the meaning of the intersection Tn`(m1,k1)∩ `(m2,k2). If (x, y) ∈
Tn`
(m1,k1)∩ `(m2,k2), then there exists a point (u0, u1) ∈ `(m1,k1)such that (x, y) =
Tn(u
0, u1) := (un, un+1) ∈ `(m2,k2). That is to say, in a CNN of n cells, its steady
state {ui}ni=1 satisfies the following Robbin’s boundary conditions
u1= m1u0+ k1
and
un+1= m2un+ k2.
In particular, (m1, k1) = (∞, 0) and (m2, k2) = (0, 0) (resp., (m1, k1) = (m2, k2) =
(1, 0)) correspond to the Dirichlet (resp., Neumann) boundary conditions. Hence, N (n, `(m1,k1), `(m2,k2), T ) denotes the number of steady-states of CNNs of n cells
with the Robbin’s boundary conditions. Likewise, we will set NM(n, `(m1,k1), `(m2,k2), T )
as the number of stable mosaic solutions of CNNs of n cells with the Robbin’s boundary conditions.
Definition 1.4. (1) The spatial entropy (resp., mosaic entropy) of the CNNs of n cells withe Robbin’s boundary conditions is defined to be
hn,`(1),`(2)(T ) = ln N (n, `(1), `(2), T ) n ( resp., h M n,`(1),`(2)(T ) = ln NM(n, ` (1), `(2), T ) n ). 4
(2) The spatial entropy h`(1),`(2)(T ) (resp., mosaic entropy h
M
`(1),`(2)(T )) of T
with respect to lines `(1) and `(2) is defined as the limit of the entropy of
the finite system, that is,
h`(1),`(2)(T ) = limn→∞ ln N (n, `(1), `(2), T ) n (1.8a) (resp., hM` (1),`(2)(T ) = limn→∞ ln NM(n, `(1), `(2), T ) n ). (1.8b)
Notation 1.1. In the case of Dirichlet (resp., Neumann) boundary conditions, we write h`(1),`(2)(T ) = hD(T ) (resp., hN(T )) and h
M
`(1),`(2)(T ) = h
M
D(T ) (resp.,
hMN(T )).
In case that the growth rate of N (n, `(1), `(2), T ) is super exponential, h`(1),`(2)(T )
is defined to be ∞. Let T be a local holomorphic mapping, preserving the origin, and two lines `(1) and `(2) passing the origin. Suppose all the images Tn`(1) are
smooth [4] or that everything is algebraic (see [2], [3]). Then h`(1),`(2)(T ) exists and
is finite. In our case, N (n, `(1), `(2), T ) ≤ 3n, see Section 2.
Let T a two dimensional map induced from a spatial discretization of a Reaction-Difussion system. In [Afraimovich and Hsu, 2003], the following open problems were raised.
(P1): Is it true that, in general, h(T ) = hD(T ) = hN(T ) = h`(1),`(2)(T ) or
hM(T ) = hM
D(T ) = hMN(T ) = hM`(1),`(2)(T )?
(P2): If it is not true, then which parameters miand ki, i = 1, 2, are
respon-sible for the values of h(T ) or hM(T ). What kind of bifurcations occurs if the lines `(m,b) move?
The purpose of this paper is to shed some light on those two problems for T being given as in (1.4). Specifically, under some mild conditions, we show that for any `(1)and n ∈ N, except possibly a few pieces of Tn`(1), Tn`(1)is contained in an
N -shaped tube for which its boundary points are ω-limit points of `(1)for T .
More-over, we show under a stronger condition, see (3.4), that the entropy h`(1),`(2)(T )
of T with respect to `(1) and `(2) is independent of the choice of `(1). It is also
shown that hD(T ) = hN(T ) = ln 3, and that h`(1),`(2)(T )(= h`(2)(T )) takes on two
distinct values ln 3 and 0. The necessary and sufficient conditions on `(2) for which
h`(2)(T ) = ln 3 are also obtained. Those results are recorded in Section 3. Similar
results for the spatial mosaic entropy are given in Section 4. In Section 2, we study the dynamics of a certain two-dimensional map induced from Tn`
(1). We conclude 5
this introductory section by mentioning some related work. Shih [2000] studied the influence of periodic, Neumann and Dirichlet boundary conditions on a problem arising in two dimensional CNNs. Since their output function f , as given in (1.1b), is flat at infinity, i.e., r = l = 0, the formulation of the problem is much different from those in [Afraimovich and Hsu 2003]. Consequently, the techniques used in both situations are also quite different. We also remark that the problem of the asymptotic behavior of the number of points on the intersection fkL
1∩ L2, where
L1, L2 are submanifolds of a smooth manifold, and f is a smooth map, is said
to be a problem of dynamics of the intersection. These problems arise in various branches of analysis. There are some general results (see, e.g., p.261 of [Arnold 1993]) obtained for such problems. However, no approaches are available to solve specific problems.
