c
World Scientific Publishing Company
SPATIAL DISORDER OF CELLULAR NEURAL
NETWORKS
WITH BIASED TERM
JUNG-CHAO BAN and SONG-SUN LIN∗ Department of Applied Mathematics,
National Chiao Tung University, Hsinchu 300, Taiwan, R.O.C CHENG-HSIUNG HSU†
Department of Mathematics, National Central University, Chung-Li 320, Taiwan, R.O.C.
Received March 28, 2001; Revised June 4, 2001
This study describes the spatial disorder of one-dimensional Cellular Neural Networks (CNN) with a biased term by applying the iteration map method. Under certain parameters, the map is one-dimensional and the spatial entropy of stable stationary solutions can be obtained explicitly as a staircase function.
Keywords: Spatial disorder; topological entropy; Bernoulli shift; transition matrix.
1. Introduction
Cellular neural networks (CNN), a large array of nonlinear circuits, consists of only locally connected cells. This work investigates the model of one-dimensional CNN proposed by Chua and Yang [1988a, 1988b]. The circuit equation of a cell is
dxi
dt =−xi+ z + αf (xi−1) + af (xi)
+ βf (xi+1) , i∈ Z1, (1) where f (x) is a piecewise-linear output function defined by f (x) = rx + m− r if x ≥ 1, mx if |x| ≤ 1, `x + `− m if x ≤ −1. (2)
Here r, m and ` are non-negative real constants and the quantity z is called threshold or biased term,
and is related to independent voltage sources in electric circuits. The coefficients of output func-tion α, a and β are real constants and called the space-invariant A-template denoted by
A≡ [α, a, β] . (3)
For simplicity, f will be denoted by fr, with ` = r and m = 1, i.e. fr(x) = rx + 1− r if x ≥ 1, x if |x| ≤ 1, rx + r− 1 if x ≤ −1. (4)
CNN is applied mainly in image processing and pattern recognition [Chua & Roska, 1993; Chua & Yang, 1988a] and [Thiran et al., 1995]. A basic and important class of solutions of (1) are the sta-ble stationary solutions of (1). In particular, the
∗Work partially supported by the NSC under Grant No. 89-2115-M-009-023, the Lee and MTI Center for Networking Research and the National Center for Theoretical Sciences Mathematics Division, R.O.C.
†Work partially supported by the NSC under Grant No. 89-2115-M-008-029 and the National Center for Theoretical Sciences Mathematics Division, R.O.C.
525 Int. J. Bifurcation Chaos 2002.12:525-534. Downloaded from www.worldscientific.com by NATIONAL CHIAO TUNG UNIVERSITY on 04/27/14. For personal use only.
complexity of stable stationary solutions of (1) must be investigated. When the output function is f0, i.e. r = 0 in (4), it is observed that much work has subsequently been done in the electrical engineer-ing community, see [Chua & Roska, 1993, 1988a] and references therein. In addition, [Juang & Lin, 2000; Hsu & Lin, 1999, 2000] and [Hsu et al., 1999] recently considered mathematical results involving the complexity of stable stationary solutions and the multiplicity of traveling wave solutions. [Juang & Lin, 2000] partitioned the parameters space (a, z) into a finite number of regions in R2 such that in each region (1) with f = f0 has the same spatial entropy.
However, for z = 0 and r∈ (0, ∞), [Hsu & Lin, 1999] proved that (1) and (4) can release infinite dif-ferent spatial entropies and the entropy function is a devil-staircase like function in r. The method used in [Hsu & Lin, 1999] considers the stationary solu-tions of (1) as an iteration map. In fact, if output v = f (x) is taken as the unknown variable, i.e. let
vi = f (xi) and ui+1= vi. (5) and if f is invertible with inverse function F , then the stationary solutions of (1) can be written as one-or two-dimensional iteration maps as follows,
T (v) = 1
β (F (v)− z − av) , (6) when α = 0 and β6= 0 and
T2(u, v) = v, 1 β (F (v)− z − αu − av) , (7) when α6= 0 and β 6= 0.
