DYNAMICS OF LATTICE DIFFERENTIAL EQUATIONS

SHUI-NEE CHOW1,4 , JOHN MALLET-PARET2,4 , AND ERIK S. VAN VLECK3,4

Abstract. In this paper recent work on the dynamics of lattice differential equations is surveyed. In

particular, results on propagation failure and lattice induced anisotropy for traveling wave or plane wave

solutions in higher space dimensions spatially discrete bistable reaction-diffusion systems are considered.

In addition, analysis of and spatial chaos in the equilibrium states of spatially discrete reaction-diffusion

systems are discussed.

Key words. lattice differential equations, traveling wave solutions, propogation failure, lattice aniso-

tropy, equilibrium solutions, stability, spatial entropy

Abbreviated title. Lattice Differential Equations

1 School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332 USA. The work of

this author was supported in part by ARO Contract DAAH04-93-G-0199 and by NSF Grant DMS-90-

05420.

2 Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912 USA. The work

of this author was supported in part by NSF Grant DMS-93-10328, by ARO Contract DAAH03-93-G-0198,

and by ONR Contract N00014-92-J-1418.

3 Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, Colorado

80401 USA. The work of this author was supported in part by NIST Contract 60NANB4D1698 and by

NSF Grant DMS-95-05049.

4 As is usual with mathematics paper, authors’ names are in alphabetical order.

0

1. Introduction. This paper is a brief survey of recent advances in the theory of

lattice differential equations (LDE’s). LDE’s are systems of ordinary differential equa-

tions with a discrete spatial structure. While they have much in common with partial

differential equations (PDE’s), which by contrast have a continuous spatial structure,

LDE’s can arise in many other contexts and need not be near a PDE continuum limit. In

many cases LDE’s often exhibit behavior not present in associated PDE’s. The LDE’s to

be considered here are similar in form to spatial discretizations of reaction diffusion and

Cahn-Hilliard type PDE’s. Indeed, lattice differential equations can be discretizations of

PDE’s, and understanding the effect of discretization is important when drawing conclu-

sions from numerical simulations. Nevertheless, much of our analysis is concerned with a

broader class of systems, and is not restricted to those near a PDE limit.

A typical LDE takes the form

uη = gη(uξξ∈Λ), η ∈ Λ.(1)

Here Λ ⊂ IRn is a lattice, that is, a discrete subset of IRn, with either finitely or infinitely

many points, and with some regular spatial structure. The variable uη for each η ∈ Λ

is the coordinate of the state vector u = uηη∈Λ, indexed by the points on the lattice,

and each gη is some function of these coordinates. The dot “ ˙ ” represents differentiation

with respect to time t, and by a solution of (1) we mean u(t) = uη(t)η∈Λ satisfying (1)

for t in some interval I. Other conditions, for example boundedness of uη(t) in η for each

fixed t (meaning that u(t) belongs to the Banach space l∞(Λ) for each t) may be imposed.

Under quite general conditions, Eq. (1) together with an initial condition

uη(t0) = u0η, η ∈ Λ,

possesses a unique solution on some interval about t = t0, just as for ODE’s. In this article

we generally take Λ = ZZn, the integer lattice in IRn, although this is just for convenience.

Many of the ideas we describe can be easily extended to other lattices, for example the

hexagonal lattice in the plane, and the crystallographic lattices in IR3.

The study of LDE’s has a number of benefits. Besides the rich structure of solutions,

LDE’s are often appropriate models in many applications. In electrical circuit theory,

much work is due to Chua and his collaborators, in particular in their studies of Cellular

Neural Networks (CNN’s). See, for example, [Chua & Roska, 1993], [Chua & Yang,

1

19881], [Chua & Yang, 19882], and [Roska & Chua, 1993]; also see [Perez-Munuzuri et al.,

1993], [Perez-Munuzuri al., 1992], and [Thiran et al., 1995]. Material science, in particular

metallurgy, is another area of interest, where LDE’s have been used to model solidification

of alloys; see, [Cahn, 1960], [Cook et al., 1969], [Hillert, 1961]. Also, models arising in

chemical reactions [Erneux & Nicolis, 1993], optics [Firth, 1988], and biology [Ermentrout,

1992], [Ermentrout & Kopell, 1994], [Keener, 1987], [Keener, 1991], [Kopell et al., 1991],

and [Winslow et al., 1993], are a rich source of LDE’s. Much has been done on chains

of coupled oscillators, usually arising in biology or electronics (Josephson junctions); see,

for example, [Aronson et al., 1991], [Aronson & Huang, 1994], [Kopell & Ermentrout,

1990], [Kopell et al., 1990], [Matthews et al., 1991], [Mirollo & Strogatz, 1990], and the

numerous references therein. The study of coupled-map lattices, namely lattice systems

with discrete time, is closely related; see, for example, the works [Afraimovich & Nekorkin,

1994], [Afraimovich & Pesin, 1993], [Chow & Shen, 1995]. We mention that in addition

to the classes of LDE’s mentioned above, there is an extensive literature on integrable

Hamiltonian systems on lattices (most notably the Toda lattice). However, we do not

consider these systems in this paper.

Our interest here is in traveling and plane wave solutions, and in the existence and

stability of equilibrium solutions. In particular, we are interested in the propagation failure

and lattice induced anisotropy in traveling wave solutions, and also in pattern formation,

ordering, stability, and spatial entropy of stationary solutions. For a general survey of

results in this direction, see also [Chow & Mallet-Paret, 1995] and [Mallet-Paret & Chow,

1995].

2. Traveling Waves, Propagation Failure, Pinning, and Lattice Anisotropy.

Throughout this section we consider spatially discrete reaction diffusion systems of the

form

uη = α(∆nu)η − f(uη), η ∈ Ω ⊂ ZZn(2)

where α is a positive constant, and ∆n is the standard 2n + 1 point discretization of the

Laplacian, that is,

(∆nu)η =

∑

|ξ−η|=1

uξ

− 2nuη,

2

where |·| denotes the usual Euclidean norm in IRn. We observe that in dimensions n = 1, 2,

and 3 respectively, we have

(∆1u)i = ui+1 + ui−1 − 2ui,

(∆2u)i,j = ui+1,j + ui−1,j + ui,j+1 + ui,j−1 − 4ui,j,

(∆3u)i,j,k = ui+1,j,k + ui−1,j,k + ui,j+1,k + ui,j−1,k

+ui,j,k+1 + ui,j,k−1 − 6ui,j,k,

where we are denoting η ∈ ZZn by η = i, by η = (i, j), and by η = (i, j, k), respectively.

