Existence and Multiplicity of Traveling Waves in
a Lattice Dynamical System
Cheng-Hsiung Hsu1
Department of Mathematics, National Central University, Chung-Li 320, Taiwan E-mail: chhsumath.ncu.edu.tw
and Song-Sun Lin2
Department of Applied Mathematics, National Chiao Tung University, Hsin-Chu 30050, Taiwan Received February 17, 1999; revised September 20, 1999
This work proves the existence and multiplicity results of monotonic traveling wave solutions for some lattice differential equations by using the monotone itera-tion method. Our results include the model of cellular neural networks (CNN). In addition to the monotonic traveling wave solutions, non-monotonic and oscillating traveling wave solutions in the delay type of CNN are also obtained. 2000 Academic Press
AMS Subject Classifications: 34A12; 34B15; 34C35; 34K10.
Key Words: traveling wave; discrete dynamical system; cellular neural networks.
I. INTRODUCTION
This work studies the existence and multiplicity of traveling wave solutions of lattice differential equations. As generally considered, lattice differential equations are infinite systems of ordinary differential equations on a spatial lattice, such as the D-dimensional integer lattice ZD. Lattice differential
equa-tions arise from many system models such as in chemical reaction theory [14, 24], biology [2, 3], image processing and pattern recognition [11, 13], and material science [4].
An underlying motivation for studying the lattice differential equations is the large array of a locally coupled first-order nonlinear dynamical system, i.e., Cellular Neural Networks (CNN). Proposed by Chua and Yang [12, 13], such an information processing system is occasionally referred to as CY-CNN.
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0022-039600 35.00
Copyright 2000 by Academic Press All rights of reproduction in any form reserved.
1Work partially supported by NSC88-2115-259-003. 2Work partially supported by NSC88-2115-M009-001.
Indeed, cellular neural networks (CNN) without input terms are of the form
dxi, j
dt =&xi, j+z+|k| d, |l | d: ak, lf (xi+k, j+l) (i, j) # Z
2 (1.1)
or
dxi
dt=&xi+z+ :|l | dalf (xi+l) i # Z
1. (1.2)
Here the nonlinearity f is an output function and a piecewise-linear func-tion in CY-CNN. The quantity z is called a threshold or bias term and the numbers ak, lcan be arranged into the (2d+1)_(2d+1) matrix A which
is called a space-invariant template.
The study of traveling wave solutions can proceed as follows. Let % # R1
be given and consider solutions of (1.1) or (1.2) of the form
xi, j(t)=x(i cos %+ j sin %&ct) or xi(t)=x(i&ct) (1.3)
for some unknown function x: R1Ä R1 and some unknown real number c.
A solution of the form (1.3) of system (1.1) or (1.2) is called a traveling wave solution of (1.1) (or 1.2). By denoting s=i cos %+ j sin %&ct (or s=i&ct), x and c satisfy the equation of the form
&cx$(s)=G(x(s+r0), x(s+r1), ..., x(s+rN)), (1.4)
where r0=0, riare real numbers for i=1 to N.
If Eq. (1.4) depends on the past and future, i.e., if
rmin#min[ri]Ni=0<0<rmax#max[ri]Ni=0, (1.5)
then (1.5) is called a mixed type. If rmin=0 or rmax=0, then (1.4) is called
an advance or delay type, respectively.
Previous studies [5, 34] have numerically observed traveling wave solutions and mathematically proven them [1, 6, 2628] in the case of the discrete reactiondiffusion equation. Chow, Mallet-Paret, and Shen [9] studied the existence and stability of traveling wave solutions in lattice dynamical systems. In a related study Mallet-Paret [28] confirmed the existence and uniqueness of a traveling wave which connects the two stable states in bistable systems. Previous investigations of [28] and [35] provide the basis for this study. Indeed, define 8(x) by
and assume x0<x+such that
8(x0)=8(x+)=0 and 8(x)>0 for x # (x0, x+). (1.7)
By assuming that G is quasi-monotone, i.e., G(u0, u1, ..., uN) is strictly
increasing in uj for 1 j N, Wu and Zou [35] verified that a family of
monotone traveling wave solutions of (1.4) satisfies the boundary conditions lim
sÄ & x(s)=x
0 and lim
sÄ x(s)=x
+. (1.8)
A monotonic iteration scheme is employed in [35]. Under certain conditions on G, Wu and Zou constructed upper and lower solutions of (1.4), thereby satisfying the boundary conditions in (1.8).
