INFLUENCE OF BOUNDARY CONDITIONS ON PATTERN
FORMATION AND SPATIAL CHAOS IN LATTICE SYSTEMS∗
CHIH-WEN SHIH†
Abstract. This work elucidates the spatial structure of lattice dynamical systems, which is
represented by equilibria of the systems.On a finite lattice, various boundary conditions are imposed. The effect from these boundary conditions on formation of pattern as well as spatial complexity, as the lattice size tends to infinity, is investigated.Two general propositions are proposed as a criteria to demonstrate that this effect is negligible.To illustrate the effectiveness of these criteria, the mosaic patterns in a cellular neural network model on one- and two-dimensional lattices are also studied. On a one-dimensional lattice, the influence of boundary conditions on pattern formation and spatial chaos for mosaic patterns is negligible.This result is justified by verifying the above-mentioned criteria and by using the transition matrices.These appropriately formulated matrices generate all the mosaic patterns on a one-dimensional infinite lattice and on any one-dimensional finite lattice with boundary conditions.On a two-dimensional lattice, two illustrative examples demonstrate that the boundary effect can be dominant.The results and analysis in this investigation have significant implications for circuit design in cellular neural networks.
Key words. lattice dynamical system, pattern formation, spatial chaos, cellular neural network AMS subject classifications. 34C35, 94C99
PII. S0036139998340650
1. Introduction. The dynamics of spatially extended systems have received considerable attention in recent years [1], [3], [4], [19]. These systems are modeled as partial differential equations or lattice dynamical systems. In the lattice models, while considering the problems on an infinite lattice is more convenient, a more practical consideration would be those problems on a large but finite domain. For a system on a finite lattice, boundary conditions must be imposed and realized. In the investigation of spatial complexity for the lattice systems, it is interesting to see the effect from various boundary conditions. Correspondingly, the following question arises: does the temporal-spatial structure on an infinite lattice differ from that on a large but finite lattice? A more fundamental question is
Q1 : h = hN = hP = hD ?
Here, h = h(U) denotes the spatial entropy of U, the set (or a significant subclass) of stationary solutions (patterns) on an infinite lattice; and hN, hP, and hD represent the
spatial entropy for the same class of solutions (patterns) which satisfy Neumann, pe-riodic, and Dirichlet boundary conditions, respectively. These notations are precisely defined in section 2. Q1 was posed by Afraimovich [2]. Let us provide a preliminary thought to this problem. In a lattice dynamical system, stationary solutions on a large lattice or infinite lattice can frequently be constructed in the following manner, cf. [29], [37]. First, one analyzes the existence of stationary solutions on a small lat-tice. The translation-invariant property (as explained in section 2) of the system is then used to attach the solutions that exist on small lattice compatibly and construct ∗Received by the editors June 22, 1998; accepted for publication (in revised form) November 26, 1999; published electronically July 19, 2000.This research was supported in part by the National Science Council of Taiwan, Republic of China.
http://www.siam.org/journals/siap/61-1/34065.html
†Department of Applied Mathematics, National Chiao-Tung University, 1001 Ta-Hsueh Road, Hsinchu, Taiwan, Republic of China (cwshih@math.nctu.edu.tw).
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stationary solutions on a large lattice or infinite lattice. Along this consideration, if the existence of the stationary state (solution) at a site of a lattice depends heavily on the states at its neighboring sites, then the total number of stationary solutions on the infinite lattice is expected to be lower. In this case, the entropy h is likely to be zero. However, owing to this heavy dependence, boundary effect is also strong and the number of stationary solutions satisfying the boundary condition is restrained. And then h = hB = 0, B = N, P , D. On the other hand, consider a situation in which the
existence of the state at a site depends weakly on the states at its neighboring sites. Under this circumstance, although more stationary solutions are likely available, the influence on the pattern forming from boundary conditions is also weaker. Therefore, an answer to Q1 tending toward the positive is (more or less) conjectured, that is, h = hB. However, a proof or an example with justification is necessary.
Throughout this paper, feasible solutions and patterns (corresponding to certain parameters) shall mean stable stationary solutions and patterns which exist for the system (with these parameters). For some fixed parameters, let ΓB
k be the number of
feasible patterns on a finite lattice of size k, which satisfy the boundary condition B. Also, let Γ∞ be the number of feasible patterns on an infinite lattice, corresponding
to the same parameters. Even if Q1 is true, one does not expect, in general, that Γ∞ coincides with lim
k→∞ΓBk. For example, suppose Γk is a constant multiple of
ΓB
k for each k, where Γk is the number of patterns projected from the patterns on
an infinite lattice onto a lattice of size k. The spatial entropy defined through these two quantities are still identical, that is, h = hB. However, the following question
arises: How do the the solutions on finite lattices with a certain boundary condition approximate, in some norm, the solutions on an infinite lattice? Or, for each solution u on an infinite lattice, can the projection of u on the main central sites coincide with some solution on these sites, which satisfies certain boundary conditions? The former one is restated more precisely as follows.
Q2: Given a solution u (or a pattern) on the infinite lattice Zd, does there exist
a solution uT (or a pattern) on T ⊂ Zd, which satisfies certain a boundary condition,
and uT approximates u in some norm, as T tends to Zd ?
Basically, what is asked in Q2 is rather strong. A positive answer to it requires that “every” feasible pattern on an infinite lattice can be approximated by feasible patterns on large finite lattices with boundary conditions. In addition, topology must be specified when considering whether or not answering one of the questions Q1and Q2implies an answer to another.
In the case of a one-dimensional (1-d) lattice, the generation of patterns for a lattice system can frequently be described by the so-called “transition matrix” M . Suppose the size of T is k (e.g., T := {i | 1 ≤ i ≤ k}). The sum of all entries in Mk−1
gives the number of patterns on T ; cf. [9], [35]. These patterns represent the patterns on T , projected from the feasible patterns on the infinite lattice, if the transition matrix is appropriately formulated. For such projected patterns, boundary conditions and the feasibility of patterns with respect to the imposed boundary conditions on T are not considered. If a certain boundary condition is imposed, then the number of feasible patterns on T may become the sum of only some of the entries in Mk−1 or
even zero. The latter case, in which the projected patterns are usually very limited, is more likely to occur in the regime of pattern formation, whereas the former can occur in the regime of spatial chaos (as defined in section 2). In some cases, the number of feasible patterns equals a single entry of Mk−1.
The studies in influence of boundary conditions upon the solution structures of
partial differential equations have been done by many scientists. For example, Mielke [34] has indicated that some typical patterns in Ginzburg–Landau equations are not dominated by boundary effects. Homotopy of boundary conditions has been used by Fiedler [23], Hale and Rocha [26], and Gardner [25] to study global attractors of a class of reaction-diffusion equations. These studies analyzed the detailed effect of boundary conditions on the structure of global attractors from different aspects. For example, Fiedler showed that the class of global attractors is independent of boundary conditions. If the equilibrium is nonhyperbolic and a bifurcation occurs, the bifurcation scenario such as the structure of attractors may vary with respect to boundary conditions. This has been observed by Dillon, Maini, and Othmer [21] in the study of pattern formation in generalized 1-d Turing systems, and by Mei and Theil [33] in the analysis of steady state bifurcations, as well as by Holder and Schaeffer [28], and Schaeffer and Golubitsky [36] on mode-jumping of von K´arm´an equations.
Lattice dynamical systems can be found in many scientific models, including the models in chemical reactions [22], [31], material sciences [7], [18], biology [6], [30], image processing and pattern recognition [11], [14], [15], [16], [20], [24], [38].
This study shall attempt to derive the solutions for Q1and Q2in a cellular neural network (CNN) model. While aiming to achieve this goal, this work also obtains some interesting results for CNN itself, which can be regarded as an independent part of this paper. The CNN model we study is proposed by Chua and Yang in 1988; cf. [13]. Its applications in image processing and pattern recognition can be found in the above-mentioned references. This study focuses mainly on exploring how boundary conditions affect the pattern formation and spatial chaos for CNN. For this purpose, the pattern forming properties for CNN with a finite number of cells (CNN on a finite lattice) are also investigated. Therefore, our results also demonstrate how the boundary conditions affect the global attractor of CNN, which has practical implications for circuit design. The so-called mosaic patterns (solutions) in CNN are considered in our investigations. These mosaic solutions are all stable for CNN on a finite lattice and on an infinite lattice. Thus, the set of all mosaic solutions is contained in the global attractor of CNN. It will be seen how the attractor varies with respect to the parameters and different boundary conditions. Notably, the mosaic solutions are called stable system equilibrium points in Chua and Yang [13].