For these maps, each bounded trajectory corre-sponds to the outputs of bounded stationary solu-tions. In practice, if the maps are chaotic, then the stationary solutions of (1) are spatially chaos. How-ever, only stable stationary solutions of (1) should be considered and the stability results can be found in [Hsu, 2000] or [Juang & Lin, 2000]. Therefore, the set of all stable bounded orbits of T must be considered, denoted by S, and the entropy h of T|S must be computed. If the entropy is positive, then the stable stationary solutions of (1) are spa-tial chaos. For convenience, T|S is denoted herein as T .
[Hsu & Lin, 1999] considered (6) with z = 0, the odd symmetry of the map T makes it much easier to investigate the complexity of T than the case of
z 6= 0. Therefore, this work focuses on the com-plexity of the one-dimensional map T with z∈ R1 by some complicated computation. According to our results, the entropy function is a staircase func-tion. As for the two-dimensional map T2, when r is positive and sufficiently small, the Smale Horseshoe structures of stable stationary solutions of (1) and (4) are constructed, for details, see [Hsu, 2000].
Carefully examining the orbits of T reveals that the entropy function h is a staircase function of r for fixed a, z and β. The main results are
Main Theorem. Assume β = 1, 0 < z < Γ(a) (see Lemma 3.1). Denote
r∞(z) = a + z− 2
a2− 2 + az (8)
and h(r) is the entropy function of T in (6). Then there exists p(z)∈ Z+ and a strictly decreasing se-quence {rp,p−1(z)}, p = 3, 4, . . . , p(z) with r∞(z) < rp,p−1 and rp< rp,p−1 < rp−1 such that (i) If 3≤ p ≤ p(z) and r ∈ [rp,p−1(z), rp−1,p−2(z)) then h(r; z) = ln λp−1,p−2.
Where λp,p−1 is the largest root of λ2[λ2p−3−
Pp−3 i=0 λi Pp−2 j=0 λj] = 0. (ii) If r ∈ (r∞(z), rp(z), p(z)−1) then h(r; z) = ln λp(z), p(z)−1. (iii) If r∈ (r∞(z), r∞(z)] then h(r; z) = 0. (iv) If r∈ [0, r∞(z)] then h(r; z) = ln 2.
Moreover, p(z) is a decreasing function of z and limz→0+ p(z) =∞.
The above results or the proof of the main the-orem in Sec. 3 indicate that the nonzero bias z causes a situation in which map T does not have enough periodic orbits when r∈ (r∞(z), r∞(z)] and it makes the entropy equal to zero. Therefore, the entropy function of T has a staircase structure as shown in Fig. 1. This differs from those results of a devil-staircase like function in [Hsu & Lin, 1999] with z = 0 as shown in Fig. 2. Additionally, the re-sults of [Hsu & Lin, 1999] recalled in the following corollary can be considered as the limiting case of the main theorem when z tends to 0.
Corollary. Assume β > 0, z = 0 and a > β + 1.
r
h(r;z) 0 r (z) r4,3 r3,2 r2,1 ln 2 rp(z),p(z)-1 r (z)
Fig. 1. Entropy of T with z6= 0.
!" $ & ' )
Fig. 2. Entropy of T with z = 0.
Denote r∞= r∞(a, β) = a− β − 1 a(a− 1) + β(a − 2), r2 = r2(a, β) = a− β − 1 a(a− 1) + β(β − 1), and h(r) is the entropy function of T in (6) with F = Fr = fr−1, r > 0. Then there exists a strictly decreasing sequence {rp}, p = 2, 3, . . . , with
lim
p→∞rp = r∞, such that
(i) If r2 ≤ r < (1/a + β), then h(r) = 0.