In this work, we first generalize the results in [35]. Indeed, we denote the characteristic equation of (1.4) at x by
2(_, c, x)=&c_& : N j=0 G uj(x) e _rj. (1.9)
The assumptions needed for this mixed type problem are: (G.1) Assume that N
j=0( G uj)(x0)>0.
(G.2) Assume that G is quasi-monotonic for uj, j 1, in [x0, x+]N+1,
i.e., G uj(u)>0, for u # [x 0, x+]N+1, j 1. (G.3) Assume that G(u) : N j=0 G uj (x0)(u j&x0) for u # [x0, x+]N+1.
The first main results are
Theorem 1.1. By assuming that rmin<0<rmax with (G.1), (G.2), and (G.3) being held, c* 0 exists such that for any c<c* Eq. (1.4) has a non-decreasing solution satisfying the boundary conditions (1.8). Herein, c* satisfies
2(_*, c*, x0)=0 and 2$(_*, c*, x0)=0 (1.10)
for some _*>0, where `` $ '' denotes the partial derivative of 2(_, c, x0) with
With Condition (G.1) we can construct lower solutions. Condition (G.2) ensures the validity of the monotone iteration scheme. Condition (G.3), a global sublinearity of G at x0, allows us to construct an upper solution.
Notably, the condition (G.3) is much weaker than that in [35]. Condition (G.3) holds in many models, such as in the reactiondiffusion equation and CNN.
In Theorem 1.1, c* is the critical velocity which verifies the existence of a monotonic traveling wave connecting x0 and x+. Furthermore, in the
delay case, our results indicate that (G.3) is redundant. Indeed, the following results are obtained.
Theorem 1.2. Assume (G.1), (G.2), and that r
max=0. Then, for any c 0,
a non-decreasing solution of (1.4) satisfies the boundary conditions (1.8). All general results of Theorems 1.1 and 1.2 can be applied to CNN, enabling us to obtain monotone traveling waves. However, owing to the simplicity of the piecewise-linear nonlinearity of CY-CNN, the solutions can be obtained explicitly in the case of the delay or advance type. In addition to monotone traveling waves, non-monotonic waves can also be obtained in the case when G is quasi-monotone. Furthermore, overshoot non-monotonic waves can be obtained in the case when G is not quasi-monotone. Previous investigations have not rigorously proved these non-monotonic waves.
The rest of this paper is organized as follows. Section 2 introduces a novel monotone iteration scheme to construct upper and lower solutions of (1.4). In Section 3 we prove the main theorems by using the monotone iteration scheme. Section 4 applies the results in Section 3 to examine the CNN problem and also obtains non-monotonic solutions when G is either quasi-monotone or not quasi-monotone.
II. MONOTONE ITERATION SCHEME In this section, we consider the differential equation (1.4) with
G(u)#G(u0, u1, ..., uN): RN+1Ä R1
being a C2-function, c<0, and r
i in R1 for i=0 to N. In general, the
smoothness of G can be relaxed, say G # C1, except at a finite set. Hereafter,
we assume (1.7) and that r0=0.
These conditions (1.7) occur quite frequently in many models, and the two zeros of 8 correspond to the homogeneous steady states of (1.4). For simplicity, we also denote x0=(x0, ..., x0) # RN+1, etc., when it does not
This section largely focuses on obtaining monotonic traveling wave solutions of (1.4). The method employed herein to study Eqs. (1.4) and (1.8) is the well-known monotone iteration method. Importantly, the characteristic equation of (1.4), which occurs with the linearization of (1.4) about some trivial solutions, e.g. x0 and x+, must be considered.
Clearly, a pair of upper and lower solutions can be constructed accord-ing to the roots of the characteristic equation of (1.4).
Herein, we denote the characteristic functions about x0 and x+ by
2(_, c, x0) and 2(_, c, x+) respectively, which are defined by
2(_, c, x0)=&c_& : N j=0 G uj (x0) e_rj (2.1) and 2(_, c, x+)=&c_& : N j=0 G uj (x+) e_rj. (2.2)
Proving the existence of a traveling wave requires that G satisfies the assumptions (G.1), (G.2), and (G.3).