2. Dynamics of Certain Maps Induced From Tn` (m,k)
We begin with the calculation of Tn`
(m,k). Now, for m 6= 0,
T (x0, mx0+ k) = (mx0+ k, F (mx0 + k) − bx0). Set x = mx0+ k, y = F (mx0+ k) − bx0, we see immediately that
y = F (x) −b(x − k) m = (a1−mb)x + (¯a1+bkm), if x ≥ 1, (a0−mb)x + (¯a0+bkm), if |x| 6 1, (a−1−mb)x + (¯a−1+ bk m), if x ≤ −1. (2.1)
From (2.1), we see immediately that T consists of three dynamics. Each dynam-ics acts on the following regions:
R1= {(x, y) : x ≥ 1}, R0= {(x, y) : |x| ≤ 1} and R−1= {(x, y) : x ≤ −1}. (2.2)
The dynamics on regions Ri, i = 1, 0, −1, are to be termed the ith-dynamics,
respectively. Given a straight line/ line segment/ half-line `, the image of ` consists of at most two half-lines/ line segments `1 and `−1 and one line segment `0. That
is, T ` = `1, if (x, y) ∈ ` and (x, y) ∈ R1, `0, if (x, y) ∈ ` and (x, y) ∈ R0, `−1, if (x, y) ∈ ` and (x, y) ∈ R−1. (2.3a) 6
Note that some of `1, `0, or `−1 could be empty. We then define the notations
`i1,i2,··· ,in−1, ij ∈ {−1, 0, 1}, j = 1, 2, · · · , n − 1, inductively as the followings.
T `i1,i2,··· ,in−1= `i1,i2,··· ,in−1,1, if (x, y) ∈ `i1,i2,··· ,in−1 and (x, y) ∈ R1, `i1,i2,··· ,in−1,0, if (x, y) ∈ `i1,i2,··· ,in−1 and (x, y) ∈ R0, `i1,i2,··· ,in−1,−1, if (x, y) ∈ `i1,i2,··· ,in−1 and (x, y) ∈ R−1. (2.3b) Therefore, Tn`
(m,k) = Tn` has at most 3n pieces of half-lines and segments.
Using the above notation, we see that Tn` = {`i1,i2,··· ,in : uj ∈ {−1, 0, 1}, j =
1, 2, · · · , n}. To understand Tn`, it is natural to first consider the cases that
i1 = i2 = · · · = in. That is the cases that ` has been applied by same
dynam-ics repeatedly. To this end, we define the following two dimensional maps of the form,
Gi(x, y) = (ai−
b x, ¯ai+
b
xy) =: (gi,1(x), gi,2(x, y)). (2.4) We call gi,1(x), i = 1, 0, −1, the slope maps of T . Since gi,1(x), i = 1, 0, −1,
denote, respectively, the slopes of `i. Here ` = `(x,y). Moreover, gi,2(x, y) are to
be termed the intercept maps. Because if we let `(x,y) = `, then gi,2(x, y) denote,
respectively, i = −1, 0, 1, the y-intercepts of `i. We next consider the dynamics of
the slope and intercept maps gi,1and gi,2.
Proposition 2.1. Let b > 0, ai > 2
√
b, i = 1, −1 and −a0 > 2
√
b. Then (i) for i = 1, −1, m±i,∞:= ai±
√
a2 i−4b
2 are two fixed points of the slope maps gi,1. For i = 0,
m−0,∞:= a0+ √ a2 0−4b 2 and m + 0,∞:= a0− √ a2 0−4b
2 are two fixed points of the slope maps
g0,1 (ii) Moreover, the attracting interval of m+i,∞, i = 1, 0, −1, is R − {m − i,∞}.
That is to say if x ∈ R − {m−i,∞}, then, for i = 1, 0, −1, lim
n→∞g n i,1(x) = m + i,∞. (iii) Suppose ai= 2 √
b. Then m+i,∞= m−i,∞ is the globally attracting fixed point of gi,1,
i = 1, 0, −1. (vi) If x /∈ (m−i,∞, m+i,∞), i = 1, −1, (resp., x /∈ (m+0,∞, m−0,∞)), then
gi,1(x), i = 1, 0, −1, converges to m+i,∞uniformly. That is, given ε > 0, there exists
an Nε, independent of x, such that |gi,1n (x) − m +
i,∞| < ε where n ≥ Nε.