I n a 1-d CNN, formation of mosaic patterns can be fully described via transition matrices. Also discussed herein is how the boundary conditions influence the tran-sition matrices. Q1 and Q2 can then be resolved completely. Results in this study demonstrate that, although subtly, Q1 is true, possibly owing to that the dimension is low and the impact from the boundary conditions is weak. In the two-dimensional (2-d) case, we have counterexamples to Q1, indicating that h = hD and h = hN,
respectively. As for the solution to Q2, some cases (parameters) can always be found so that it is true or false, in 1-d and 2-d.
Mosaic patterns have been studied in discrete reaction diffusion models [9], [10], [12], as well as in CNN [29], [37]. The considerations in these works centered around patterns on an infinite lattice. Results in this study are presented for the first time in the following aspects. A family of transition matrices is formulated. All mosaic patterns of 1-d CNN with a general 1×3 template, on an infinite lattice and on a finite lattice with boundary conditions can be generated from these matrices. Boundary conditions (on 1-d and 2-d) have not been taken into account in the notion of spatial chaos, the formulation of a transition matrix and the existence of patterns on a finite lattice, in all the aforementioned studies [9], [10], [12], [29], [37].
The rest of this paper is organized as follows. Section 2 reviews the lattice dynam-ical system with some of its basic features. Two general propositions (2.1 and 2.2) are given which provide a criteria toward the positive answers to Q1and Q2, respectively. We then focus our discussion on CNN. Section 3 describes some general properties of CNN. In section 4, the solutions of Q1 and Q2 are derived for 1-d CNN with general templates, using both the setting of transition matrix and Propositions 2.1 and 2.2. Finally, in section 5, the solutions of Q1 and Q2 are derived for 2-d CNN with a two-parameter symmetric template, mainly using Propositions 2.1, 2.2. Taking this template and another horizontally symmetric template, some counterexamples are given.
2. General formulation and propositions. This study largely concerns itself with a class of lattice dynamical systems (LDS) which can be described as follows. Let d ≥ 1 be an integer and Λ be the d-dimensional integer lattice or a subset of it. That is, Λ ⊆ Zd⊆ Rd. The LDS considered herein is a system with continuous time,
and the spatial variable takes values in the discrete lattice Λ. The state of the system is represented by the vector {uα}α∈Λ with uα ∈ R for each α. In the case Λ = Zd,
the phase space X is an infinite-dimensional Banach space. For example, X = {u = {uα} : α ∈ Zd, u < ∞},
where uα∈ R and the norm · could be · ∞, the ∞norm or · 2, the 2norm
or · 2
q, the 2q norm. The 2q norm is defined as follows:
u 2 q = α∈Zd q−|α||u α|2 1/2 ,
where q > 0 is a fixed number, |α| = max{|αl|, l = 1, . . . , d}, and α = (α1, . . . , αd) ∈
Zd. These spaces are represented by X = ∞, 2, 2
q, respectively. Notably, 2= 2q for q = 1. A previous investigation has demonstrated that the norm · 2
q is quite useful
in investigating a space-time chaos in LDS; cf. [3] and the references therein. The LDS with continuous time can be expressed by
du/dt = F(u(t)), where F : X → X. (2.1)
Or in coordinate form
duα
dt = Fα({uγ}γ∈Λ), α ∈ Λ.
Generally, if F is locally Lipschitz in X, the existence and uniqueness of the initial value problem for LDS hold; see [5], [8]. Furthermore, the systems of interest, for example, spatially discrete version of Allen–Cahn equation (cf. [7]) and CNNs, have the property of finite-range interaction. Let a finite subset G ⊆ Zd be fixed (it
describes the range of coupling). Then F can be written as F(u)α= F ({uα+˜α}α∈G˜ ), α ∈ Zd,
where F : {{vα˜}α∈G˜ | vα˜ ∈ R} → R is a smooth function. These systems also have the property of invariance under translations of the lattice. Let {Sγ}γ∈zdbe the group
of translations acting on the space X, which is defined by (Sγu)α = uγ+α. A direct
consequence of this invariance is
u ∈ E∞⇐⇒ S
γu ∈ E∞,
where E∞is the set of all stationary solutions of (2.1). Further details can be found
in [10]. Herein, we denote by (LDS)∞ the infinite-dimensional initial value problem
for the LDS on Zd.
If Λ is only a proper subset of Zd, then boundary conditions are natually imposed
on the boundary of Λ in Zd. For example, consider Λ to be the following finite lattice Tk:
Tk= {(α1, α2, . . . , αd) ∈ Zd: 1 ≤ αl≤ kl},
where k = (k1, . . . , kd) is a d-tuple of positive integers. The width of the frame of
boundary sites is usually equal to the neighborhood size r. The latter one is related to the coupling range G. For instance, if d = 2, G = {(i, j) | i, j = −1, 0, 1}, then r = 1. In this case, the boundary sites are
b := {(k1+ 1, j), (0, j), (i, k2+ 1), (i, 0), 0 ≤ i ≤ k1, 0 ≤ j ≤ k2}. (2.2)
The following three types of boundary conditions for LDS on Tk are considered:
• (LDS)k-N, LDS with Neumann boundary condition on Tk;
• (LDS)k-P , LDS with periodic boundary condition on Tk;
• (LDS)k-D, LDS with Dirichlet boundary condition on Tk.
These boundary conditions are discrete analogues of the ones in PDEs; cf. [16]. For the case d = 2, r = 1, let us describe precisely these boundary conditions on Tk. The
Neumann boundary condition is the zero flux boundary condition. The states of the boundary sites are set equal to the states at the corresponding neighboring sites in Tk. Namely, for 0 ≤ i ≤ k1+ 1, 0 ≤ j ≤ k2+ 1,
uk1+1,j= uk1,j, u0,j = u1,j, ui,k2+1= ui,k2, ui,0= ui,1.
The periodic boundary condition identifies the first and the last rows (respectively, columns) of the array Tk, thereby forming a torus. Namely, for 0 ≤ i ≤ k1+ 1, 0 ≤ j ≤ k2+ 1,
u1,j = uk1,j, u0,j = uk1−1,j, u2,j = uk1+1,j, ui,1= ui,k2, ui,0= ui,k2−1, ui,2= ui,k2+1.
The Dirichlet boundary condition means that certain boundary data (fixed constants) are prescribed on the boundary sites b, that is, ub= ˆub:= {ˆui,j, (i, j) ∈ b}.
Notably, LDS on Tk with any above boundary condition is a system of ODEs on
a finite-dimensional phase space. Its existence and uniqueness theorems follow from the regular fundamental theorems of ODEs.
To illustrate the spatial complexity of (LDS)∞, the definition of spatial entropy
is recalled. Let A be a finite set of elements (symbols) which are used to represent the patterns at each site on the lattice. Let AZd
represent the set of all functions y : Zd→ A. There is a natural projection
πk: AZd→ ATk,
given by restricting any y ∈ AZd
to finite subset Tk. Assume that U is a translational
invariant subset of stationary solutions E∞ in (LDS)∞, or of AZd
, which represents a class of patterns in (LDS)∞. Set
Γk(U) := card(πk(U)).
This quantity refers to the number of distinct patterns among the elements of U, when restricting one’s observation to the subset Tk. The spatial entropy h(U) is defined as
h(U) := lim
k→∞
1
k1k2· · · kdln Γk(U).
(2.3)
On the other hand, when considering the problems (LDS)k-B on finite lattice Tk,
boundary condition B should be taken into account. Denote by UB
k (B = N, P, D)
a class of patterns (or stationary solutions) for (LDS)k-B. We can define the
spa-tial entropy, depending on boundary condition B, by using Γ(UB
k) := card(UkB) in
(2.3). The spatial entropy defined in this setting is denoted by hN = hN(UN), hP = hP(UP), hD = hD(UD), corresponding to boundary condition N (Neumann), P
(pe-riodic), and D (Dirichlet), respectively.