We recall the definition of upper and lower solutions of (1.4).
Definition 2.1. A continuous function U: R1Ä R1 is called an upper solution of (1.4) if it is differentiable almost everywhere and satisfies
&cU$(s) G(U(s+r0), ..., U(s+rN)). (2.3)
Similarly, the lower solution L(s) satisfies
&cL$(s) G(L(s+r0), ..., L(s+rN)). (2.4)
To construct the upper and lower solutions of (1.4), we need some properties of characteristic function 2(_, c, x0). Indeed, by differentiating
with respect to _, we have 2(_, c, x0) _ =&c& : N j=1 G uj (x0) e_rjr j (2.5) and 22(_, c, x0) _2 =& : N j=1 G uj (x0) e_rjr2 j . (2.6)
Lemma 2.2. By assuming that (G.1) and (G.2) hold, c* 0 exists such that for any c<c*, _0(c)>0 and =
0(c)>0 satisfy
2(_0, c, x0)=0
and
2(_0+=, c, x0)>0 for 0<=<= 0.
Proof. According to (G.2) and (2.6), 2(_, c, x0) is a concave function of
_. Hence, (G.1) implies that c* 0 exists such that for any c<c*, _0(c)>0
and =0(c)>0 satisfy the results. Therefore, the proof is complete.
In the following proposition, the construction of upper and lower solutions in mixed type resembles that in [35]. The construction of the lower solution in the delay case is new.
Proposition 2.3. (i) Under the assumptions of Theorem 1.1, for the given positive numbers `, h, and =, define functions
U(s)=
{
x + x0+(x+&x0) e_0s if s 0, if s 0, (2.7) and L(s)={
x 0 x0+`(1&he=s) e_0s if s s0, if s s0, (2.8) where s0<0 is such that he=s0=1. Then U(s) is an upper solution of (1.4),and positive numbers h0, `0, and =0 in R1 exist such that if h>h0>1,
0<`<`0, and 0<=<=0, L(s) is a lower solution of (1.4).
(ii) Under the assumptions of Theorem 1.2, for given positive numbers `, h, and =, we define the function
L(s)=
{
x 0+`(1&he=s1) e_0s 1 x0+`(1&he=s) e_0s if s s1, if s s1, (2.9) with s1=(1=) ln(_0h(_0+=))<0. Then, positive numbers h0, `0, and =~0exist such that if h>h0>1, 0<`<`0, and 0<=<=~0, then L(s) is a lower
solution of (1.4).
Proof. To demonstrate that U(s) is an upper solution, note that if s 0 then U$(s)=0, and by (G.2) we have
Hence,
&cU$(s) G(U(s+r0), ..., U(s+rN)).
If s 0, according to the definition of U we have U$(s)=_0(x+&x0) e_0s
. Now, applying (G.3), we have
G(U(s+r0), ..., U(s+rN)) : N j=0 G uj (x0)(U(s+r j)&x0), : N j=0 G uj (x0)(x+&x0) e_0(s+r j). (2.10) Since 2(_0, c, x0)=&c_0& : N j=0 G uj (x0) e_0r j, (2.10) implies &cU$(s) G(U(s+r0), ..., U(s+rN)),
for s 0. Hence, U(s) is an upper solution of (1.4).
Next, we prove L is a lower solution. If s s0, we have L$(s)=0 and
(G.2) implies that
G(L(s+r0), ..., L(s+rN)) G(x0, ..., x0)=0.
Hence, for s s0,
&cL$(s) G(L(s+r0), ..., L(s+rN)).
If s s0, then from the definition of L we have
L$(s)=`(_0&h(_0+=) e=s) e_0s .
Now, applying Taylor's expansion of G about x0, if ` is small then we can
write G(u0, ..., uN)=8(x0)+ : N j=0 G uj(x 0)(u j&x0)+Q(u&x0) (2.11) for u in [x0, x+]N+1 and |Q(u&x0)| K
Thus, by (2.11) and direct computation, we have G(L(s+r0), ..., L(s+rn))+cL$(s) =`e(_0+=) s h 2(_0+=, c, x0)+Q(L(s+r 0)&x0, ..., L(s+rN)&x0). (2.13) From (2.12), the constant K>0 exists such that
|Q(L(s+r0)&x0, ..., L(s+rN)&x0)| K`2e2_
0s .