Proof. We illustrate only i = 1. Clearly, two fixed points of g1,1 are m±1,∞. The
attracting interval of g1,1 can be easily concluded by using graphical analysis on
Figure 2.1. To prove (vi), let x = a1. For ε > 0, then there exists an N such
that |gn
1,1(a1) − m+1,∞| < ε whenever n ≥ N . Let x ∈ (m +
1,∞, a1), clearly, for ε > 0, 7
Figure 2.1 |gn
1,1(x)−m +
1,∞| < |gn1,1(a1)−m+1,∞| < ε whenever n ≥ N . Now for x ∈ (−∞, m − 1,∞),
we see that g1,13 ∈ (m +
1,∞, a1). Thus, the assertion of Proposition 2.1-(vi) for i = 1
holds by choosing Nε= N + 3. The other part of the proof is similar and is thus
omitted.
Remark 2.1. Given `(m,k)= `, we see from Figure 2.1, that the slopes of `i1,i2,··· ,in
remain positive (resp., negative) for all n ≥ 3. Here i1, i2, · · · , in ∈ {−1, 1} (resp.,
∈ {0}).
Proposition 2.2. Suppose
b > 0, ai > 1 + b, i = 1, −1 and − a0> 1 + b. (2.5)
For fixed x = m+i,∞, i = 1, 0, −1, then ki,∞:= m+i,∞¯ai
m+ i,∞−b
is a globally attracting fixed point of the intercept maps gi,2(m+i,∞, y).
Proof. It suffices to show that 0 < b
m+i,∞ < 1, i = 1, −1, and −1 < b m+0,∞ < 0.
We illustrate only i = 1. Now, 0 < b m+1,∞ = 2b a1+pa21− 4b =a1−pa 2 1− 4b 2 < 1. (2.6) The last inequality is justified by the fact that a1> 1 + b ≥ 2
√
b > 0.
Theorem 2.1. Suppose (2.5) holds. (i) The two dimensional map Gi, as defined
in (2.4), i = 1, 0, −1, have two fixed points (m±i,∞, m
± i,∞¯ai
m±i,∞−b) =: A ±
i . (ii) Moreover,
the attracting regions of A±i , i = 1, 0, −1, are R2− {(x, y) : x = m−
i,∞}. That is to
say, for any (m, k) ∈ R2− {(x, y) : x = m−i,∞}, i = 1, 0, −1, lim
n→∞G n
i(m, k) = A + i .
Proof. We only illustrate i = 1. The cases for i = 0, −1 are similar. Define gn
1,1(m) = m1,n and Gn1(m, k) = (m1,n, k1,n). If m 6= m−1,∞, then given ε > 0, there
exists an Nε∈ N such that for every n ≥ Nε, we have
m+1,∞− ε < m1,n< m+1,∞+ ε. (2.7)
It follows from (2.7) that for any k ∈ R, and n sufficiently large, min{¯a1+ bk m+1,∞− ε, ¯a1+ bk m+1,∞+ ε} < ¯a1+ bk m1,n < max{¯a1+ bk m+1,∞− ε, ¯a1+ bk m+1,∞+ ε}. (2.8) It follows from (2.6) and Proposition 2.2 that for all sufficiently small ε > 0, lim
n→∞g n 1,2(m
+
1,∞± ε, k) exist and that
lim n→∞g n 1,2(m + 1,∞± ε, k) = ¯ a1(m+1,∞± ε) m+1,∞± ε − b =: k1±ε. Using (2.8), we see inductively that
min{g1,2n (m+1,∞+ ε, k), g1,2n (m+1,∞− ε, k)} < gn 1,2(m1,n, k) < max{gn1,2(m + 1,∞+ ε, k), g n 1,2(m + 1,∞− ε, k)}.