To discuss approximating a solution on infinite lattice by a solution on a finite lattice, there can be various considerations. One consideration is to identify the coinci-dence on the main central sites for a solution on an infinite lattice and a corresponding solution on a (large) finite lattice with boundary conditions. This is the main content in the following conditions: (H1) and (H2). Alternatively, a suitable norm can be used to measure the difference between two corresponding solutions. Which norm is appropriate depends on the nature of the problem (LDS). To be more explicit in this consideration, assume that u = {ui,j} is a solution on infinite lattice Z2 and uT is a
solution on finite lattice T = Tk ⊂ Z2. The expression u − uT is meaningful only
if u, uT belong to the same space. However, u ∈ X = ∞= ∞(Z2) or 2q = 2q(Z2),
whereas uT ∈ {v = {vi,j}(i,j)∈T : vi,j ∈ R}. Therefore, one has to be more
spe-cific in discussing the approximation in terms of the norm. An option is to consider πk(u) − uT ∞(T ) (or πk(u) − uT 2
q(T )), where v ∞(T ) = sup{|vi,j| : (i, j) ∈ T }
and v 2 2
q(T ) =
(i,j)∈Tq−|(i,j)||vi,j|2. Equivalently, uT can be extended to an
ele-ment ˜uT that is defined on Z2 by setting (˜uT)i,j = ui,j for (i, j) ∈ Z2\ T and, then,
considering u − ˜uT ∞ (or u − ˜uT 2
q). For · = · ∞ or · 2q, we state Q2 in
this consideration precisely as follows:
Let u ∈ U ⊆ E∞. Given ε > 0, do there exist a positive integer ˜k and ˜vk∈ X for
every d-tuple of positive integers k = (k1, . . . , kd) with every kl> ˜k such that πk( ˜vk)
is a solution of (LDS)k-B and u − ˜vk < ε ?
In this setting, obtaining this ˜vk ∈ X amounts to obtaining a solution vk of
(LDS)k-B such that πk(u) − vk ∞(T ) or πk(u) − vk 2
q(T )< ε. I f ˜k is independent
of u, the approximation in Q2is said to be uniform. Notably, Q2 is discussed in the context of stationary solutions (not patterns in symbols) of (LDS)k-B and (LDS)∞.
In the following, we give two criteria which indicate positive answers to Q1 and Q2, respectively. Assume that the patterns on the boundary sites (as b in (2.2) for case d = 2), which are reflected from previously described boundary conditions, are the same elements (symbols) that represent the patterns in U at each site. The following notations will be used. By k > s, we mean kl> s, for all l = 1, . . . , d and by πk−swe
mean the projection onto Tk−s, where Tk−s = {(α1, α2, . . . , αd) ∈ Zd: 1 + s ≤ αl ≤ kl− s} ⊂ Tk. The basic conditions are as follows:
(H1) There is a fixed positive integer s such that for every u ∈ U and for each k > s , there exists wk ∈ X (or AZd
) with πk−s(wk) = πk−s(u) and πk(wk) is a
solution (or pattern) of (LDS)k-B.
(H2) For each u ∈ U ⊆ E∞, there are constants c1 > 0, c2 > 1 such that for each d-tuple of integers k, there exists a solution vk of (LDS)
k-B with |(vk)α| < c1, α ∈ Tkand πk−s(vk) = πk−s(u) for some integer s < (1/c2) · min{kl: l = 1, . . . , d}.
Notably, (H2) implies that u ∞ < c1. I n (H1), s does not depend on u, while
s and c1, c2 are allowed to depend on u in (H2). Formulation of the conditions in (H1), (H2) involves the notion of boundary reconstruction scheme, as described later for CNN in sections 4 and 5. Basically, for u ∈ U, if we can reconstruct the solutions (or output patterns) on the sites in and near the boundary of Tk for πk(u) to fit
the boundary condition, then we have positive answers to Q1 and Q2. If this can be achieved for certain parameters, then one establishes a sense that the strength of effect from boundary conditions is weak for these parameters. Assume that the range of interaction (e.g., the radius of aforementioned G) has length r. In the propositions, recall that Γk(U) = card(πk(U)) =: Γ∞k denotes the number of solutions (patterns)
on Tk projected from U and Γ(UkB) = card(UkB) =: ΓBk represents the number of
solutions (patterns) in UB k.
Proposition 2.1. Let U be a translational invariant subset of stationary solu-tions E∞, or of AZd
. Assume (H1) and ΓBk ≤ pc· Γ∞k−r for some p > 0 and c = c(k)
with limk→∞[c/(k1k2· · · kd)] = 0, then h = hB, where B = N or P or D.
Proposition 2.2. Consider X = 2
q with q > 1. Let U be a translational invariant subset of E∞, then (H2) implies that Q2 is true.
Proposition 2.2 is formulated for the phase space X = 2
q with q > 1, in general
cases. In fact, Q2is true for CNN on X = ∞with B = N, P, D for certain parameter regions. This space is more practical for CNN from the application viewpoint. In sections 4 and 5, we elaborate on these results. The proofs of Propositions 2.1 and 2.2 are given successively as follows.
Proof of Proposition 2.1. We show only the case d = 2. Note that ΓB
k ≥ Γ∞k−s,
according to (H1). Thus, on the one hand, hB(UB) = limk→∞k1k21 ln ΓBk ≥ lim k→∞ 1 k1k2ln Γ∞k−s = lim k→∞ (k1− 2s)(k2− 2s) k1k2 ln Γ∞ k−s (k1− 2s)(k2− 2s) = h(U),
and, on the other hand, hB(UB) = lim k→∞ 1 k1k2ln Γ B k ≤ lim k→∞ 1 k1k2ln(p c· Γ∞ k−r) = lim k→∞ (k1− 2r)(k2− 2r) k1k2 c ln p + ln Γ∞ k−r (k1− 2r)(k2− 2r) = h(U).
Proof of Proposition 2.2. Assume that u ∈ U. Given ε > 0, let m be a positive integer such that
max |α|>m q−|α||u α|2 1/2 , |α|>m q−|α|c2 1 1/2 < ε/2.
Choose ˜k > mc2/(c2− 1). For any d-tuple of positive integers k with kl > ˜k, l =
1, . . . , d, let vk, s = s(k) be as in the assumption (H2). Note that k
l− s > kl(c2− 1)/c2> m. Set ( ˜vk)α= uαfor α ∈ (Zd\Tk), ( ˜vk)α= (vk)αfor α ∈ Tk. Then ˜vk∈ X,
πk( ˜vk) is a solution of (LDS)k-B, and u − ˜vk 2
q < ε. The proof is completed.
In sections 4 and 5, we locate the parameters for CNN for which the boundary reconstruction schemes can be developed and, in doing so, the conditions (H1), (H2) can be fulfilled. Thereafter, we can answer Q1, Q2for CNN via Propositions 2.1, 2.2. In the proof of Proposition 2.1, the equality hB(UB) = h(U) comes from two
inequalities. One follows from (H1) and the other follows from the assumption ΓBk ≤
pc· Γ∞
k−r. Let us give the motivation for this assumption. In counting the number
of feasible patterns on Tk, there are three quantities, namely Γ∞k, ΓBk, and Γk. The
first two were introduced earlier. The last one is the number of patterns on Tk,
without considering any boundary condition. Although the feasibility for such kinds of patterns is not well defined, they can still be obtained from a construction process: attaching feasible patterns on a lattice of smaller size compatibly to form patterns on a larger lattice, as is the methodology in [29], [37]. However, one knows for sure only that Γ∞
k ≤ Γk, ΓBk ≤ Γk, and not the relation between Γ∞k and ΓBk. Under normal
circumstances, the assumption ΓB
k ≤ pc· Γ∞k−r in Proposition 2.1 or even ΓBk ≤ Γ∞k is
expected to hold, at least for some kind of boundary condition B. For example, in the case d = 2, fix k, every site of Tk−ris interior in Tk. The number of patterns on Tk−r,
without considering any boundary condition, is expected to be the same as Γ∞ k−r. The
number of sites on Tkand outside Tk−ris 2·r·k1+2·r·k2−4·r2=: cr. In addition, the
maximal possible number of patterns on these sites is pcr, where p = card(A). Thus,
ΓB
k ≤ pcr · Γ∞k−r. Moreover, some boundary condition should exist so that pattern
forming is no less restrictive in the case with that boundary condition imposed than the case which requires extending the patterns to infinite lattice. In the following discussion of mosaic patterns in CNN, ΓB
k ≤ pc· Γ∞k−r is satisfied for the boundary
conditions B = N, P , and D.