Since (G.1) holds and by Lemma 2.2 we know that there exists an =0>0
such that
2(_0+=, c, x0)>0 for 0<=<= 0,
there exists an h0>1 such that if h>h0, the right-hand side of (2.13) is positive, i.e.,
&cL$(s) G(L(s+r0), ..., L(s+rN)).
for s s0. Hence, L(s) is a lower solution of (1.4).
Finally, we show that L(s) is a lower solution when rmax=0. First, we
choose positive numbers `, h, and = such that L(s) is a lower solution and define L(s) as in (2.9). Let L(s1) be the maximum of L(s) in R1, i.e.,
s1=1 =ln
_0
h(_0+=)<0,
then L$(s)=0 for s s1. From (G.1), we have
G u0 (x0)+ : n j=1 G uj (x0) e_0r j>0,
and this implies
G(L(s+r0), ..., L(s+rN)) = : N j=0 G uj (x0)(L(s+r
j)&x0)+Q(L(s+r0)&x0, ..., L(s+rN)&x0)
K
\
G u0 (x0)+ : N j=1 G uj (x0) e_0r j+
>0,for some positive constant K. Hence,
&cL$(s) G(L(s+r0), ..., L(s+rN)),
for s s1. If s<s1, then by an argument similar to that used in proving
that L(s) is a lower solution we can also obtain &cL$(s) G(L(s+r0), ..., L(s+rN)),
for s s1. By combining these results, L(s) is a lower solution of (1.4).
The proof is complete.
After construction of upper and lower solutions of (1.4), using the quasi-monotonicity of G, we present a novel monotone iteration scheme to obtain the non-decreasing solutions of (1.4) and (1.8).
From (G.2), a +>0 exists such that the function H(u0, ..., uN): RN+1Ä
R1 defined by
H(u0, ..., uN)=&
1
cG(u0, ..., uN)++u0 (2.14)
is monotonic in uj# [x0, x+] for each j 0. Thus we rewrite (1.4) as
x$(s)=H(x(s+r0), ..., x(s+rN))&+x(s). (2.15)
Then x(s) is easily verified to be a solution of (2.15) if and only if x(s) satisfies
x(s)=e&+s
|
& s e+tH(x(t+r0), ..., x(t+rN)) dt. (2.16)
If we define the operator T by (T.)(s)=e&+s
|
& se+tH(x(t+r
0), ..., x(t+rN)) dt, (2.17)
then by (2.16) the fixed point of T satisfies (2.15), and vice versa.
In the following, we apply the monotonic iteration method to find the fixed point of T. Clearly, .(s) is an upper (lower) solution of (1.4) if and only if
.(s) ( ) (T.)(s). (2.18)
Denote the set by
and the set of profiles by
1=[. # 1 | . is non-decreasing and satisfies (1.8)], then T has the following properties on 1.
Lemma 2.4. Assume that (G.2) holds, then
(i) If .(s), .~(s) # 1 and .(s) .~(s) for all s in R1, then
(T.)(s) (T.~)(s) for all s in R1.
(ii) If . is an upper (or lower) solution of (1.4), then (T.)(s) is also an upper (or lower) solution of (1.4).
(iii) if . # 1 then (T.)(s) # 1, too.
Proof. Since H is non-decreasing, (i) follows. Next, assume that . is an upper solution of (1.4). By (2.18) we have (T.)(s) .(s) for all s in R1. By
(i), we obtain
T(T.)(s) (T.)(s) for all s in R1.
Hence, (T.)(s) is also an upper solution of (1.4), and (ii) follows.
To prove (iii), note that H is non-decreasing. Hence, . # 1 obviously implies that T. is also non-decreasing. To demonstrate that T. satisfies (1.8), note that
H(x0)=+x0 and H(x+)=+x+.
Now, according to L'Hospital's rule, it is easy to verify that lim
sÄ & (T.)(s)=x
0 and lim
sÄ (T.)(s)=x +.
Hence, (T.)(s) lies in 1. The proof is complete.
III. PROOF OF THE MAIN THEOREMS
Proof of Theorem 1.1. By assuming that (G.1), (G.2), and (G.3) hold, then by Proposition 2.3 U and L are the upper and lower solutions of (1.4), respectively. For any positive integer n, define Un(s) and Ln(s) by
with U0=U and L0=L. Then using (2.18) and Lemma 2.4, we have
x0 } } } U
n(s) } } } U1(s) U(s) x+.