However, it is easy to see that the single limits lim
n→∞g n 1,2(m + 1,∞ ± ε, k) and lim ε→0g n 1,2(m +
1,∞ ± ε, k) exist and the convergence of limn→∞g n 1,2(m
+
1,∞± ε, k) is
uni-form for all sufficiently small ε > 0. So the double limit and both iterated limits of gn
1,2(m +
1,∞±ε, k) exist and all three limits are equal. However, limε→0n→∞lim g n 1,2(m
+ 1,∞±
ε, k) = lim
ε→0k1±ε= k1,∞. Taking the double limit on (2.8), we see that the double
limit of gn 1,2(m
+
1,n, k) exists and equals to k1,∞. It is then easy to see that, for
(m, k) ∈ R2− {(x, y) : x = m− i,∞}, limn→∞G n 1(m, k) = (m + 1,∞, m+1,∞¯a1 m+1,∞− b). We thus complete the proof of theorem. It then follows from Theorem 2.1 that for any ` = `(m,k), m 6= m−i,∞, i = 1, 0, −1,
then `i,i,··· ,i,···, i.e., applying the ith-dynamics on ` infinitely many times, are
well-defined. The resulting images are denoted by `i∞, where, for i = 1, 0, −1,
`i∞ = `(m0,k0), (m
0, k0) = A+
i . (2.9)
Figure 3.1
3. Boundary Influence on the Spatial Entropy
The following lemma is very useful in determining how the order of the line segments and half-lines `i1,i2,··· ,in ij ∈ {−1, 0, 1}, j = 1, 2, · · · , n, is. The proof is
trivial and, thus, skipped.
Lemma 3.1. Let b > 0. For fixed y, if x1≥ x2, then the y-coordinate of T (x1, y)
is no greater than that of T (x2, y).
Remark 3.1. Since our objective here is to study how the number of points in the intersection Tn`
(m1,mk)∩`(m2,k2)grows as n increase, we may assume from here on
by Remark 2.1 that the slopes of `1 and `−1 are positive and that of `0 is negative.
Using Remark 3.1, lemma 3.1 and the fact that T is one-to-one, we have the following principle. See Figure 3.1 for one special case.
Proposition 3.1. Suppose (2.5) holds. Let ` and k be lines or line segments or half lines, and ` ∩ k = ∅. If k is to the right of `. Then so are kito `i, for i = 1, −1.
However, `0 is to the right of k0. Here ki, `i, i = 1, 0, −1 are defined in (2.3a).
Note that the reverse of the ordering in k0 and `0 is due to the fact that, in
R0, F (y) has a negative slope. We next give the ordering of `i1,i2,··· ,in in terms of
their locations to each others. Moreover, we will show that all `i1,i2,··· ,in, except
6 possibly line segments/ half lines, lie in an N -shaped tube whose boundaries are given in Figure 3.2, where `i∞, i = 1, 0, −1, are defined in (2.9) and `i∞,j,
j = 1, 0, −1, means the j-th dynamics is applied to `i∞.
Figure 3.2
Proposition 3.2. Suppose (2.5) holds. Then the following holds true.
(1) Let {im}nm=1 and {jm}nm=1 be two distinct finite sequences, im and jm ∈
{1, 0, −1}, m = 1, 2, · · · , n. Suppose k is the first index such that i` = j`
for all ` ≥ k. Then `i1,i2,··· ,in is to the right of `j1,j2,··· ,jn provided that the
following hold.
(a) ik= 1 or −1, and ik−1> jk−1.
(b) ik= 0 and ik−1< jk−1.
(2) For any straight ` and n ∈ N, `i1,i2,··· ,in, ij ∈ {−1, 0, 1}, j = 1, 2, · · · , n,
lie in the N -shaped tube, see Figure 3.2, except possibly for those `1,··· ,1,in,
`−1,··· ,−1,in, in = 1, 0, −1. Here the boundaries of the N -shaped tube are
`1∞, `1∞,0, `1∞,−1, `−1∞, `−1∞,0 and `−1∞,1.
Proof. The assertions for (1) follows directly from Proposition 3.1. To see (2), let `i1,i2,··· ,in−1,1 6= `1,1,··· ,1. Then there exists 1 ≤ j ≤ n − 1 such that ij 6= 1 and
so `i1,i2,··· ,ij−1,ij ∈ R−1 or R0. Thus, `i1,i2,··· ,ij is to the left of `1∞. Therefore, it
follows from Proposition 3.1 that `i1,i2,··· ,ij,ij+1,··· ,in−1,1is to the left of `1∞,1,··· ,1=
`1∞. Hence, all `i1,i2,··· ,in−1,1, ij ∈ {1, 0, −1}, 1 6 j 6 n − 1, are to the left of `1∞
except possibly `1,1,··· ,1. The proof for the other parts of (2) is similar.