(We shall sometimes employ this notation for η.) Of course, Eq. (2) is the spatially discrete

analog of the PDE

ut(t, x) = α∆u(t, x) − f(u(t, x)), x ∈ Ω ⊂ IRn,(3)

where in (3) the subscript denotes the partial derivative, and where ∆ is the usual Lapla-

cian. Let us note that, in contrast to the PDE (3), the LDE (2) enjoys a local existence

and uniqueness theorem both for forward and backward time, irrespective of the sign of

α, as long as f : IR→ IR is locally Lipschitz and we take a bounded initial condition.

In cases where Ω ⊂ ZZn is bounded we will consider (2) with respect to appropriate

boundary conditions, typically of Neumann or periodic type. In many cases we take

Ω = ZZn and we take f of bistable type, both symmetric and nonsymmetric, as typified

by the cubic polynomial

f(u) = (u2 − 1)(u− a).(4)

The parameter a in (4) generally satisfies −1 < a < 1, and is known as the detuning

parameter. With this f , observe that for the ODE u = −f(u) which arises as the

spatially homogeneous system, the equilibria u = ±1 are stable, while u = a is an unstable

equilibrium.

2.1. Existence of traveling waves in one space dimension. Traveling wave so-

lutions are an important class of solutions both for PDE’s, and also for lattice dynamical

systems. Indeed, traveling waves have been observed quite generally in numerical works.

For example, in [Cahn et al., 19951] traveling interfaces between two patterns (such as

between areas of checkerboard with opposite phase) were observed. Chua has observed,

both numerically and in experimental (electronic) simulations, many quite exotic trav-

eling waves, including spiral waves; see, for example, [Perez-Munuzuri et al., 1993] and

3

[Perez-Munuzuri et al., 1992]. Spiral waves for a different class of lattice systems (coupled

oscillators) were constructed in [Paullet & Ermentrout, 1994].

For the PDE (3) with Ω = IRn, a traveling wave solution takes the form u(t, x) =

ϕ(σ · x − ct) for some function ϕ : IR → IR, where σ ∈ IRn is a unit vector, |σ| = 1,

representing the direction of motion of the wave, and where c ∈ IR is the wave speed.

Both the function ϕ, and the quantity c are unknown. In particular, c is not given in

advance, but must be determined, and it can be either zero or nonzero. Substitution of

the above traveling wave formula into (3) leads to the well-known second order ODE

− cϕ′(ξ) = αϕ′′(ξ) − f(ϕ(ξ)), ξ ∈ IR.(5)

Observe that both the dimension n and the direction σ are absent from Eq. (5). One

typically imposes boundary conditions

ϕ(−∞) = q−, ϕ(∞) = q+,(6)

so that the wave joins two equilibria q− < q+ which are stable for the ODE, that is,

with f(q±) = 0 and f ′(q±) > 0. (For the nonlinearity (4) these are q± = ±1 of course.)

With these conditions, one determines c, often uniquely, as the parameter for which the

boundary conditions can be realized. At least for f as in (4), and similar N-shaped

functions, there exists a unique (up to translation ϕ(ξ+ a) of ξ) solution (ϕ, c) to (5), (6);

see [Fife & McLeod, 1977].

By contrast, the situation in LDE’s is by no means as simple. Early work of Zinner

[Zinner, 1991], [Zinner, 1992] (see also [Hankerson & Zinner, 1993], [Zinner et al., 1993])

was concerned with theoretical issues of traveling waves for a class of one-dimensional

problems including (2) with n = 1 and Ω = ZZ. By a traveling wave solution of (2) in this

case, we mean a solution of the form

ui(t) = ϕ(i − ct),(7)

for some function ϕ : IR → IR, and some quantity c ∈ IR. Again, the wave speed c must

be determined as part of the solution. Substitution of (7) into (2) yields

− cϕ′(ξ) = α

(ϕ(ξ + 1) + ϕ(ξ − 1) − 2ϕ(ξ)

)− f(ϕ(ξ)), ξ ∈ IR,(8)

which, if c 6= 0, is a differential-difference equation. If c = 0, then (8) is a difference

equation, and only the discrete values ϕ(i) for i ∈ ZZ on the lattice are relevant. In any

4

case, just as for the PDE, one typically takes boundary conditions such as (6) for Eq. (8).

In [Hankerson & Zinner, 1993] existence of a solution pair (ϕ, c) to (8), (6), was proved

under the condition that α > 0, for fairly general f that included the cubic polynomial (4).

2.2. Pinning and propagation failure. The term propagation failure refers to

the inability of a solution (generally as a wave) to move along a lattice, and that moreover

this inability persists over an open set of parameters. For the systems (8), (6), arising

from (2), this means that one has wave speed c = 0 identically, as the detuning parameter

a ranges over a nontrivial interval a− ≤ a ≤ a+. This is a phenomenon peculiar to LDE’s,

which generally does not occur for PDE’s, and in some literature is known as pinning (one

thinks of the wave being pinned to the lattice). The foundations of propagation failure,

in the theoretical literature, go back to [Bell, 1981], [Bell & Cosner, 1984], and [Keener,

1987].

For the system (5), (6), arising from the PDE (3), with the nonlinearity (4), it is the

case that the unique c = c(a) is a smooth function of a ∈ (−1, 1), and satisfies

ac(a) > 0 for a 6= 0.

That is, no matter how close a is to zero, the resulting wave moves with nonzero speed

c(a) 6= 0 as long as a 6= 0, and so propagation failure does not occur. The situation

is decidedly different for LDE’s, where propagation failure is generally the rule. Indeed,

this can be seen from several points of view. First, note that when c = 0 the difference

equation (8), with ξ = i ∈ ZZ, can be written as a planar map

ϕ(i+ 1) = 2ϕ(i) − ψ(i) + α−1f(ϕ(i)),

ψ(i+ 1) = ϕ(i),(9)

where ψ(i) = ϕ(i − 1), and where the boundary conditions (6) mean that one seeks

a heteroclinic orbit joining the two saddle equilibria (ϕ,ψ) = (q−, q−) as i → −∞, to

(ϕ,ψ) = (q+, q+) as i → ∞. The map (9) is very similar to the well-known Henon map,

and it shares many of its features. In particular, one expects that for most values of the

parameters α > 0 and a ∈ (−1, 1), any such heteroclinic orbit would occur as a transverse

intersection of the stable and unstable manifolds of (q−, q−) and (q+, q+). As such, this

intersection would persist over an open subset of values of α and a, that is, c = 0 would

persist for some families of solutions of (8), (6), for an open set of α and a.