According to Lebesgue's dominated convergence theorem, the limiting function U
*(s) defined by
U
*(s)= limnÄ Un(s)
exists and is a fixed point of T. Moreover, U
*(s) is non-decreasing and satisfies (1.4). Therefore, it must be verified that U
*(s) satisfies the boundary conditions (1.8). However, L(s), constructed in (2.8), is a non-trivial lower solution. Since U L in R1, it is also easy to verify that U
n L for all n, hence U* L. Since
Unis non-decreasing and satisfies (1.8), U*lies in 1 and is a non-decreasing
solution of (1.4) and (1.8).
It remains to show that c* satisfies (1.10). Since 2(0, c, x0)<0 and 2"(_, c, x0)<0,
it is clear that there are a unique c*<0 and _*>0 that satisfy (1.10). Indeed, c* satisfies c*=& : N j=1 G uj (x0) e_*rjr j (3.2) with _*=inf
{
_>0}
: N j=1 G uj (x0) e_rj (_rj&1)> G u0 (x0)=
. (3.3)The proof is complete.
Remark 3.1. According to Theorem 1.1, the critical velocity c* exists, thereby ensuring a monotone traveling wave solution connecting x0 and
x+ for any c # (& , c*). When c=c*, it is not easy to construct the lower
solution as (2.8) to show the existence of a solution of (1.4) and (1.8). However, we believe that such a solution exists. For example, in [38], Zinner et al. studied the discrete Fisher equation and obtained the traveling wave solutions when c c*.
For another example, consider one-dimensional cellular neural networks by
dxi
with f (x)=( |x+1| & |x&1| )2. Define L(s) by L(s)=
{
x 0+$ x0+$e_*s for s 0, for s 0, (3.5)then L(s) is a lower solution of (3.4) when a+;&1>0 and $ is positive and small enough. Hence, we have a traveling wave solution of (3.4) and (1.8) when c=c*. In addition, the global structure of the traveling wave solutions of (3.4) is completely classified in [19]. Of relevant interest is whether or not a traveling wave of (1.4) exists which may be non-monotone for c>c*.
Proof of Theorem 1.2. Since rmax=0, by (2.5) and (2.6), we have that
2(_, c, x0) is a concave function in _ and 2$(_, c, x0)>0 for any c<0.
Hence (G.1) holds for any c<0. Now from Proposition 2.3(ii), we know that L(s) is a lower solution of (1.4). If we denote Ln(s) by
Ln(s)=(TnL)(s). (3.6)
with L0=L and apply Lemma 2.4, we obtain
x0 L
0(s) L1(s) } } } Ln(s) } } } x+.
By the Lebesgue dominated convergence theorem again, the limiting function L
*(s) defined by L
*(s)= limnÄ Ln(s)
is the fixed point of T. It remains to be shown that L
*(s) satisfies the boundary conditions. Clearly, L
*(s) is non-decreasing due to the monotonicity of T. This is because we do not have a non-trivial upper solution as a barrier function to separate L
*(s) from a trivial solution x
+; to overcome this
difficulty, we need to show L
*(s) Ä x
0 as s Ä & . This can be achieved
inductively on Ln(s). By (3.6), we have Ln+1=e&+s
|
s e+tH(L n(t+r0), ..., Ln(t+rN)) dt.We begin with the study of L1(s) as s Ä & . By (2.11), L1can be written as
L1(s)=e&+s
|
s & e+t_
&1 c : N j=0 G uj (x0)(L(t+r j)&x0)++L(t)As s tends to & , we have L1(s)=x0+`e_ 0s &`h
\
1&2(_ 0+=, c, x0) ++_0+=+
e (_0+=) s +e&+s|
& s e+tQ(L(t+r 0)&x0, ..., L(t+rN)&x0)] dt. (3.7)However, (2.12) implies that a positive constant K exists such that
e&+s
|
& s e+tQ(L(t+r 0)&x0, ..., L(t+rN)&x0) dt K ++2_0e 2_0s . (3.8) Define ! and \ as !=1&2(_ 0+=, c, x0) ++_0+= and \= K ++2_0.In addition, by combining (3.7) with (3.8), we have 0<!<1 and \<1 2, for
+ large enough. Hence, L1(s) can be written as
L1(s)=x0+`(1&!he=s) e_ 0s +r1(s), where r1(s)=e&+s
|
s & e+tQ(L(t+r 0)&x0, ..., L(t+rN)&x0) dt and |r1(s)| \e2_ 0s . Let ` \, and by induction Ln(s) can be written asLn(s)=x0+`(1&!nhe=s) e_0s +rn(s) (3.9) and |rn(s)| \e2_ 0s . (3.10)
Hence, Ln and L* tend to x0 as s tends to & . Thus L*is not the trivial
solution x+. Since L
* is monotonously increasing, according to (1.7), we have
lim
sÄ L*(s)=x +.