Concerning the implication from answering one of the questions Q1 and Q2 to answering another, we make the following remarks. Consider X = 2
q, q > 1, and U ⊆ E∞. On the one hand, (H
1) implies that Q2 is true in the uniform sense if we further assume that in (H1), for each u, the corresponding wk is uniformy bounded for all k. On the other hand, a positive answer to Q2does not imply positive answer to Q1. This can be seen by respecting (H1), (H2) and the proofs of Propositions 2.1 and 2.2. Indeed, in (H1), s has to be a fixed integer for all u ∈ U, which is needed in the proof of Proposition 2.1, whereas s can depend on each u ∈ U in the proof of Proposition 2.2.
3. Cellular neural networks. A CNN is a large array of nonlinear analogue circuit, which is made of only locally connected cells. We consider the CNN model proposed by Chua and Yang in 1988; cf. [13], [14]. In this section, we present the equations of the model on the integer lattice in two dimensions. Some fundamen-tal dynamic properties for this CNN model are summarized and verified. To our
knowledge, previous literature has not derived these properties. The definitions and notations associated with the stationary solutions of CNN are also provided. We then discuss the model on finite lattice along with consideration of the boundary conditions mentioned in section 2. The stationary equation for CNN on 1-d lattice is presented in the next section.
Consider CNN on Z2, under space-invariant coupling and without an input control term. The circuit equation of a cell is
dxi,j dt = −xi,j+ z + |k|≤1,||≤1 ak,f(xi+k,j+), (i, j) ∈ Z2. (3.1)
Herein, the node voltage xi,j at (i, j) is called the state of the cell at (i, j). z is an
independent current source, which is called a bias term. The output function f (a nonlinearity) is given by
f(ξ) = 12(|ξ + 1| − |ξ − 1|).
The model (3.1) we consider has coupling range N1(i, j) for the cell at (i, j). Here, N1(i, j) = {(k, ) ∈ Z2| max{|k − i|, | − j|} ≤ 1}.
The elements in N1(i, j) are called the nearest and the next-nearest neighbors of (i, j). The real numbers ak,, |k|, || ≤ 1 describe the coupling weights between cells. They
can be arranged into a 3 × 3 matrix A :=
aa−1,1−1,0 a0,1a0,0 a1,1a1,0 a−1,−1 a0,−1 a1,−1
.
Such a matrix is called a (space-invariant) template. Furthermore, A is called sym-metric if ak,= a−k,−. This symmetry notion represents symmetric coupling weights
between cells.
The dynamical system (3.1) on an infinite lattice is an infinite system of ODEs. Equation (3.1) can be written in the form
dx dt = F(x), x ∈ X, where Fi,j(x) = F ({xi+k,j+}(k,)∈N1(0,0)), F ({xk,}(k,)∈N1(0,0)) := −x0,0+ z + |k|≤1,||≤1 ak,f(xk,). Proposition 3.1. Let X = ∞ or 2 q, q > 1.
(i) The vector field F : X → X is Lipschitz.
(ii) On X, the initial value problem for (3.1) is well posed, that is, for each x0 ∈ X, there exists a unique solution x(t) with x(0) = x0 for forward and backward time t near t = 0.
(iii) (3.1) is dissipative and every solution of it exists globally in time.
Proof. We only present the 2 − d case.
(i) Let ˆa = max{|ak,| : k, = −1, 0, 1}. Then |Fi,j(x)| ≤ |xi,j| + |z| + 9ˆa. This
verifies that F maps X into X for X = ∞. It can also be verified that F maps 2 q
into 2
q for q > 1. This assertion does not hold for 0 < q ≤ 1 (2q = 2as q = 1). Recall
the definition of |(i, j)| = max{|i|, |j|} in section 2. To verify the Lipschitz condition for F, the following computations are elaborated on. First, f is Lipschitz. Indeed, |f(ξ) − f(η)| ≤ |ξ − η|. Using this fact, we have
|Fi,j(x) − Fi,j(w)| = wi,j− xi,j+ |k|≤1,||≤1 ak,[f(xi+k,j+) − f(wi+k,j+)] ≤ |wi,j− xi,j| + ˆa
|k|≤1,||≤1
|xi+k,j+− wi+k,j+|
(3.2)
≤ (9ˆa + 1) x − w ∞.
Since this is true for every (i, j), it follows that
F(x) − F(w) ∞≤ (9ˆa + 1) x − w ∞. For X = 2 q, (3.2) yields that |Fi,j(x) − Fi,j(w)|2≤
|wi,j− xi,j| + ˆa |k|≤1,||≤1 |xi+k,j+− wi+k,j+| 2 ≤ 10
|wi,j− xi,j|2+ ˆa2 |k|≤1,||≤1 |xi+k,j+− wi+k,j+|2 . Note that q−|(i,j)|= q · q−|(i+k,j+)| or (1/q) · q−|(i+k,j+)| for k, = −1, 1. Therefore,
for q > 1,
F(x) − F(w)22
q =
(i,j)∈Z2
q−|(i,j)||Fi,j(x) − Fi,j(w)|2
≤
(i,j)∈Z2
10q−|(i,j)|
|wi,j− xi,j|2+ ˆa2 |k|≤1,||≤1 |xi+k,j+− wi+k,j+|2 ≤ 10x − w2 2 q+ 10ˆa 2 (i,j)∈Z2 |k|≤1,||≤1 q · q−|(i+k,j+)||x i+k,j+− wi+k,j+|2 = 10x − w2 2q+ 10ˆa2q |k|≤1,||≤1 (i,j)∈Z2 q−|(i+k,j+)||x i+k,j+− wi+k,j+|2 = 10(9ˆa2q + 1)x − w2 2 q. This verifies that F is Lipschitz on X = 2
q, q > 1.
(ii) The existence and uniqueness proof for finite-dimensional ODEs can be carried over to our infinite-dimensional systems here. Discussions for these fundamental properties in a general setting can be found in [5], [9].
(iii) It can be concluded by similar arguments as in [5] that the evolution operator for (3.1) exists for all t ∈ R. In fact, for each (i, j)
Fi,j(x)xi,j< 0 if |xi,j| > 9ˆa + |z|.
(3.3)
Let x0be an arbitrary point in X, then the solution φ(t, x0) of (3.1) satisfies
φ(t, x0)∞ ≤ max{2(9ˆa + |z|), x0∞}.
Thus, every local (in time) solution can be extended. Equation (3.3) also implies that the bounded set {x ∈ X : x∞ ≤ 2(9ˆa + |z|)} attracts every point of X, therefore (3.1) is dissipative. A similar conclusion holds for X = 2
q, q > 1. Details regarding the notion of
dissipative dynamical system can be found in [27]. This completes the proof.
Let x = {xi,j} be a stationary solution of (3.1). The associated output y = {yi,j} = {f(xi,j)} is called a (stationary) pattern. The stationary solutions and
patterns can be classified into four types: mosaic, defect, interior, and transitional, as defined in [29], [37]. Herein, the mosaic and the defect ones on the 2-d lattice are recalled. They can be easily generalized to another dimension.
Definition 3.2. A stationary solution x = {xi,j} of (3.1) is called nontransi-tional if |xi,j| = 1 for all (i, j) ∈ Z2. In particular, x is called a mosaic solution if |xi,j| > 1 for all (i, j) ∈ Z2. Its associated pattern is called a mosaic pattern. If |xi,j| = 1 for all (i, j) ∈ Z2 and there are (m, n) and (k, ) such that |xm,n| < 1 and |xk,| > 1, then x and y = {f(xi,j)} are called, respectively, a defect solution and a defect pattern.