5

A second point of view is offered in [MacKay & Sepulchre, 1995]. Let a ∈ (−1, 1) be

fixed, and, beginning at α = 0, let the coupling parameter α be increased slightly. At

α = 0 one immediately has the solution

ϕ(i) =

q− for i < 0,

q+ for i ≥ 0,(10)

to (8), (6). A perturbation argument using the implicit function theorem in a Banach

space then shows that for small α > 0, there exists a solution to (8), (6), with c = 0,

which is near (10), and which varies smoothly with α and a over an open set of these

parameters. Again we have propagation failure.

2.3. Plane wave solution in higher space dimensions. For the higher-dimen-

sional LDE (2) with Ω = ZZn, by a traveling wave solution, we mean a solution u(t) =

uη(t)η∈ZZn of the form

uη(t) = ϕ(σ · η − ct),(11)

where again ϕ : IR→ IR and c is the (unknown) wave speed, and again with the unit vector

σ ∈ IRn, |σ| = 1, representing the direction of motion, given beforehand. Substitution the

traveling wave Ansatz (11) into Eq. (2) now gives

− cϕ′(ξ) = αLσϕ(ξ) − f(ϕ(ξ)), ξ ∈ IR,(12)

where Lσ is the difference operator

Lσϕ(ξ) =

(n∑

k=1

(ϕ(ξ + σk) + ϕ(ξ − σk))

)− 2nϕ(ξ),(13)

where we denote σ = (σ1, σ2, . . . , σn). In contrast to the case of the PDE (3), Eq. (12),

with (13), does depend on the direction of motion σ of the wave. This is of course due to

the underlying anisotropy of the lattice, which is absent in the PDE.

A clearer understanding of the effects of this anisotropy can be obtained by an analysis

of such traveling waves where f is the idealized piecewise linear function

f(u) = u− h(u− a) =

u− 1 if u > a,

u if u < a.(14)

Here h denotes the Heaviside function and a ∈ (0, 1) is the detuning parameter. We regard

this f as set-valued, with f(u) single valued for u 6= a, and multivalued, f(a) = [a− 1, a],

6

for u = a. Thus, we consider a differential equation with this piecewise linear f as

a differential inclusion. The function f in (14) is a cartoon of the cubic nonlinearity

f(u) = u(u − a)(u − 1), which is equivalent to (4) by a simple change of variables. One

hopes that the piecewise linear nature of (14) will serve to simplify, and make explicit,

the calculation of the traveling wave (ϕ, c). One hopes that at the same time the essential

structure of the problem will be preserved. This approach for the PDE was taken in

[McKean, 1970].

An extensive and detailed analysis of traveling wave solutions for (2), with f as in

(14), was carried out in [Cahn et al., 19953]. For Eq. (12) with boundary conditions

ϕ(−∞) = 0, ϕ(∞) = 1,(15)

we seek solutions for which ϕ(ξ) = a at only one value ξ = ξ0 of the argument, at least if c

turns out to be non-zero. Without loss we may assume ξ0 = 0 (by translating ξ → ξ− ξ0),and so ϕ(ξ) < a for ξ < 0, while ϕ(ξ) > a for ξ > 0. This implies that f(ϕ(ξ)) = ϕ(ξ)−h(ξ)for all ξ 6= 0, and so necessarily ϕ satisfies the inhomogeneous linear equation

− cϕ′(ξ) = αLσϕ(ξ) − ϕ(ξ) + h(ξ), ξ ∈ IR.(16)

Equation (16) with the boundary conditions (15) may now be solved by means of Fourier

transform techniques to determine explicitly the function ϕ. Indeed, one obtains

ϕ(ξ) =1

2+

1

π

∫ ∞

0

Aσ(s) sin sξ

s(Aσ(s)2 + c2s2)ds+

c

π

∫ ∞

0

cos sξ

Aσ(s)2 + c2s2ds,(17)

where

Aσ(s) = 1 + 2α

(n−

n∑

k=1

cos(σks)

).(18)

We must caution the reader at this point, that the analysis here is perhaps not as

straightforward as it may seem. While the above formulas (17), (18), are the result of a

fairly routine (albeit messy) calculation, their derivation is predicated on our assumption

that ϕ(ξ) = a at exactly one value of ξ ∈ IR. This assumption must be verified a posteriori,

for the formula (17). It is in fact the case that the function ϕ given in (17) is strictly

monotone, in fact ϕ′(ξ) > 0 for all ξ 6= 0. With this the desired assumption is verified.

However, is not at all obvious from the formula (17) that ϕ is monotone, and it is by no

means easy to establish this fact.

7

We required in our derivation that ϕ(0) = a, and substituting this into (17) determines

the relation between the wave speed c, and the parameter a, namely

a− 1

2=c

π

∫ ∞

0

ds

Aσ(s)2 + c2s2=: Γσ(c).(19)

The function Γσ is well-defined for c 6= 0, and is clearly an odd function. It is shown in

[Cahn et al., 19953], at least in the case of dimension n = 2, that Γσ(c) depends smoothly

on c for c 6= 0, and is strictly increasing there, in fact that Γ′σ(c) > 0 for all c 6= 0. Also in

[Cahn et al., 19953] the limits

limc→∞

Γσ(c) =1

2, γσ(α) := lim

c→0+Γσ(c) = lim

T→∞1

2T

∫ T

0

ds

Aσ(s),(20)

are established, where γσ(α) > 0. It follows that if γσ(α) < |a− 1/2| < 1/2, then there is

a unique c 6= 0 for which a = 1/2+Γσ(c), that is, a unique nonzero wave speed. A further

analysis of the system (12) carried out in [Cahn et al., 19953] shows that if

|a− 1/2| ≤ γσ(α)(21)

then (12), (15), possesses a solution with c = 0 which is nondecreasing in ξ. Moreover,

this solution is not unique if |a− 1/2| < γσ(α).

Let us summarize some of these facts in the following theorem, proved for n = 2 in

[Cahn et al., 19953].

Theorem 2.1. Consider the system (2), with the nonlinearity (14). For γσ(α) <

|a − 1/2| < 1/2 there exists a unique c ∈ IR for which (12), (15), has a solution passing

through the value ϕ(ξ0) = a exactly once, at ξ0 = 0. Moreover c 6= 0, with a = 1/2+Γσ(c),

and ϕ is given by (17) and satisfies ϕ′(ξ) > 0 for all ξ 6= 0.