We note that the boundary points of the N -shaped tube are ω-limit points ω(`1; T ) of `1 for T . That is, if B ∈ ω(`1; T ), then there exists an A ∈ `1, and a
sequence {nk}∞k=1, nk∈ N, such that Tnk(A) → B as k → ∞. 11
To ensure that each `i1,i2,··· ,in is nonempty, we need the following lemma.
Lemma 3.2. Let
min{a1, a−1} ≥ −a0> 1 + 2b, and ¯a0 is sufficiently small. (3.1)
Then the y-coordinate (`−1∞,0∩ `−1∞,1)y of (`−1∞,0∩ `−1∞,1) is less than -1, and
(`1∞,−1∩ `1∞,0)y > 1.
Proof. We illustrate only (`1∞,−1∩ `1∞,0)y > 1. The other assertion is similarly
obtained. Note that the equation of the line `1∞ is y = m
+ 1,∞x + k1,∞. Letting y = −1, we have that x = −k1,∞−1 m+1,∞ . Clearly, (`1∞,−1∩ `1∞,0)y= the y-coordinate of T (−k1,∞− 1 m+1,∞ , −1) = −a0+ ¯a0+ b(k1,∞+ 1) m+1,∞ =: t. (3.2) Now, taking ¯a0= 0, we have
−k1,∞− 1 m+1,∞ < −k1,∞ m+1,∞ = a1− a0 m+1,∞− b ≤ 2a1 m+1,∞− b ≤ 2(1 + 2b) 1 +√1 + 4b2 ≤ 2. (3.3)
The fact that 2a1
m+1,∞−bis decreasing in a1has been used to justify the above
inequal-ities. Thus, for ¯a0= 0, we have t ≥ −a0− 2b > 1. We just completed the proof the
lemma.
Remark 3.2. Since our objective is to study the spatial entropy of the system, without loss of generality, we may assume, via Proposition 3.2 and Lemma 3.2, that all `i1,i2,··· ,in, ij ∈ {1, 0, −1}, 1 ≤ j ≤ n, are nonempty provided that (3.1)
holds.
We next give stronger conditions on ai, i = 1, 0, −1 so as to ensure that for any
j ∈ N, `i1,i2,··· ,ij,··· ,in+j, where ij= · · · = ij+n= 1 or ij= · · · = ij+n= −1 become
unbounded as n grows larger for any j. Lemma 3.3. Suppose
1
2min{a1, a−1} ≥ −a0≥ 3 + 2b and that ¯a0 is sufficiently small. (3.4) Let A , see Figure 3.2, be any point in the line segment for which its both end-points are `−1∞∩`−1∞,0and `1∞,−1∩`1∞,0(resp., `1∞∩`1∞,0and `−1∞,0∩`−1∞,1).
Then the limit of both coordinates of Tn(A) approaches to +∞ (resp., −∞).
Proof. We first note that T has a fixed point B = (a1−a0−¯a0
a1−1−b ,
a1−a0−¯a0
a1−1−b ) in R
1 for
which its stable (resp., unstable) direction is (1,a1−
√ a2 1−4b 2 ). (resp., (1, a1+ √ a2 1−4b 2 )).
Since (`−1∞∩ `−1∞,0)y> (`1∞,−1∩ `1∞,0)y> 1, via lemma 3.2, it suffices to show
that Tn(`1∞,−1∩ `1∞,0) → (+∞, +∞) as n → ∞. To this end, we need to show
that T (`1∞,−1∩ `1∞,0) = T (−1, t), t as given in (3.2), lies on the upper half of the
stable line (y −a1− a0− ¯a0 a1− 1 − b ) = m−1,∞(x − a1− a0− ¯a0 a1− 1 − b ), or, equivalently, F (t) + b −a1− a0− ¯a0 a1− 1 − b − m−1,∞t + m−1,∞a1− a0− ¯a0 a1− 1 − b =: h(¯a0) > 0. Now, h(0) = a1t + a0− a1− a1− a0 a1− 1 − b + b − m−1,∞t + m−1,∞ a1− a0 a1− 1 − b . (3.5) We also have that with ¯a0= 0,
b − m−1,∞t = b − 2b a1+pa21+ 4b t ≥ b − 2b a1+pa21+ 4b (−a0) ≥ b − b(−a0) a1 > 0, (3.6) and a1− a0 a1− 1 − b ≤ a1− a0. (3.7)
It then follows from (3.3), (3.5), (3.6) and (3.7) that
h(0) > a1(−a0− 2b) + 2(a0− a1) = a1(−a0− 2b − 2) + 2a0≥ a1+ 2a0≥ 0
We thus complete the proof of the lemma. The first main results of the paper are stated in the following.