This study focuses mainly on mosaic solutions and patterns. For generic param-eters, an above-mentioned pattern, if it exists, corresponds to an isolated equilibrium of (3.1). Notably, our mosaic pattern takes value (pixel value in image processing) 1 or −1 at each (i, j), whereas the mosaic pattern in Chow, Mallet-Paret, and Van Vleck [9] takes value 1 or 0 or −1. Notice that the global mosaic solutions, an essential type of equilibrium in CNN, do not belong to 2
q, 0 < q ≤ 1. For this and for the reason in
Proposition 3.1, it is inappropriate to formulate CNN on X = 2
q, 0 < q ≤ 1 (2q = 2,
if q = 1).
Definition 3.3. x is called a global solution if it is a stationary solution of (3.1) on Z2. In this case, y (= {f(x
i,j)}) is called a global pattern. Given any (proper) subset T ⊆ Z2, x(≡ x
T) is called a local solution if xT is a restriction of some global solution on T . Similarly, y(≡ yT) is called a local pattern if it is an output of some (local) solution xT on T . A set T ⊂ Z2 is called basic with respect to the template A if T = {(i + k, j + ) ∈ N1(i, j) | ak, = 0} for some (i, j) ∈ Z2. A basic (mosaic) pattern y is a local mosaic pattern defined on the basic set.
Our recipe for finding the global patterns or local patterns is to attach the patterns on a smaller lattice compatibly and construct patterns on a larger lattice, as found in [29], [37]. The global and local mosaic patterns are constructed from the basic patterns. Following from our partitioning of parameters (note: the next two sections provide more details), the set of “tentative” basic patterns can be characterized with respect to parameter subregions. The set of “tentative” basic patterns contains two types of patterns defined on basic sets. Some of these patterns can appear only on the cells near the boundary of a finite lattice. These are referred to herein as boundary basic patterns. If a tentative basic pattern can be expanded by attaching to it some tentative basic patterns (including itself) in all directions (east, west, south, north in the 2-d case) to form a global pattern, then this pattern is called a basic (mosaic) pattern, in respecting our Definition 3.3.
The term “feasible” generally refers to specifying “stable” stationary solutions; cf. [29]. According to a previous investigation, −1 is the only eigenvalue for the lineariza-tion of F at a mosaic solulineariza-tion. Therefore every global mosaic solulineariza-tion (pattern) is stable in the spaces X = ∞, 2
q, q > 1. In this study, since the solutions and patterns
we are discussing are already stable, “feasible” solutions and patterns (corresponding to certain parameters) are adopted to emphasize the existence of the stable stationary solutions and patterns for the system (with these parameters).
The following dynamic properties for CNN on a finite lattice can be found in [13], [32]. If the parameters are given, every solution x(t) of CNN is bounded for all time t ≥ 0. Furthermore, if the template is symmetric, every orbit converges to an equilibrium. In addition, with further restriction on the central element of template (a0,0> 1 in (3.1)), almost every orbit tends to a mosaic solution.
Definition 3.4. For any two integers k < l, denote I[k, l] = {k, k + 1, . . . , l}, the set of all integers that are no smaller than k and no greater than l.
Definition 3.5. Let U be a translation-invariant set of feasible global mosaic patterns. U is said to exhibit spatial chaos if the spatial entropy h(U) is greater than zero. Otherwise, we say U exhibits pattern formation.
Next, we discuss CNN on a finite lattice T . Every cell of the CNN has the same neighborhood of interaction except the ones located on the edge of T . I t was suggested in [17] that one can surround the rectangular array with boundary cells to compensate for the absent neighbors of these cells. The output for each of these boundary cells is chosen to be between −1 and 1 (in most applications, it has been set to zero) and remains constant with respect to time t. These are the boundary conditions of Dirichlet type. There is also a design of circular array; see [39]. In that proposed boundary condition, two edges of a 1-d array are connected to form a circular circuit. Our periodic boundary condition fits into this setting. In the remaining discussion of patterns on a finite lattice, we shall consider patterns on a square lattice T = Tk := {(i, j) ∈ Z2 | −k ≤ i, j ≤ k} in the 2-d case and T = Tk := {i ∈ Z1| −k ≤ i ≤ k} in the 1-d case. This differs from Tk in section 2,
and is merely for convenience of discussion. The Dirichlet boundary value problems of CNN on Tk can be further classified into two types. Let (CNN)k–D1 denote the one with saturated boundary data. That is, each boundary cell has an output of magnitude equal to one. Let (CNN)k–D0 denote the one with defect boundary data. That is, each boundary cell has output of magnitude less than one and, in most cases, we take the zero boundary data.
The equations associated with the interior cell are the same as (3.1). If the Neumann boundary condition is imposed, the equation for the boundary cell at (k, j), −k < j < k is
dxk,j
dt = −xk,j+ z + (a0,0+ a1,0)f(xk,j) + (a1,1+ a0,1)f(xk,j+1) +(a0,−1+ a1,−1)f(xk,j−1) + a−1,1f(xk−1,j+1)
+a−1,0f(xk−1,j) + a−1,−1f(xk−1,j−1).
In addition, the equation for the corner cell (k, k) is dxk,k
dt = −xk,k+ z + (a0,0+ a1,0+ a0,1+ a1,1)f(xk,k) + (a0,−1+ a1,−1)f(xk,k−1) +(a−1,1+ a−1,0)f(xk−1,k) + a−1,−1f(xk−1,k−1).
The equations of the other boundary cells and the other corner cells of Tkcan be
anal-ogously obtained. The equations for the other boundary conditions can be obtained in a similar manner.
In the remainder of this presentation, if not in an arithmetic computation, the symbols “+” and “−” are used to represent the positive and negative saturated states
as well as their output patterns, respectively. Thus, the elements in the set AZd
, A = {+, −}, give all possible global mosaic patterns. Nevertheless, to save the notations, we shall use M∞ = M∞(z, A) to represent both the sets of feasible global mosaic
solutions and patterns corresponding to the parameters z, A = (ak,). We also denote
the set of feasible mosaic solutions (patterns) on Tk with boundary condition B by MB
k = MBk(z, A) and abbreviate it by Mk, when no confusion arises.
To obtain patterns on finite lattice, the boundary basic patterns should also be taken into account. However, these boundary basic patterns cannot exist if boundary conditions N, P, D1 are imposed, as will be seen later. The following two sections are devoted to deriving the solutions to Q1 and Q2 for the mosaic patterns of 1-d and 2-d CNN, respectively. The notation MB
k (respectively, M∞) plays the role of UB
k (respectively, U) in section 2 and πk is the projection on Tk now. Conditions
(H1), (H2) for CNN are reduced to the following: (H
1) There is a fixed positive integer s such that for every u ∈ M∞and for each k > s there exists wk ∈ X with π
k−s(wk) = πk−s(u) and πk(wk) is a solution (or
pattern) of (CNN)k-B.
(H
2) Let x ∈ M∞ be a global mosaic solution. There is a constant c2> 1 such that for each positive integer k there exists a mosaic solution vk of (CNN)
k-B with πk−s(vk) = πk−s(x) for some integer s < (2k + 1)/c2.
Here, in (H
2), the constant c1in (H2) is unnecessary, since the mosaic solution vk in (H
2) can always be considered bounded (its components have bounded magnitude). Notably, the condition other than (H1) in Proposition 2.1 holds. Indeed, Γ(MBk) ≤
Γk(M∞) if B = N, P, D1, Γ(MD0k ) ≤ 24k−4· Γk−1(M∞) if d = 2, and Γ(MD0k ) ≤
22· Γ
k−1(M∞) if d = 1, according to our approach of forming patterns.
4. CNN on a one-dimensional lattice. The stationary equation of 1-d CNN with the general template A1= [α, a, β] is
0 = −xi+ z + αf(xi−1) + af(xi) + βf(xi+1), i ∈ Z1.