If |a−1/2| ≤ γσ(α) then (12), (15), possesses no such solution with c 6= 0, but instead

for c = 0 possesses a solution ϕ for ξ ∈ ZZ, which is nondecreasing.

The function Γσ(c) depends smoothly on c for c 6= 0, and we have Γ′σ(c) 6= 0 there,

with the limits (20).

Define the set D by

D = σ · η | η ∈ ZZn.

If the coordinates σk of σ span a one-dimensional subspace over the rationals (equivalently,

if all σk are rational multiples of some common nonzero quantity), then there exists ν > 0

8

such that

D = mν |m ∈ ZZ,(22)

If, on the other hand this is not true, then σj and σk are linearly independent over the

rationals for some j 6= k, and then the set D is dense in IR. In [Cahn et al., 19953] the

following was proved for n = 2.

Theorem 2.2. Assume the setting of Theorem 2.1. Let cm 6= 0 with cm → 0,

and let ϕm denote the solution of (12), (15), with c = cm, for am = 1/2 + Γσ(cm), as in

Theorem 2.1. Then the limit

limm→∞

ϕm(ξ) = ϕ∗(ξ)

exists for each ξ ∈ IR \ D, and is independent of the sequence cm for such ξ. The limits

ϕ∗(ξ) → 0, 1, as ξ → −∞, ∞, respectively, hold, the nondecreasing function ϕ∗ is constant

on each connected component of IR \D, and ϕ∗ suffers a nontrivial jump at each ξ ∈ D,

that is, the left- and right-hand limits of ϕ∗ at each ξ ∈ D differ.

By Theorem 2.2 then, if the coordinates σ1, σ2, . . . , σn of σ are rationally related, so

that (22) holds for some ν > 0, then ϕ∗ is a step function which is constant on each interval

(kν, (k + 1)ν), with jumps at each ξ = kν. If on the other hand σj/σk is irrational for

some j and k, then ϕ∗ is a strictly increasing function, with a dense set of discontinuities,

namely the points ξ ∈ D.

The interval (21) thus determines the range of a for which the wave speed is zero,

that is, the range of propagation failure. One may explicitly calculate this interval, by

calculating the quantity γσ(α). Denoting

A∗(s1, s2, . . . , sn) = 1 + 2α

(n−

n∑

k=1

cos sk

),

we see that Aσ(s) = A∗(σ1s, σ2s, . . . , σns). The function 1/A∗, which is analytic for each

sk real, can be written as an absolutely convergent Fourier series

1/A∗(s1, s2, . . . , sn) =∑

η∈ZZn

βηei(η1s1+η2s2+...+ηnsn),

where βη > 0 for each η ∈ ZZn. Thus, one sees directly that

γσ(α) =1

2

∑

σ·η=0

βη,(23)

9

so the range of propagation failure depends (see [Hale, 1980]) on the rational dependence

or independence of the coordinates σk of σ.

It is a direct consequence of the formula (23) and the positivity of the βη, that γσ(α) de-

pends continuously on σ at precisely those values σ at which the coordinates σ1, σ2, . . . , σn

are rationally independent, and is discontinuous at all other points. Thus, σ → γσ(α) is

discontinuous at a dense set of σ ∈ Sn−1, and is continuous at a dense set of σ ∈ Sn−1,

where Sn−1 ⊂ IRn denotes the unit sphere.

In Fig. 1, we plot some typical graphs of a = 1/2+Γσ(c). As one can see, there is a large

range of a values for which the wave speed is zero, that is, where propagation failure occurs.

Also, note the dependence of these graphs on σ, which is a reflection of the lattice induced

anisotropy. For Curve 1, (σ1, σ2, σ3) = (1, 0, 0), for Curve 2, (σ1, σ2, σ3) = ( 1√2, 1√

2, 0), for

Curve 3, (σ1, σ2, σ3) = ( 1√3, 1√

3, 1√

3), for Curve 4, (σ1, σ2, σ3) = (1

2 ,12 ,

1√2), and for Curve

5, (σ1, σ2, σ3) = ( 1√2, 1√

3, 1√

6). In Figs. 2 and 3 we plot waves forms obtained for α = 1

2

and the same values of σ used in Fig. 1. In Fig. 2 we choose the wave speed c = 0.01,

while for Fig. 3 we plot the limiting function ϕ∗, that is, with c = 0. Note the agreement

with Theorem 2.2 in Fig. 3.

For more extensive calculations in this direction, see [Elmer & Van Vleck, 1996].

2.4. Existence and uniqueness of traveling wave solutions for smooth non-

linearities via global continuation. While the results above for the idealized nonlin-

earity (14) may appear to be largely a matter of brute force computation, there is in fact

a significant amount of general theory that must be developed. For example, the proof

of the monotonicity Γ′σ(c) > 0 of the relation between c and a is proved by a Mel’nikov

technique, rather than by differentiating the explicit formula (19). Roughly, one regards

the equation (12), (13), as an operator equation in an appropriate function space, and

takes Frechet derivatives with respect to c and a. An application of the implicit function

theorem in a Banach space then provides not only the relation c = c(a), but also the

desired derivative c′(a) > 0.

In [Mallet-Paret, 1996], such an approach is in fact carried over to the case of quite

general smooth f , including N-shaped nonlinearities such as the cubic polynomial (4). If,

for such f , the system (12), (13), with the boundary conditions (6) possesses a solution

ϕ for some c 6= 0, then in fact ϕ′(ξ) > 0 for all ξ ∈ IR. Moreover, the solution pair

(ϕ, c) is locally unique, and varies smoothly with any parameters (including the detuning

10

0

0.2

0.4

0.6

0.8

1

-4 -3 -2 -1 0 1 2 3 4

a

c

Curve 1Curve 2Curve 3Curve 4Curve 5

Fig. 1. Plots of c versus a for α = 1

2and various values of σ.

parameter) in the system. In fact, if ∂f/∂a > 0 between q− and q+, then c′(a) > 0 for

the derivative of c = c(a), just as above. Finally, this local continuation of the solution

can be extended (in the range where c = 0) to a global continuation, to prove existence

and uniqueness of solutions (ϕ, c). Such an approach is in the spirit of [Bates et al., 1995],

[Ermentrout & McLeod, 1993], and [Chow et al., 1989].