The proof is complete.
Remark 3.2. By using a comparison theorem obtained in [28], we can prove that the monotone solution obtained in Theorems 1.1 and 1.2 is unique for c # (& , c*). We only sketch the proof in the following and omit the details.
It is not difficult to prove that if x(s) is a monotonic solution of (1.4) and (1.8) then we have x(s)=x0+O(e_0s
), as s Ä & . On the other hand, if N
j=0( G uj)(x+)<0 then (1.4) satisfies the hyperbolicity at x+; see [27].
Hence, as in [27], we have x(s)=x0+O(e_+s
), as s Ä . Here _+<0 and
satisfies 2(_+, c, x+)=0. By an argument similar to that used in proving
Proposition 6.5 of [28], the uniqueness result follows.
IV. APPLICATIONS TO CNN
In this section, we initially apply the above results to obtain a monotonic traveling wave solution in CNN. For a CY-CNN with delay or advance type, we demonstrate that the solutions can be obtained explicitly. In addition to the non-decreasing traveling waves, we obtain non-monotonic traveling waves. The various results obtained for CY-CNN allow us to study the general case of (1.4) even when G is not quasi-monotonic.
For simplicity we only study the one-dimensional CNN; the higher dimensional cases can be treated analogously. Consider
dxi
dt=&xi+z+:f (xi&1)+af (xi)+;f (xi+1) (4.1)
where z, :, a, and ; are constants. Here f is a non-decreasing continuous function which is differentiable except for finite points. A typical case is
f (x)= f0(x)#1
2( |x+1| & |x&1| ). (4.2)
In this case, it is called CY-CNN. Assuming
where s=i&ct and .(s) is in C1(R1, R1), then .(s) satisfies
&c.$(s)=&.(s)+z+:f (.(s&1))+af (.(s))+;f (.(s+1)), (4.4) and the boundary conditions are
lim sÄ & .(s)=x 0 and lim sÄ .(s)=x +. (4.5) Now, 8(x)=&x+z+(:+a+;) f (x). (4.6)
Assume that x0<x+ are the two zeros of 8(x) such that 8(x)>0 for
x # (x0, x+). For CY-CNN we have
x0= &z
&1+a+:+; and x
+=z+a+:+;,
whenever a+:+;&1{0.
Applying Theorem 1.1, we obtain
Theorem 4.1. Suppose :, a, and ; are real numbers and f is a continuous function differentiable except for finite points. If f $(x) 0, f $(x0) exists and
satisfies the conditions
(i) &1+(:+a+;) f $(x0)>0,
(ii) :>0 and ;>0,
(iii) f (x) f (x0)+(:+a+;) f $(x0)(x&x0), for x in [x0, x+],
then c*<0 exists such that for each c<c* there is a non-decreasing solution satisfying (4.4) and (4.5). Moreover, c* satisfies
&c*= f $(x0)(&:e&_*+;e_*) and
&c*_*=&1+ f $(x0)(a+:e&_*+;e_*)
for some _*>0.
Proof. It is easy to verify that the assumptions (i), (ii), and (iii) imply (G.1), (G.2), and (G.3) with
2(_, c, x0)=&c_+1& f $(x0)(a+:e&_+;e_).