(4.1)
If a = 0, and x = {xi}∞i=1 is a mosaic solution of (4.1), then for each i ∈ Z1,
(u, v) = (xi, yi= f(xi)) satisfies v = f(u), v = 1 a[u − (z + σ1α + σ2β)] (4.2)
for σ1, σ2 = 1 or −1. The expression in (4.2b) represents four straight lines on u-v plane if a, α, β = 0, α = ±β, z are given. Herein, these lines are labeled by Lσ1σ2, σ1, σ2 = 1 or −1. Notably, on u-v plane, the configuration for the curve of output function in (4.2a) and the four lines in (4.2b) are fixed once α, β, z, a are given. For template A1with α, β = 0, there are at most eight basic (mosaic) patterns (refer to Definition 3.3): + + +, + + −, − + +, − + −, − − −, − − +, + − −, + − +. We collect them into two groups, w + e or w − e, where w (west), e (east) =“+” or “−”. If the parameters α, β, z, a are given, then the corresponding tentative basic patterns can be determined from (4.2) (the term “tentative” was explained after Definition 3.3). More precisely, if α, β, z, a are the parameters with which Lσ1σ2 in
(4.2b) intersects the piecewise linear curve (4.2a) at u > 1 (respectively, u < −1), then w + e (respectively, w − e ) is a tentative basic pattern, corresponding to (α, β, z, a), where w = “ + ”, “−” if σ1 = 1, −1, respectively, and e = “ + ”, “−” if σ2 = 1, −1, respectively, as illustrated in Figure 4.1. Therefore, characterizing tentative basic patterns with respect to the parameters α, β, z, a amounts to examining how the
curves in (4.2) intersect. These intersections can actually be classified into finitely many types. This classification then yields a partitioning of the parameter space. In the following, we describe this partitioning along with its associated notations.
The relative position of these four lines depends on the region in Figure 4.2 in which (α, β) lies. These relative positions will be used to demonstrate the existence of the basic patterns corresponding to (z, a) in each region of Figure 4.4. This will be explained after we introduce all necessary notations. The notations in Figure 4.2 are of assistance in spelling out the relative position of these four lines. The overhead bar there means “minus sign of ” a real number or a symbol. Indeed, these four regions in the half plane α > β are described in (4.3):
Ωαβ = {(α, β) | α + β > α − β > −α + β > −α − β};
(α, β) ∈ Ωα ¯β ⇔ (α, −β) ∈ Ωαβ,
(α, β) ∈ Ωβα¯ ⇔ (−β, α) ∈ Ωαβ,
(α, β) ∈ Ωβ ¯¯α ⇔ (−β, −α) ∈ Ωαβ.
(4.3)
Notably, Ωβα is symmetric to Ωαβ with respect to α = β on α-β plane. Thus, y
is a pattern corresponding to (z, a) with (α, β) ∈ Ωβα if and only if a 180-degree
rotation of y is a pattern corresponding to the same (z, a) with (α, β) ∈ Ωαβ. The
same situations hold for the other symmetric regions. Therefore, only (α, β) in the half plane α ≥ β needs to be considered.
There are 25 regions in Figure 4.4, which are denoted by [1, 0; 1, 0], [m; 1, 0], [1, 0; n], and [m; n] for m, n ∈ I[0, 3]. These notations are interpreted as follows. For example, assume that (α, β) ∈ Ωαβ, if (z, a) ∈ [m; n], the tentative basic patterns are
exactly those with at least (3−m) “+” in the nearest neighbors of a positive saturated state and with at least (3 − n) “−” in the nearest neighbors of a negative saturated state. If m = 3 (respectively, n = 3), then it means that there is no restriction for w and e in w + e (respectively, w − e ). If m = 0 (respectively, n = 0), then any w + e (respectively, w − e ) with w, e=“+”, “−”, is not feasible. (z, a) ∈ [1, 0; 1, 0] indicates that, w + e with w= “+” (one “+” for w) and no restriction for e, and w − e with w= “−” and no restriction for e, are tentative basic patterns. Similar interpretations apply to the notations [m; 1, 0], [1, 0; n]. The following expressions provide the details of these notations. For α, β ∈ Ωαβ, the notation Bαβ([∗; ·]) represents the sets of
tentative basic patterns corresponding to each of these parameter region [∗; ·] in z-a plane, where “ ∗ ”, “ · ” = m or n or “1, 0”, m, n ∈ I[0, 3]. Patterns in each of these sets can be collected into two categories: the ones with center element “+” and the ones with center element “−”. Herein, this arrangement is denoted by Bαβ([∗; ·]) = B+
αβ(∗) ∪ Bαβ− (·). Similar notations B•([∗; ·]) = B•+(∗) ∪ B•−(·) are used for (α, β) in
the other regions of α-β plane. Let us describe these B+
•(∗), B•−(·) now. The following m, n ∈ I[1, 3]. For (α, β) ∈ Ωαβ
B+
αβ(m) := {w + e | at least (3 − m) of w and e are “+” }, B−
αβ(n) := {w − e | at least (3 − n) of w and e are “−” }, B+αβ(1, 0) := {w + e | w = “ + ”},
B−
αβ(1, 0) := {w − e | w = “ − ”}.
The overhead bar is used to represent the “negative sign of ” symbol “+” or “−” in the following expressions, for example, ¯w = “ − ” if w = “ + ”. I f (α, β) ∈ Ωα ¯β, then
w + e ∈ B+
α ¯β(m) ⇔ w + ¯e ∈ B+αβ(m),
w − e ∈ Bα ¯β(n) ⇔ w − ¯e ∈ B− αβ(n), B+ α ¯β(1, 0) = B+αβ(1, 0), B−α ¯β(1, 0) = B−αβ(1, 0). I f (α, β) ∈ Ωβα¯ , then w + e ∈ B+ ¯ βα(m) ⇔ ¯e + w ∈ Bαβ+ (m), w − e ∈ B− ¯ βα(n) ⇔ ¯e − w ∈ Bαβ− (n), w + e ∈ B+ ¯ βα(1, 0) ⇔ e = “ − ”, w − e ∈ B− ¯ βα(1, 0) ⇔ e = “ + ”. I f (α, β) ∈ Ωβ ¯¯α, then w + e ∈ B+ ¯ β ¯α(m) ⇔ ¯e + ¯w ∈ B+αβ(m), w − e ∈ B− ¯ β ¯α(n) ⇔ ¯e − ¯w ∈ B−αβ(n), B+ ¯ β ¯α(1, 0) = B+βα¯ (1, 0), B−β ¯¯α(1, 0) = B−βα¯ (1, 0). In these expressions, w, e could be either “+” or “−” in B+
•(m) (respectively, B−•(n))
if m = 3 (respectively, n = 3).
As to how the relative position of the slant lines in (4.2b) relates to the feasibility of the basic patterns corresponding to (z, a) in each region of Figure 4.4, the following instance is used as an illustration. The first line on the far right side of Figure 4.1 is L1,1 if (α, β) ∈ Ωαβ and is L1,−1 if (α, β) ∈ Ωα ¯β. The tentative basic pattern
corresponding to (z, a) ∈ [1; 0] of Figure 4.4 is + + + for the first case, and is + + − for the second case.
Notably, the four lines in (4.2b) degenerate into three lines if α = β or α = −β with α, β = 0. The tentative basic patterns for α = β > 0 (respectively, α = β < 0, α = −β > 0 ) and (z, a) ∈ [m; n] in Figure 4.5 are exactly the same for (α, β) ∈ Ωαβ
(respectively, Ωβ ¯¯α, Ωα ¯β ) and (z, a) ∈ [m; n] in Figure 4.4. Furthermore, these four lines in (4.2b) degenerate into two lines if one of α, β is zero. The tentative basic patterns for α > 0, β = 0 and (z, a) ∈ [3; 3], [1, 0; n], [n; 1, 0], n = 0, 3 in Figure 4.6 are exactly the same for (α, β) ∈ Ωαβ or Ωα ¯β and (z, a) ∈ [3; 3], [1, 0; n], [n; 1, 0] of
Figure 4.4, respectively. The same situation holds for α = 0, β < 0 and (α, β) ∈ Ωβα¯ (or Ωβ ¯¯α). The regions Ωαβ, Ωα ¯β, Ωβα¯ , and Ωβ ¯¯α, therefore, serve as the representative
cases in the α-β plane.