Theorem 3.1. Let (3.4) hold. (1) If a1> a−1, (resp., a1< a−1), let `(2) be a line
satisfying the following (i) (`(2)∩ `(∞,1))y ≤ (`1∞∩ `1∞,0)y(resp., (`(2)∩ `(∞,−1))y≥
(`−1∞ ∩ `−1∞,0)y. (ii) m + −1,∞ ≤ m ≤ m + 1,∞ (resp., m + 1,∞ ≤ m ≤ m + −1,∞), where
m is the slope of `(2), then h`(1),`(2)(T ) = 0; otherwise, h`(1),`(2)(T ) = ln 3. (2) If
a1= a−1, let `(2)be a line with the slope m = a1and the y-intercept ¯y of k satisfying
¯
y ≥ k−1,∞ or ¯y ≤ k1,∞, then h`(1),`(2)(T ) = 0; otherwise, h`(1),`(2)(T ) = ln 3. (3)
h`(1),`(2)(T ) is independent of the choice of `(1).
Proof. Let a1> a−1. We will break down `(2)= k into three cases.
(a) (`−1∞,0∩ `−1∞,1)y> (k ∩ `(∞,1))y> (`1∞∩ `1∞,0)y. See Figure 3.3.
(b) (k ∩ `(∞,1))y ≥ (`−1∞,0∩ `−1∞,1)y. (c) (k ∩ `(∞,1))y ≤ (`1∞∩ `1∞,0)y and m > m + 1,∞ or m < m + −1,∞. 13
Figure 3.3
To prove case (a), we note, via Proposition 2.1-(iii) and Proposition 3.1, that for N sufficiently large, `0,i2,··· ,iN, where i2= i3= · · · = iN = 1, we have that
(k ∩ `(∞,1))y> (`0,i2,··· ,iN ∩ `(∞,1))y for any `. (3.8)
In particular, the natural number N is independent of the choice of `. We also note that `0 has a negative slope. Therefore, for any n ≥ N , by identifying
` = `j1,j2,··· ,jn−N, we see that
`j1,j2,··· ,jn−N +1,··· ,jn, where jk∈ {1, 0, −1}, for 1 ≤ k ≤ n − N,
jn−N +1= 0 and jn−N +2= · · · = jn = 1, satisfying (3.8). (3.9)
Hence, k must intersects with `j1,j2,··· ,jn−1,0, where j1, · · · , jn−1are given as in (3.9),
see Figure 3.3. Hence, for any n ≥ N , the number N (n, `(1), k, T ) of intersections
of Tn`
(1)∩ k satisfies
3n−N ≤ N (n, `(1), k, T ) ≤ 3n.
Thus, h`(1),`(2)(T ) = ln 3. If a1 > a−1 and case (b) or case (c) holds, then k must
intersect with `−1∞and `1∞,−1or `−1∞,0and `1∞,0or `−1∞,1and `1∞. Upon using
Lemma 3.3, we conclude, in any of three cases, that h`1,`2(T ) = ln 3. The proof for
a1< a−1 is similar and thus omitted. We have just completed the proof of the first
part of the theorem. The second part of the theorem is obvious and thus omitted. The last part of the theorem is a direct consequence of the first and the second assertions of the theorem.
4. Stable Mosaic Solutions and Their Corresponding Boundary Influence
In this section, we will show that hM(T ) = hM
D(T ) = h M
N(T ) = ln 2 in some
reasonable parameters range. To this end, we first give a sufficient condition on ai, b, i = 1, 0, −1 so that the mosaic solutions are stable. In real applications, r
and l are “small” positive constants. To have a1, a−1, −a0 and b are all positive,
we shall assume that β > 0, a > 1, and α > 0. Lemma 4.1. If
ai> 1 + b, i = 1, −1, (4.1)
then the stability condition (1.6) for the mosaic solutions of (1.1) is satisfied. Proof. Dividing β on both side of (1.6) yields that
min{ 1 βr, 1 β`} > a β + α β + 1. Using (1.4d), we get ai+ a β > a β + b + 1, i = 1, −1.
Thus, the assertion of the lemma holds as asserted. To show that hM
N(T ) = ln 2, we need the following lemma.
Lemma 4.2. Suppose
−a0> 2 + 2b, min{a1, a−1} > 4 + b and ¯a0 is sufficiently small. (4.2)
Then T (−1, t) = T (`1∞,−1∩ `1∞,0) (resp., T (`−1∞,0∩ `−1∞,1)) stays above (resp.,
below) the line y = x, where t is given as in (3.2).