The following result immediately occurs. Therefore, the proof is complete. In the following theorem, we observe whether the assumption (G.2) fails. A traveling wave solution may exist which satisfies (4.5) which overshoots
the steady state x+. For simplicity, we consider the delay type of CY-CNN,
i.e., ;=0. Now, the monotonic traveling wave can be solved explicitly. Theorem 4.2 (Delay case of CY-CNN). Assume that ;=0 and f =f
0in
(4.2). If &1+:+a>0, then:
(i) If :>0, then for any c<c*, monotonic traveling wave solutions of (4.1) and (4.5) exist.
(ii) If a>1, :<0 and we define c*=(ln(&:a))&1, then for any
c c*, monotonic traveling wave solutions of (4.1) and (4.5) exist.
(iii) If a>1, :<0, and c>c*, then a solution , of (4.1) and (4.5) exists which has a single maximum. In this case, , is not monotonic.
Furthermore, in any case the solution ,(s) can be expressed as x0+(1&x0) e_0s
for s 0,
,(s)=
{
x0+le_0s+al(1&e&$s)& al$
_0+$(e _0s
&e&$s) for s # [0, 1],
x++e&$(s&1)(,(1)&x+) for s # [1, ),
(4.7) where l=1&x0, $=&1c, x0=&z(&1+a+:), x+=z+a+:, and
2(_0, c, x0)=0.
Proof. Since f is piecewise linear, the problems (4.4) and (4.5) can be decomposed into the equations
&c,$(s)=
{
&,(s)+z+:,(s&1)+a,(s)&,(s)+z+:,(s&1)+a,(s) &,(s)+z+:,(s&1)+a &,(s)+z+:+a if s # ( , 1], if s # [&1, 0], if s # [0, 1], if s # [0, ). (4.8)Herein, assume that ,(0)=1, ,(& )=x0, and ,( )=x+. Now, (4.8) can
be solved and the solution is given in (4.7). Since the proof is elementary but lengthy, the detail is omitted. Therefore, the proof is complete.
Remark 4.3. In the case (iii) of Theorem 4.2, the assumption (G.2) does not hold and the solution is now nonmonotonic, as shown in Fig. 1. Remark 4.4. According to Theorem 4.2, a bifurcation diagram can be drawn which exhibits how the monotonic traveling wave changes into a non-monotonic traveling wave when : changes from positive to negative.
FIGURE 1
Indeed, if we assume that a>1 and c<0 will be given fixed numbers, we define :*(a, c) and c*(a) by
:*(a, c)=&ae1c and c*(a)=lna&1
a .
If c>c*(a) then the bifucation pictures with respect to : are given as follows.
In Case (ii) of Fig. 2, the traveling wave solution ,(s) is equal to x+for
s greater than some s*.
Finally, the oscillating traveling wave solution of CY-CNN is considered as follows
Theorem 4.5. While considering (4.4) and (4.5) with ;=0, a solution ,osc(s) exists which is given by
,osc(s)=
{
x0+le*scos(&s)+me*ssin(&s)
x0+le*scos(&s)+me*ssin(&s)+, osc(s)
if s # (& , 0], if s # [0, 1],
(4.9)
FIG. 3. Oscillating wave.
where *>0, l and m are non-zero real numbers, and $=&1c, ,osc(s)=al&ale
&$s
&al$e&$sg(s)&am$e&$sv(s),
g(s)=(*+$)(e (*+$) s cos(&s)&1)+&e(*+$) s sin(&s) &2 +(*+$)2 , v(s)=(*+$) e (*+$) s sin(&s)&&(e(*+$) s cos(&s)&1) &2 +(*+$)2 , and 2(*+&i, c, x0)=0.
Proof. The proof of the theorem is the same as that used in proving Theorem 4.2 by solving (4.8) with suitable l and m. Since the proof is elementary but lengthy, the details are omitted here.
Example 4.6. Within a certain parameter range of a, :, z, and appropriate choices of l and m, we can prove that ,oscsatisfies lims Ä ,osc(s)=x
+. Here
is an example with the aid of numerical computation: If we choose a=200, :=144.14?, z=0, &=1.1?, c=&17.3277, l=1, and m=0.2, then the oscillat-ing wave ,osc(s) is given in Fig. 3.
ACKNOWLEDGMENTS
The authors thank Professors S.-N. Chow and J. Mallet-Paret for several interesting and stimulating discussions with the second author when they attended a conference at Augas de Lindoia, Brazil, Oct. 1996. We also thank J. Mallet-Paret for sending us his preprints [27, 28] which were very helpful.
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