The technical matters which must be resolved in obtaining such results rest on estab-

lishing a Fredholm alternative for a class of linear differential-difference equations, of the

form

ψ′(ξ) =k∑

i=1

Ai(ξ)ψ(ξ + ri),(24)

where the ri are of either sign, and the equation is asymptotically autonomous at ξ → ±∞,

with a hyperbolic limit.

2.5. Speed of interfaces. We now consider the spatially discrete reaction diffusion

equation (2) in one space dimension with the cubic nonlinearity (4) for a = 0. In this

11

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-4 -2 0 2 4

Waveform 1Waveform 2Waveform 3Waveform 4Waveform 5

Fig. 2. Plots of waveforms for α = 1

2, c = 0.01, and various values of σ and a.

case there are only standing wave solutions in both the spatially discrete and spatially

continuous equations. We consider (2) on the finite domain Ω = 0, ..., N where N is a

positive integer with the discrete analog of Neumann boundary conditions so that

u−1 = u0, uN = uN+1.(25)

If we normalize the problem to the unit interval, then 1/N may be thought of as the

spacing between meshpoints, and we define ε as α = N 2ε2.

It was shown in [Carr & Pego, 1989], [Fusco & Hale, 1989], [Bronsard & Kohn, 1990]

that for one space dimension reaction diffusion PDE’s with the cubic nonlinearity (4) for

a = 0 there exist solutions that evolve at exponentially slow rates. This is important

because although these types of solutions are not equilibrium solutions their motion is

exponentially slow and hence the solution does not change form on very long time scales.

In [Grant & Van Vleck, 1995] the speed of motion of interfaces for (2) for n = 1 was

considered (related work appears in [Estep, 1994]. We summarize here the results that

were obtained in [Grand & Van Vleck, 1995]. They involve first finding bounds on the

12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-4 -2 0 2 4

Waveform 1Waveform 2Waveform 3Waveform 4Waveform 5

Fig. 3. Plots of waveforms for α = 1

2, c = 0, and various values of σ and a.

speed of motion, that are analogous to the exponential rates in the continuous case, and

then giving criteria that ensure that the solution is in fact evolving.

In order that our results may be stated precisely, we introduce some notation and

terminology. Given a continuous function u : [0, 1] → IR, let Z(u) be the set of zeros of u.

For u : Ω → [−1, 1] we let Z(u) be the set of zeros of the piecewise linear interpolation of

the data (i/N, ui) | i = 0, . . . , N. We will measure the speed of motion by determining

the distance between the zeros of solutions at different instances of time. To determine

the distance between solutions we let d(·, ·) be the Hausdorff distance between sets; that

is,

d(A,B) = max

supa∈A

d(a,B), supb∈B

d(b, A)

.

Let v : [0, 1] → −1, 1 be piecewise constant with precisely k discontinuities at the points

x1, x2, . . . , xk ⊂ (0, 1), and let v be its spatial discretization, that is, vi = v(i/N). In

this way v denotes a canonical initial condition with k transition layers.

The following theorem obtained in [Grant & Van Vleck, 1995] provides bounds on the

13

speed of motion of interfaces of (2).

Theorem 2.3. There exists discretized initial data u(0) within O(√ε) of v that gen-

erate solutions u(t) to (2) with transition layers moving so slowly that the time necessary

for d(Z(v), Z(u(t))) to exceed a fixed value ρ is greater than

1

C1ε

(1 +

C2

Nε

)C3N−C4

.

Theorem 2.3 provides bounds on the speed of motion while the following theorem

proved in [Grant & Van Vleck, 1995] provides criteria to ensure that a solution is not in

fact pinned, that is, to ensure that a solution is not an equilibrium. In the presentation of

this result, we say that ui is a local maximum (minimum) if ui ≥ (≤)uj for all lattice

sites j adjacent to i.

Theorem 2.4. Let u be a nonconstant equilibrium of (2), and let ui and uj be a

local minimum and a local maximum, respectively. If α ≥ 12 then

|i− j| ≥⌊

π

cos−1(1 − (2α)−1)

⌋− 1.

Consequently, the transition layers of a solution u to (2), (25), whose speed is bounded by

Theorem 2.3, cannot become pinned (at least) until every interval of length

⌊π

cos−1(1 − (2α)−1)

⌋− 1

contains no more than two members of 0, N ∪NZ(u).

Remark. Results analogous to those obtained in Theorems 2.3 and 2.4 have been

obtained in [Grant & Van Vleck, 1995] for one-dimensional spatially discrete Cahn-Hilliard

equations.

2.6. Stability and coordinate systems for traveling waves. For the PDE (3),

one can study the stability, and other dynamical properties, of a traveling wave, by intro-

ducing a coordinate system which moves with the wave. That is, in place of x ∈ IR, one

considers ξ = x− ct (say in one dimension), so that (3) becomes

vt(t, ξ) = ∆v(t, ξ) + cvξ(t, ξ) − f(v(t, ξ)),

14

where v(t, ξ) = u(t, ξ + ct). In this new system, the traveling wave u = ϕ(x − ct) has

become an equilibrium solution v = ϕ(ξ), and standard local theory can be invoked.

Obviously, such an approach does not work for waves on a lattice. Nevertheless, in

[Chow et al., 1996] a moving coordinate system on a lattice, analogous to the above, is

constructed, at least on the one-dimensional lattice. Suppose for some c 6= 0 that (7),

satisfying appropriate boundary conditions, is a solution to (2) with n = 1. Denoting the

infinite vector (considered as an element of the Banach space l∞) by u(t) = ui(t)i∈ZZ =

ϕ(i − ct)i∈ZZ , and letting τ = c−1 we have for this solution that u(t + τ) = S−10 u(t),

where S0 : l∞ → l∞ denotes the shift operator (S0v)i = vi+1.

Now take any smooth curve S : IR → GL(l∞) such that S(0) = I and S(t + τ) =

S(t)S0, where GL(l∞) is the space of all linear isomorphisms of l∞. The existence of such

S(t) is a consequence of the connectedness of GL(l∞); see [Edelstein et al., 1970]. With the

change of variables v = S(t)u, the traveling wave u(t) becomes a τ -periodic solution v(t)

of an τ -periodic system. Existing theory of periodic solutions of ODE’s in a Banach space,

including linearized stability theory, invariant manifold theory, and bifurcation theory, can

now be applied.