Proof. We only illustrate the proof of the first assertion of the lemma. To this end, we first note that (4.2) implies (3.1). We then need to show that
F (t) + b > t. (4.3) Letting ¯a0= 0, (4.3) becomes
(a1− 1)t + a0− a1+ b := L > 0. (4.4)
Using (3.3), we see, via (4.2) that
L ≥ (a1− 1)(−a0− 2b) + a0− a1+ b = (−a0− 1 − 2b)(a1− 2) − 2 − b > 0.
We just complete the proof of the theorem.
Figure 4.1. Here we denote by K1 = T (K). We use similar
notations to denote points under the first iteration of T . Let S be a square defined as
S = {(x, y) ∈ R2: |x| ≤ p, |y| ≤ p}, where p > 1. Then T (S) ∩ S = S−1∪ S0∪ S1. See Figure 4.1.
Inductively, we see that Tn(S) ∩ S consists of 3n nested pieces of S
i1,i2,··· ,in,
ij = 1, 0, −1, j = 1, 2, · · · , n. Likewise, backward iterations: T−n(S) ∩ S will
produce 3nnested pieces of ¯S
i1,i2,··· ,in, ij= 1, 0, −1, j = 1, 2, · · · , n with each piece
¯
Si1,i2,··· ,in crossing the east and west side of the rectangle S. Let A be a point
in R2. T (A) is denoted by A1. Let K = (p, p), ¯L = (p, −1), ¯N = (−p, 1) and
M = (−p, −p). If
K1 and ¯L1 stay above y = p, (4.5a)
and
¯
N1 and M1 stay below y = −p, (4.5b) 16
then each of Si1,i2,··· ,in is nonempty. The following lemma gives a sufficient
condi-tion on the parameters ai, i = 1, 0, −1 and b so that (4.5) holds.
Lemma 4.3. Suppose
(−a0−1−b)(min{a1, a−1}−2(1+b)) > 2(1+b)2 and ¯a0 is sufficiently small . (4.6)
Then there exists a p > 1 such that the following holds.
F (p) − bp > p, (4.7a) F (1) + bp < −p, (4.7b) F (−1) − bp > p, (4.7c) and F (−p) + bp < −p. (4.7d) .
Proof. Equations (4.7) are equivalent to min{−a0+ ¯a0 1 + b , −a0− ¯a0 1 + b } > p > max{ a1− a0− ¯a0 a1− 1 − b ,a−1− a0+ ¯a0 a−1− 1 − b }. (4.8) Letting ¯a0= 0, (4.8) reduces to −a0 1 + b > p > max{ a1− a0 a1− 1 − b , a−1− a0 a−1− 1 − b }. (4.9) Clearly, − a0 1+b > 1. Thus, if −a0 1 + b > max{ a1− a0 a1− 1 − b , a−1− a0 a−1− 1 − b }, (4.10) then there exists a p > 1 such that (4.9) holds. However, (4.6) implies (4.10). The proof of the lemma is thus complete. Remark 4.1. Note that T−1(x, y) = (1bF (x) − yb, x). Replacing ai, i = 1, 0, −1,
by ai b, b by 1 b and ¯a0 by ¯ a0
b, we see (4.6) is invariant. That is (− a0 b − 1 − b)(min{a1 b a−1 b } − 2(1 + 1 b)) > 2(1 + 1 b) 2 and ¯a0
b is sufficiently small if and only
if (4.6) holds. Thus, (4.6) is not only to ensure that each of Si1,i2,··· ,ın is nonempty
but also that each of ¯Si1,i2,··· ,ın is nonempty.
To show that T has a Smale-Horseshoe, we need to show that each of Si1,i2,··· ,in,
and ¯Si1,i2,··· ,in shrinks to a line segment as n → ∞. To this end, we first need
the following notations. Let the slope and intercept pair of two straight lines ` and ¯` are (m, k) and (m, ¯k), respectively. Here k 6= ¯k. Let the slope and intercept
pair of `i1,i2,··· ,in and ¯`i1,i2,··· ,in, i1= i2 = · · · = in ∈ {1, 0, −1}, are, respectively,
(mn,i, kn,i) and (mn,i, ¯kn,i). Define
d0,i= |k − ¯k|, i = 1, 0, −1, (4.11a)
and
dn,i= |kn,i− ¯kn,i|, i = 1, 0, −1. (4.11b)
Lemma 4.4. Let m > m+i,∞, i = 1, −1, and m < −m+0,∞, respectively. Suppose ai ≥ 1 + 2b, i = 1, −1, and − a0≥ 1 + 2b, respectively. (4.12)
Then, respectively, dn+1,i≤ 12dn,i, i = 1, 0, −1, for all n ∈ N. Moreover, if −∞ <
m < m+i,∞, i = 1, −1, and m−0,∞ < m < ∞, then dn+1,i ≤ 12dn,i for i = 1, 0, −1,
respectively, for all n ≥ 3.