The coordinates of these four points B, C, D, E in Figure 4.4 depend on which region (α, β) lies in. For example, B = 1 − α − β, C = 1 − α + β, D = 1 + α − β, E = 1 + α + β, if (α, β) ∈ Ωαβ. The location of z-axis in Figure 4.4 (similarly in Figures
4.5 and 4.6) is either below B or between B and C or between C and D, depending on where (α, β) is located in the further partitioned subregions in Figure 4.3.
In association with these partitions of parameter space, we denote by M∞([∗; ·] α,β),
M∞
k ([∗; ·]α,β) := πk(M∞([∗; ·]α,β)), and MBk([∗; ·]α,β), respectively, the set of feasible
global mosaic patterns, the set of feasible local mosaic patterns (projected from global patterns) on Tk, and the set of feasible mosaic patterns on Tk with boundary
con-dition B, corresponding to parameters (z, a) ∈ [∗; ·], [∗; ·] = [m; n], [1, 0; n], [m; 1, 0],
Fig. 4.1. (α, β) ∈ Ωαβ, (z, a) ∈ [2; 1, 0].
with given α, β. Occasionally, α, β or even [∗; ·]α,β is omitted in these notations when
the parameters have been specified.
We have described the tentative basic patterns which contain the feasible basic patterns corresponding to each parameter region. The global mosaic patterns can be constructed by attaching the feasible basic patterns, as mentioned in section 3. Therefore, the regime of spatial chaos and pattern formation for the mosaic patterns of CNN on a 1-d infinite lattice with template [α, a, β] can be completely determined. Theorem 4.1. Consider CNN on Z1with template [α, a, β]. h(M∞(z, a, α, β)) >
0 if and only if (z, a, α, β) lies in one of the following regions:
(i) (α, β) ∈ Ωαβ or Ωβα or α = β, α > 0 and (z, a) ∈ [m; n], m, n ≥ 2.
(ii) (α, β) ∈ Ωα ¯β or Ωβ ¯α or α > 0, β = 0 or β > 0, α = 0 and (z, a) ∈ [m; n],
min{m, n} ≥ 2, max{m, n} = 3.
(iii) (α, β) ∈ Ωβα¯ or Ωαβ¯ or α = −β and (z, a) ∈ [m; n], min{m, n} ≥ 2, max{m, n} = 3, or (z, a) ∈ [3; 1, 0], [1, 0; 3].
(iv) (α, β) ∈ Ωβ ¯¯α or Ωα ¯¯β or α = β, α < 0 or α < 0, β = 0 or β < 0, α = 0 and (z, a) ∈ [m; n], min{m, n} ≥ 1, max{m, n} ≥ 2, or (z, a) ∈ [1, 0; n], [m; 1, 0], m, n ≥ 2.
The proof of this theorem follows directly from computing the maximum eigen-value of the corresponding transition matrix in each case. The notion for spatial entropy here is as the topological entropy for Markov shifts. For an account of how the maximal eigenvalue of the transition matrix relates to the spatial entropy of these mosaic patterns, please see [35, p. 341].
Taking the following identification between the indices of a matrix and 1 × 2 patterns
1 ↔ ++, 2 ↔ +−, 3 ↔ −+, 4 ↔ −−, (4.4)
Fig. 4.2. Primary partition of α-β plane.
Fig. 4.3. Secondary partition of α-β plane.
we consider the transition matrix M in (4.5): M = M(r) := r1 r2 0 0 0 0 r3 r4 r5 r6 0 0 0 0 r7 r8 , (4.5)
where r = {rj}8j=1, rj = 0 or 1, j ∈ I[1, 8]. The formation of mosaic patterns can be
described as follows: the (i, j)-entry of M is one if and only if the jth 1 × 2 pattern
Fig. 4.4. Partition of z-a plane for given α, β = 0, α = ±β.
Fig. 4.5. Partition of z-a plane for given α, β, α = ±β = 0.
Fig. 4.6. Partition of z-a plane for given α, β, α = 0 or β = 0.
in (4.4) can be joined, with one site overlapped, to the right of ith 1 × 2 pattern in (4.4) to form a 1×3 feasible pattern. We collect some properties about the maximum eigenvalue of M in the following lemma, which can be used to verify some cases in Theorem 4.1. Moreover, the motivation of and behind this elementary lemma is to observe how many and what kind of basic mosaic patterns can contribute to positive spatial entropy in this 1-d case.
Lemma 4.2. (i) The minimal number of nonzero entries for the matrix M to have eigenvalue greater than 1 is 4. (ii) The maximal number of nonzero entries for the matrix M to have all eigenvalues no greater than 1 is 6. (iii) M cannot have more than one eigenvalue greater than 1 for any choice of rj= 0 or 1, j ∈ I[1, 8].
Assume that the parameters are fixed. There corresponds a set of tentative basic patterns (each of them is a 3 × 1 pattern) from our previous formulation. In the set, some basic pattern can be attached to itself or other elements in the set, from the left and the right, to form a global pattern. These patterns are the feasible basic (mosaic) patterns, with respect to our Definition 3.3 and its following discussion. On the other hand, some patterns in the set cannot be constructed to form a global pattern by attaching to it any tentative basic patterns. Such a pattern is called a boundary basic pattern since it can only appear near the boundary sites of a finite lattice. Owing to our definition of spatial entropy (without considering boundary conditions), only the feasible basic patterns should be considered. However, it can be seen that, for example, in the following proof (case (iii)) of Theorem 4.1, in constructing the transition matrix, excluding or including these boundary basic patterns does not affect the spatial entropy. That is, the maximal eigenvalues are the same for the corresponding two different transition matrices. One of them is formulated with respect to the tentative basic patterns and the other is for feasible basic patterns only.
To provide more details to the proof of Theorem 4.1, we discuss the critical cases in each item of the theorem as follows. These cases (the following (i)–(iv)) are the
ones with a minimal amount of feasible basic patterns in each item so that the system exhibits spatial chaos. Herein, we only present the results for the representative regions Ωαβ, Ωα ¯β, Ωβα¯ , and Ωβ ¯¯α on α-β plane. In each case, we give the entries of
matrix M, which can be verified by inspecting the set of feasible basic patterns. (i) If (α, β) ∈ Ωαβ, (z, a) ∈ [2; 2], then r3= r6 = 0, rj = 1, j ∈ I[1, 8], j = 3, 6.
The set of tentative basic patterns is given in (4.6). Each of its elements is feasible: B+ αβ(2) = { + + +, + + −, − + + }, B− αβ(2) = { − − −, − − +, + − − }. (4.6)
(ii) If (α, β) ∈ Ωα ¯β, (z, a) ∈ [3; 2], then r4= 0, rj = 1, j ∈ I[1, 8], j = 4. The set
of tentative basic patterns is given in (4.7). Each of its elements is feasible: B+
α ¯β(3) = { • + •, • = “ + ” or “ − ”}, B−
α ¯β(2) = { − − +, − − −, + − + }.
(4.7)
(iii) If (α, β) ∈ Ωβα¯ , (z, a) ∈ [3; 1, 0], then r4 = r7 = r8 = 0, rj = 1 otherwise.
The set of tentative basic patterns is given in (4.8): B+ ¯ βα(3) = { • + •, • = “ + ” or “ − ” }, B− ¯ βα(1, 0) = { + − +, − − + }. (4.8)
Note that the basic pattern − − + in B− ¯
βα(1, 0) is a boundary basic pattern,
since no element in the set B+ ¯
βα(3) ∪ Bβα−¯ (1, 0) can be attached to the left of
it. However, the matrix M with r4= r7= r8= 0, rj = 1, j = 4, 7, 8 has the
same maximal eigenvalue as the one with r4= r8= 0, rj = 1, j = 4, 8.