3. Equilibrium Solutions. In this section we consider the LDE

uη = −β+(∆+u)η − β×(∆×u)η − f(uη), η ∈ ZZ2,(26)

on the two-dimensional lattice ZZ2, where β+ and β× are real parameters. The Laplacian

type operators ∆+ and ∆× are given by

(∆+u)η =

∑

ξ∈N+(η)

uξ

− 4uη, (∆×u)η =

∑

ξ∈N×(η)

uξ

− 4uη ,

where N+(η), N×(η) ⊂ ZZ2, with η = (i, j), are given by

N+(i, j) = (a, b) ∈ ZZ2 | |a− i| + |b− j| = 1,N×(i, j) = (a, b) ∈ ZZ2 | |a− i| = |b− i| = 1.

Thus N+(η) denotes the set of nearest neighbors of η, and N×(η) denotes the set of

next-nearest neighbors of η. Observe that if β× = 0 then Eq. (26) reduces to Eq. (2) on

Ω = ZZ2 with α = −β+. We will also consider the associated spatially discrete Cahn-

Hilliard equation (see [Cahn et al., 19951])

uη = −(∆+ + ∆×)

(− β+(∆+u)η − β×(∆×u)η − f(uη)

).(27)

15

We write (26) and (27) with minus signs in front of the terms involving the Laplacians.

In one respect this is mere notation: in this section, the coefficients β+ and β× can be

of either sign, positive or negative, and so no restriction is imposed. Nevertheless some

of the most interesting phenomena of LDE’s arise for positive β+ and β×, as it is here

that there is no continuum analog, at least as a well-posed PDE. Indeed, one encounters

completely new phenomena which have no counterpart in PDE’s, when β+ and β× are

positive, and much of our analysis is focused in this direction.

We take f (see [Elliott et al., 1994] and [Oono & Puri, 1988]) to be the so-called

“double-obstacle” function, namely the set-valued function

f(u) =

γu if |u| < 1,

φ if |u| > 1,f(1) = [γ,∞), f(−1) = (−∞,−γ],(28)

which is linear for |u| < 1, with vertical lines in the “graph” of f at u = ±1. Again,

solutions to (26) or (27), with f as in (28), are interpreted as differential inclusions. In

a metallurgical context such f represents deep quenching; that is, for T > 0 and Tc ∈ IR

the function f can be thought of as an approximation to

g(u) = log

(1 + u

1 − u

)+ (Tc/T )u(29)

when T/|Tc| 1. The parameter γ can have either sign. We think of f in (28) as a cartoon

of the symmetric cubic nonlinearity f(u) = γu+ u3, which has a bistable character when

γ < 0, and a monostable character when γ > 0. For the choice of positive β+ and

β×, one finds the richest array of new phenomena by taking γ > 0, as this makes for

a competition: the Laplacian with negative coefficients −β+ and −β× tends to separate

values of neighbors uη and uξ for ξ ∈ N+(η), N×(η) (“negative diffusion”), while the

nonlinearity f draws them toward the stable origin. The outcome of this competition is

decided by the relative sizes of β+, β×, and γ, in a way that we make precise below.

Let us remark here that the usual existence and uniqueness theorem does not apply

to (26) with the nonlinearity (28). Nevertheless, existence and uniqueness was proved in

[Chow et al., 19951] for forward time t ≥ t0 only, irrespective of the signs of β+ and β×.

In general existence and uniqueness fail for backward time with the nonlinearity (28).

3.1. Mosaic solutions. For the nonlinearity (28), no matter what the signs of β+,

β×, and γ, it is natural to seek equilibrium solutions u of (26) for which uη ∈ −1, 0, 1for all η ∈ ZZ2. For this, we make some formal definitions.

16

Definition. An n-mosaic, or simply a mosaic, is a mapping u : ZZn → −1, 0, 1,that is, an assignment of −1, 0, or 1, to each point of the lattice ZZn. We let Mn denote

the set of all n-mosaics.

Definition. A mosaic solution of (26), (28) is a 2-mosaic u ∈ M2 which is an

equilibrium solution of this system.

In order to characterize mosaic solutions we employ the quantities

σ•η =∑

ξ∈N•(η)

uξ, τ•η = 4 − uησ•η ,

z•η = cardξ ∈ N •(η) | uξ = 0,

associated to any u ∈ M2. Here and below • denotes either + or ×. The following result,

whose proof is mostly a matter of checking definitions, is given in [Chow et al., 19951].

Theorem 3.1. A mosaic u ∈ M2 is a solution of (26) with (28) (that is, is a mosaic

solution) if and only if

σ+η β

+ + σ×η β× = 0 whenever uη = 0,(30)

and also

τ+η β

+ + τ×η β× ≥ γ whenever uη = ±1.

The next result is also given in [Chow et al., 19951]. However, its proof is not as

simple, and is based on comparison arguments for the differential equation. By a stable

equilibrium we mean an equilibrium solution which is stable in the usual Lyapunov

sense, with respect to perturbations in the initial condition, measured with respect to the

supremum (l∞) norm ‖u‖ = supη∈ZZ2 |uη| over the lattice.

Theorem 3.2. If a mosaic solution u, as in Theorem 3.1, satisfies in addition the

strict inequality

τ+η β

+ + τ×η β× > γ whenever uη = ±1,(31)

17

and also satisfies

(4 + z+η sgn(β+))β+ + (4 + z×η sgn(β×))β× < γ whenever uη = 0,(32)

then u is a stable equilibrium.

Definition. A mosaic u ∈ M2 satisfying the conditions of Theorem 3.2, is called an

S-solution. We let S(β+, β×, γ) ⊂ M2 denote the set of all S-solutions.

With Theorem 3.2, the classification of all S-solutions u ∈ M2 now reduces to con-

sideration of a finite (but large) number of cases. More precisely, for any mosaic u ∈ M2

the quantities σ• = σ•η and τ• = τ•η , and also z• = z•η , are integers in the range

−4 ≤ σ• ≤ 4, 0 ≤ τ • ≤ 8, 0 ≤ z• ≤ 4.

For a fixed γ, the lines determined by replacing the inequalities in (31) and (32) with

equalities, together with the equality (30), partition the (β+, β×)-parameter space into

finitely many pieces. For convenience, let us avoid values (β+, β×) lying on the 25 lines

σ+β+ + σ×β× = 0 for −4 ≤ σ+, σ× ≤ 4 and (σ+, σ×) 6= (0, 0), that is, let us restrict

(β+, β×) to the open dense complement of these lines. In this case, necessarily σ+η = σ×η =

0 for any solution of the equation in (30). Consider now, with γ 6= 0 fixed, the 92 − 1 = 80

lines τ+β+ + τ×β× = γ for 0 ≤ τ+, τ× ≤ 8 and (τ+, τ×) 6= (0, 0). It is shown in [Chow

et al., 19951] that these lines partition the plane into exactly 2041 polygonal regions, each

representing a different set of pairs (τ+, τ×) and (z+, z×) that are permitted to occur in

an S-solution. In each such region, the set S(β+, β×, γ) of S-solutions is independent of

β+ and β×. Some of these regions, for γ = 1, are shown in Fig. 4.