Proof. We illustrate only the case i = 1. Using (2.4), we see that dn+1,1= b mn,1 dn,1≤ b m+1,∞dn,1
The inequality above is justify by the fact that mn,1 > m+1,∞ for all n ∈ N, see
Figure 2.1. However, b m+1,∞ = 2b a1+pa21− 4b ≤ 2b 1 + 2b +√1 + 4b2 = 2 1 b+ 2 + q 1 b2 + 4 ≤ 1 2, we thus complete the proof of the first part the lemma. Upon using Figure 2.1, we see immediately that the remaining assertions of the lemma holds true. Remark 4.2. In the case of T−1, if one reverses the role of x and y in the slope and intercept pair (m, k), then the assertions of Lemma 4.4 hold provided that (4.12) is replaced by a1 b ≥ 1 + 2 b or equivalentlly ai≥ 2 + b, i = 1, −1 and − a0≥ 2 + b. Thus, if min{a1, a−1, a0} ≥ max{1 + 2b, 2 + b}, (4.13)
then the size of each of Si1,i2,··· ,in or ¯Si1,i2,··· ,in shrinks by a factor no greater than
1
2 as one applies the i-th dynamics on them.
We are now in the position to state the second main results of the paper.
Theorem 4.1. (i) Suppose (3.1) holds. Then hMD(T ) = ln 2. (ii) Suppose (4.2) holds. Then hM
N(T ) = ln 2. (iii) Suppose
−a0> 2(1 + b) and min{a1, a−1} > 4(1 + b). (4.14)
Then hM(T ) = hM
N(T ) = h M
D(T ) = ln 2.
Proof. Suppose (3.1) holds. Then the mosaic solutions under consideration are all stable. The first assertion of the theorem follows from Lemma 3.2. Suppose (4.2) holds. Let Γn = the number of intersections points of `i1,i2,··· ,in, ij ∈ {1, −1},
1 ≤ j ≤ n, and the line y = x. We see, via Lemma 4.2, that 2n− 4 ≤ Γn ≤ 2n.
Thus, hM
N(T ) = ln 2. To prove (iii), we first note that if (4.14) holds, then (4.2),
and (4.6) and (4.13) are satisfied. Applying Lemmas 4.3 and 4.4, we see that
∞
\
n=−∞
Tn(S) ∩ S =: Λ is a cantor set of infinite points. Let Λ2= Λ ∩ {(x, y) ∈ R2:
|x|, |y| > 1}, and Σ2 be the space the two sided sequences of 1’s and −1’s. Define
the itinerary map i : Λ2→ Σ2
i(P ) = (· · · s−2s−1s0s1s2· · · ),
where
p ∈ Λ2 and sj= k if and only if Tj(P ) ∈ Sk.
Impose a metric on Σ2 by defining
d[(si)∞i=−∞, (ti)∞i=−∞] = ∞ X i=−∞ |si− ti| 2|i| .
Define the shift map σ by
σ((si)∞i=−∞) = (ti)∞i=−∞, where ti= si+1.
It is then not difficult to show, see e.g., [Devaney, [10]] and [Robinson, [14]], that the dynamics of T on the invariant set Λ2is conjugate to the shift map on Σ2. Note
that any trajectory of T in Λ2 is a bounded, stable mosaic steady state of (1.1).
We just complete the proof of the theorem. We conclude this paper with the following remarks.
(1) If (3.4) holds, then (4.14) is also satisfied. Consequently, all assertions in Theorem 3.1 hold true for the spatial mosaic entropy hM
`1,`2(T ).
(2) In the language of CNNs, condition (4.14) means that the slopes r and l of the output function f are chosen to be small and so is the bias term z. The self-interaction weight a has to be relatively strong.
(3) It is also of interest to see if our techniques developed here can be applied to the cases when F (y) is a cubic polynomial, such as those in p.163 of Afraimovich and Hsu [2003] or a quadratic map for which the resulting T is a Henon map.
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Department of Applied Mathematics, National Chiao Tung University, Hsin Chu, Taiwan, R.O.C.