(iv) If (α, β) ∈ Ωβ ¯¯α, (z, a) ∈ [2; 1], then r1 = r4 = r7 = r8 = 0, r2 = r3 = r5 = r6= 1. The set of tentative basic patterns coincides with the one of feasible basic patterns and is given in (4.9):
B+ ¯ β ¯α(2) = { − + −, + + −, − + + }, B− ¯ β ¯α(1) = { + − + }. (4.9)
Next, the patterns on a finite lattice are discussed. Notably, the patterns projected onto a finite lattice T from all global mosaic patterns are exactly the same as the set of patterns on T constructed from attaching the feasible basic patterns compatibly. These transition matrices subsequently give us the number of local mosaic patterns on a finite lattice of any size. For example, consider Tk := {i | −k ≤ i ≤ k}. The
number of local mosaic patterns on Tk (of size 2k + 1) is
Γk(M∞) = 4 i=1 4 j=1 (M2k−1) i,j.
While attempting to describe patterns on a finite lattice, the transition matrix should be formulated with consideration of both the feasible basic patterns and the boundary basic patterns. However, it is not difficult to verify that if boundary condition N or P , or D1 is imposed, the boundary basic patterns fail to exist.
In fact, the notion of transition matrix that describes the pattern forming tells us more. The (i, j)-entry of M2k−1 gives the number of feasible mosaic patterns on
Tk with ith 1 × 2 pattern in (4.4) at the two sites to the far left of Tk and jth 1 × 2
pattern in (4.4) at the two sites to the far right of Tk. For instance, the (1, 2)-entry
of M2k−1 gives the number of mosaic patterns on T
k with the left-hand side having
“++” and the right-hand side having “+−”, that is, patterns of the form in (4.10): + + · · · + − .
(4.10)
Using this information and our formulation of basic mosaic patterns allow us to count the number of mosaic patterns on Tk with any boundary conditions. Q1 can then be answered completely for the mosaic patterns of CNN. Q2 can also be resolved completely for the mosaic patterns of CNN, with our formulation of basic mosaic patterns.
Next, the mosaic patterns on Tk, which satisfy various boundary conditions are
considered. Since the pattern forming described by transition matrix works only in a 1-d lattice in general, we attempt to derive the solution for Q1 via the setting of transition matrix as well as Proposition 2.1. The latter gives us experience in treating the problems in two and higher-dimensional lattices. Notably, the condition other than (H
1) in Proposition 2.1 has been verified at the end of section 3. In deriving the solution for Q2, basically, we verify if (H2 ) in section 3 is satisfied. If a boundary reconstruction scheme can be developed for every local pattern projected from a global pattern onto Tk (abbreviated as local pattern or projected pattern),
for given parameters, then Q2 is true for these parameters. In the 1-d case, if this boundary reconstruction scheme cannot be obtained for some projected pattern, then at the same time, Q2 cannot be true. Though (H2 ) and Q2 are discussed in the context of stationary solutions, it is more convenient to manipulate the boundary reconstruction in terms of the output patterns which correspond to the stationary solutions. Our discussions are divided into four parts (I), (II), (III), (IV). They correspond to (α, β) ∈ Ωαβ, Ωα ¯β, Ωβα¯ , Ωβ ¯¯α, respectively. Herein, only the typical and
representative cases in each item are presented.
Notably, if (z, a) ∈ [3; 3], for any given α, β, formation of patterns is unaffected by any boundary condition. Moreover, if h(M∞) = 0 in some parameter region, then hB= 0 for boundary condition B = N, P, D1, D0.
(I) For (α, β) ∈ Ωαβ, Neumann B.C. is very natural in the sense that πk(x) is
a solution of (CNN)k-N for every mosaic solution x of (CNN)∞. Therefore, Q1, Q2
are true for Neumann B.C. and any (z, a); see Theorem 4.3 and Table 4.1 in Theorem 4.4.
Consider (z, a) ∈ [2; 2]. Since periodic boundary condition requires coincidence of the patterns at the sites to the far right and the ones to the far left, some projected patterns πk(y) (solutions πk(x)) certainly fail to be feasible patterns of (CNN)k-P .
For example, patterns πk(y) of the form in (4.11): − − − · · · − + · · · + ++ . (4.11)
Also, imposing certain prescribed boundary data obviously affects the formation of patterns. For instance, imposing positive saturated state at the left end, then any pattern of the form enclosed in (4.12) cannot exist:
+ − + + · · · . (4.12)
Furthermore, if a < 1 + β, imposing zero data on the left end, the pattern enclosed in (4.13) cannot exist:
0 + − − · · · . (4.13)
On the other hand, given any mosaic pattern y of (CNN)∞, (πk(y))i, i = ±k, ±(k−1), ±(k − 2), can always be amended so that they become (πk(y))i = 1. That is, the
outputs near the two ends of any local pattern πk(y), y = {f(xi)}, can be changed,
without losing the feasibility, into the one in the form (4.14): w + + + · · · + ++ e. (4.14)
A pattern of this form certainly satisfies the periodic B.C. and Dirichlet B.C., with any prescribed boundary values w and e imposed on the left and the right ends. Thus, (H
1) (see section 3) is satisfied, and h = hN = hP = hD1 = hD0 can be verified, by
Proposition 2.1. Q2 is also true for all these boundary conditions, by Proposition 2.2. I f (z, a) ∈ [3; 1, 0], the set of projected patterns πk(M∞) consists of elements in
(4.15):
{ − − − · · · − + · · · + ++ , − − − · · · − −− , + + + · · · + ++ }. (4.15)
In this set, patterns of the form (4.11) do not and cannot be amended to satisfy the periodic boundary conditions, so Q2is false. If positive saturated state at the left end is imposed, the “−” strip pattern in (4.15) fails to exist. If negative saturated states are imposed at both ends, then MD1k = πk(M∞). Thus, Q2 can be true or false in this case, depending on different prescribed data. The other results in Theorem 4.4 can be similarly obtained.
(II) For (α, β) ∈ Ωα ¯β, (z, a) ∈ [3; 2]. Imposing Neumann B.C. forces a projected pattern on Tk of the form in (4.16) fail to exist:
· · · − +− . (4.16)
However, the outputs near the right edge can be changed, without losing the feasibility. In doing so, the pattern becomes the form in (4.17), which satisfies the Neumann B.C.:
· · · − ++ . (4.17)
Thus, Q1, Q2 are true for Neumann B.C., by Propositions 2.1 and 2.2. It is more complicated for periodic boundary conditions. Here, the setting of transition matrix is adopted. Many (actually, the number grows exponentially in the size of the lattice) patterns on Tk do not satisfy periodic B.C. However, the spatial entropy remains
unchanged if we only count the patterns that satisfy periodic B.C. This can be verified by computing the transition matrix to the power 2k−1, that is, M2k−1. As mentioned earlier Γk(M∞([3; 2])) = 4 i=1 4 j=1 (M2k−1) i,j,
where M is given in the proof (ii) of Theorem 4.1. If periodic boundary condi-tion is demanded, then the number of mosaic patterns on Tk is Γ(MPk([3; 2])) =
Γk(M∞([3; 2])) − (M2k−1)4,2. This number can be estimated or actually computed so that h = hP can be verified. However, Q2 is false because some projected pat-terns do not and cannot be amended to satisfy the periodic boundary conditions. For Dirichlet B.C., if some positive saturated state or the zero boundary data (with parameters satisfying z −a ≥ β −1) at the left end is imposed, then the local patterns on Tk of the forms in (4.18) cannot exist:
+ − − + · · · ·, + − − − · · · ·, 0 − − + · · · ·, 0 − − − · · · ·. (4.18)
![Fig. 5.2. β > 0, (z, a) ∈ [4; 3].](https://thumb-ap.123doks.com/thumbv2/9libinfo/7651805.139385/30.918.109.662.119.720/fig-β-gt-z.webp)
![Fig. 5.5. β < 0, (z, a) ∈ [4; 2].](https://thumb-ap.123doks.com/thumbv2/9libinfo/7651805.139385/31.918.118.649.145.992/fig-β-lt-z.webp)
![Fig. 5.8. Formation of patterns on T k under Neumann B.C. with HA-template, β < 0, (z, a) ∈ [4; 2].](https://thumb-ap.123doks.com/thumbv2/9libinfo/7651805.139385/32.918.97.663.141.251/fig-formation-patterns-t-neumann-b-ha-template.webp)