The simplest nontrivial mosaics are stripes, ui,j = (−1)i and ui,j = (−1)j , and the

checkerboard ui,j = (−1)i+j . For either vertical or horizontal stripes one checks that

(τ+i,j, τ

×i,j) = (4, 8), for all (i, j) ∈ ZZ2, while for the checkerboard one has (τ+

i,j , τ×i,j) = (8, 0)

for all (i, j). Thus if 4β+ + 8β× > γ, both the vertical and the horizontal stripe mosaics

are stable equilibria of (26), (28), while if 8β+ > γ then the checkerboard is a stable

equilibrium.

In Figs. 5 and 6 we illustrate other stable mosaic solutions that exist for nonempty

polygonal sets in (β+, β×) parameter space. We denote

= −1, = 0, = 1.

18

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Fig. 4. All regions in (β+, β×) parameter space contained in the square [−1, 1] × [−1, 1] for γ = 1.

(a) (b)

Fig. 5. Checkerboard type mosaic solutions.

Fig. 5 depicts checkerboards with various interfaces, while in Fig. 6 we see stripes with

interfaces. The range of parameters for which these are S-solutions can be determined

from Theorem 3.2, and was given in [Chow et al., 19951].

19

(a) (b)

Fig. 6. Striped type mosaic solutions.

3.2. Simulations of spatially discrete equations. In [Cahn et al., 19951] numer-

ical results were obtained for a spatially discrete Cahn-Hilliard equation (27) on a finite

square lattice with the discrete analogue of periodic and Neumann boundary conditions,

and with the nonlinearity f replaced with g as in (29). In Fig. 7 are snapshots of simula-

tions of the time varying solutions u(t) performed on a Connection Machine. These are not

equilibrium solutions, but rather belong to orbits u(t) which are approaching equilibria as

t→ ∞. Nevertheless, there is in some cases a notable resemblance to the stable equilibria

given by Theorem 3.2. Similar results appear in the papers [Cahn & Van Vleck, 1995] and

[Cahn & Van Vleck, 1996] on the existence of quadrijunctions (see Fig. 7(a)). The value

Tc/T in (29) is denoted by γ in Fig. 7.

3.3. Bifurcation phenomena. To understand how regular patterns appear in such

models with smooth f , bifurcation theory is used in [Cahn et al., 19952] to provide a

rigorous analysis. Although bifurcation theory deals with local phenomena, we shall see

that already it reveals a complex and rich structure.

Let us return to the simpler system (26). (Note that any equilibrium of (26) is also

an equilibrium of (27).) Assume that f(0) = 0, that f is odd, let γ = f ′(0), and define

g : IR→ IR by

f(u) = γu+ g(u),

20

(γ = 8.0, β+ = −0.5, β× = 2.0) (γ = −2.1, β+ = 0.25, β× = −0.5)

(a) (b)

(γ = 4.0, β+ = −1.0, β× = 2.0) (γ = −1.0, β+ = 0.25, β× = −0.5)

(c) (d)

Fig. 7. Simulations of a spatially discrete Cahn-Hilliard equation.

so that g(0) = g′(0) = 0 and g is odd. We assume for definiteness that γ > 0. Consider

now solutions u : ZZ2 → IR which have spatial period 2 in both the horizontal and vertical

directions, that is, belong to the set

P = u : ZZ2 → IR | ui,j = ui+2,j = ui,j+2 for all (i, j) ∈ ZZ2.

Clearly, P is a four-dimensional vector space which is invariant under the flow of (26). A

21

Fig. 8. The bifurcation diagram near (γ/8, γ/16), indicating the number of local equilibria.

natural coordinate system (w, x, y, z) ∈ IR4 is given by writing any u ∈ P as

ui,j = (−1)i+jw + (−1)jx+ (−1)iy + z,

and we observe that for any nonzero scalar quantities w, x, y, and z, that

ui,j = (−1)i+jw is a checkerboard,

ui,j = (−1)jx are horizontal stripes,

ui,j = (−1)iy are vertical stripes, and

ui,j = z is a constant pattern.

In these coordinates, Eq. (26) restricted to P becomes

w = (8β+ − γ)w + g1,

x = (4β+ + 8β× − γ)x+ g2,

y = (4β+ + 8β× − γ)y + g3,

z = −γz + g4,

(33)

where gk = gk(w, x, y, z), for 1 ≤ k ≤ 4, are higher order terms, that is, terms at least

quadratic (in fact cubic) near the origin, and which depend only on g. At the point

(β+, β×) = (γ/8, γ/16), the linearization of (33) at the origin has three zero eigenval-

22

ues and one negative eigenvalue −γ. One concludes that there exists locally a three-

dimensional attracting center manifold

z = Φ(w, x, y, β+, β×)(34)

through (w, x, y, z) = (0, 0, 0, 0) near (β+, β×) = (γ/8, γ/16), which slaves the variable z

to the other variables. Upon inserting (34) into the first three equations of (33), we obtain

a three-dimensional system. In particular, the methods of bifurcation theory yield the

existence of many equilibrium solutions, both stable checkerboards and stripes, as well as

more complex solutions which are hybrids of these, and which in fact are always unstable.

If, for example, g′′′(0) > 0, as with g(u) = u3, one finds pure checkerboards for 8β+ > γ,

and both horizontal and vertical stripes for 4β+ + 8β× > γ, at least near the bifurcation

point.

Fig. 8 depicts the complete bifurcation diagram near (β+, β×) = (γ/8, γ/16), with the

number in each sector denoting the number of local equilibria that have bifurcated from

u = 0. Of the many different types of local equilibria, the checkerboards and stripes are

the only stable ones, and only for certain sectors in Fig. 8.

With further analysis, it is possible to describe the dynamics of the three-dimensional

ODE in terms of orbits connecting the various equilibria. The system (33) possesses

a global Lyapunov function, so all solutions tend to equilibria as t → ∞, at least if

u−1f(u) → ∞ as u → ±∞. This gives a rather complete though complicated picture of

the dynamics on the center manifold.

3.4. Spatial entropy – theory and numerics. Here we define the spatial entropy

of the set S(β+, β×, γ) of S-solutions. We then use this definition to give a rigorous

description of the concepts of spatial chaos, and pattern formation.

We begin abstractly by considering any non-empty set V ⊂ M2 of mosaics, which we

assume is translation invariant, that is, SH(V) = SV (V) = V, where SH , SV : M2 → M2

denote the horizontal and vertical shifts of a mosaic. We shall eventually take V =

S(β+, β×, γ). For each pair of positive integers (a, b), let ca,b denote the number of different

patterns that one observes in V, by viewing the elements in this set through a window of

size a× b in the lattice ZZ2. That is, for each u ∈ V we restrict ui,j to the range 0 ≤ i < a

and 0 ≤ j < b, thereby giving a mosaic on the finite lattice Fa,b = (i, j) ∈ ZZ2 | 0 ≤ i <

23

a and 0 ≤ j < b of ab points. The number ca,b is simply the number of such finite-lattice

mosaics that one obtains from all elements u ∈ V. The translation invariance of V implies

that only the size of the a × b window, and not its position in the infinite lattice ZZ 2,

determines the number ca,b.

Certainly, we have that 0 < ca,b ≤ 3ab. We define the spatial entropy h = h(V) of

the set V to be the limit

h(V) = limm,n→∞

1

ablog ca,b,(35)

which can be shown always to exist. In fact, each of the terms in the right-hand side of

(35) is an upper bound for h, and so

h(V) = infa,b≥1

1

ablog ca,b.

It is clear that one always has 0 ≤ h ≤ log 3.

The above construction holds quite generally for the n-dimensional lattice ZZ n, and

for arbitrary finite alphabets A (above we have A = −1, 0, 1). For n ≥ 2 there is no

way in general to calculate h, however, when n = 1 and the set V is a Markov shift then

h can be explicitly given. A Markov shift or a subshift of finite type on an alphabet

A = 1, 2, . . . , d of d elements is determined by a d× d Boolean matrix B, the so-called

transfer matrix. By definition, u = uii∈ZZ belongs to V if and only if bui,ui+1= 1

for each i, for the relevant entry of B = (bj,k). With V so defined, the entropy V is

h(V) = log λ, where λ is the largest eigenvalue of the matrix B.

For each choice of (β+, β×, γ) we may take V = S(β+, β×, γ), the set of all S-solutions,

and calculate its entropy h(S(β+, β×, γ)). This leads to a convenient definition, whereby

we distinguish between two opposite types of behavior.

Definition. For a choice (β+, β×, γ) of parameters, we say that the system (26), (28),

exhibits pattern formation in case h(S(β+, β×, γ)) = 0, and we say it exhibits spatial

chaos in case h(S(β+, β×, γ)) > 0.

Pattern formation means there are relatively few stable equilibria, and so the spatial

variations that they display are limited. Spatial chaos, on the other hand, means that a

wide variety of stable disordered patterns occur.

There is no simple formula for calculating h(S(β+, β×, γ)). Nevertheless, rigorous

estimates of the entropy are made in [Chow et al., 19951], valid over a wide range of

24

parameter space, and sufficient conditions both for pattern formation, and for spatial

chaos, are given. As an illustration of the approach, suppose β+ > γ and β× > γ. Then

by Theorem 3.2 the set S(β+, β×, γ) consists precisely of all mosaics for which ui,j = ±1

and (τ+i,j , τ

×i,j) 6= (0, 0) for all (i, j). Equivalently, ui,j = ±1 for all (i, j), and neither of the

3 × 3 blocks

(36)

occurs anywhere in the mosaic u. Since all of the 29 possible 3×3 arrays of ±1 are allowed

in the mosaic except for the above two, we have that c3,3 = 29 − 2. This gives the upper

bound

h(S(β+, β×, γ)) ≤ 1

9log c3,3 =

1

9log(29 − 2) ≈ (.99937) log 2.

A lower bound on h(S(β+, β×, γ)) can be obtained by considering all tilings of a (2a) × b

sub-lattice with the two “bricks” of the form

and

It is impossible for the forbidden patterns (36) to occur anywhere in such a mosaic, and

so the number 2ab of such tilings of F2a,b is a lower bound for c2a,b, that is, c2a,b ≥ 2ab.

With (35), this gives

h(S(β+, β×, γ)) ≥ 1

2log 2.

In fact, it is possible to make a numerical estimate of h, which in this case is

h ≈ (.9904) log 2.

To numerically determine the spatial entropy h(S(β+, β×, γ)) in general, we approxi-

mate the system (26) on ZZ2 with a related system on an infinite strip 0, 1, 2, . . . , a−1×ZZof width a. We employ the “a-corkscrew” boundary conditions so that u(i + a, j) =

u(i, j + 1) for all (i, j) ∈ ZZ2. A basic configuration consists of 2a+ 2 contiguous locations

that we label r1, r2, . . . , r2a+2, where r1 = (i, j), r2 = (i − 1, j), . . . , r2a+2 = (i− 2a− 1, j)

modulo the boundary conditions. The transfer matrix, B, is a boolean matrix where the

(p, q)-entry of B is 1 if there exists a transition from the pth configuration of r2, . . . , r2a+2, s

25

Fig. 9. Approximate spatial entropy in the (β+, β×) plane for γ = 1.

where s = (i − 2a − 2, j) modulo the boundary condition to the qth configuration of

r1, . . . , r2a+2.

If we let ca,b denote the number of S-solutions satisfying a-corkscrew boundary con-

ditions, then we approximate the spatial entropy by

ha = limb→∞

1

ablog ca,b = log λ(a)

where λ(a) is the largest eigenvalue of the transfer matrix defined using a-corkscrew bound-

ary conditions. We note that this approximation to the spatial entropy gives us neither an

upper bound nor a lower bound. The approximation error given because ca,b ≤ ca,b “gives

a lower bound,” while not taking the limit a→ ∞ “gives an upper bound.” We determine

the dominant eigenvalue for increasing values of a, the width of the infinite strip, to obtain

the sequence λ(a). The power method is used to find each λ(a). In Fig. 9 we illustrate

approximations to the spatial entropy obtained in [Chow et al., 19952] for γ = 1. In Fig.

9 Blue is used to denote low entropy or pattern formation and Red is used to denote high

entropy or spatial chaos.

4. Acknowledgments. Many of the authors’ theoretical and numerical results pre-

sented here were obtained in various collaborations with V.S. Afraimovich, J.W. Cahn,

26

C.E. Elmer, C.P. Grant, and W. Shen.

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