(ii) If r ∈ [rp, rp−1), p = 3, 4, . . . , then h(r) is ln λp where λp is the largest root of λ2p−2 −
(Ppi=0−2 λi)2 = 0. Moreover, λp is strictly in-creasing in p with
1 +√5
2 = λ3 < λp < 2 , for p = 4, 5, . . . (iii) If r∈ [0, r∞], then h(r) = ln 2.
The rest of this paper is organized as follows. Section 2 introduces the basic properties of the one-dimensional map T in some range of parameters. Section 3 proves the main theorem by symbolic dy-namics, indicating that the entropy function h(r) is a step function under certain parameters range.
2. Iteration Map
This section considers the one-dimensional map T in (6) with z6= 0. If a > 1, β > 0, and m = 1, then the inverse function F of fr is
F (v; r) = 1 rv− 1 r + 1 if v≥ 1, v if |v| ≤ 1, 1 rv− 1 + 1 r if v≤ −1, (9)
and the map T can be rewritten as T (v; a, β, r) = 1 β 1 r v− 1 r + 1− av − z if v≥ 1, 1 β (v− av − z) if |v| ≤ 1, 1 β 1 r v + 1 r − 1 − av − z if v≤ −1. (10) Instead of F (v; r) and T (v; a, β, r), F (v) and T (v) will be used if it does not cause any confusion. For simplicity, assume that β = 1 and z≥ 0 hereinafter. The graph of T can be found in the following figure.
An elementary computation produces that A = (A1, A2) = rz− r + 1 1− ra − r, rz− r + 1 1− ra − r , B = (B1, B2) = (1, 1− a − z) , C = (C1, C2) = (−1, a − 1 − z) , D = (D1, D2) = rz + r− 1 1− ra − r, rz + r− 1 1− ra − r .
According to [Hsu, 2000] and [Juang & Lin, 2000], any orbit {Tk(v)} of T with |Tk(v)| ≤ 1 for some k≥ 0 is unstable. Hence, only trajectories of T ly-ing outside the unit rectangle in (u, v) plane should be considered. Therefore, assume that B2 < −1 and C2 > 1 while these conditions are equivalent to 2− a < z < a − 2. For further computation, we give the following notations.
Definition 2.1. Assume a > 2.
(i) Define functions r∞(z) and r∞(z) by r∞(z) = a+z−2
a2−2+az and r∞(z) =
a−z−2 a2−2−az.
(11) (ii) Let m, n ∈ Z+, if the slope of f , r = rm,n
satisfies
Tm−1(B2) =−1 and Tn−1(C2) = 1 , (12) then we call map T is of (m, n)-type and de-note rm,m, km,n and ξm,nby
rm,m= rm,
km,n= rm,n1 − a and ξm,n= k−1m,n. (iii) Define polynomials E(x; m) and U (x; m) by
E(x; m) = a m X i=1 xi− a + 2 , (13) U (x; m, n) = (a + z) m X i=n+1 xi + 2a n X i=1 xi− 2a + 4 . (14)
From Fig. 3, the relative positions of A, B, C and D are easily obtained in the following.
Lemma 2.1. Assume a > 2, then r∞(z) and r∞(z) are increasing and decreasing functions of z, respec-tively. Moreover, we have
(1) If r∈ (r∞(z), ∞), then A2 > C2 and B2 > D2. (2) If r = r∞(z) then A2 > C2 and B2 = D2. (3) If r ∈ (r∞(z), r∞(z)), then A2 > C2 and D2 > B2. (4) If r = r∞(z), then A2 = C2 and D2 > B2. (5) If r∈ (0, r∞(z)), then A2 < C2 and D2 > B2. u=T(v) u=v v u A B D C 1 -1 0 Fig. 3. Graph of T .
Proof. By elementary computation, we have r0∞(z) = 2a− 2
(a2− 2 + az)2 and
r0∞(z) = 2− 2a (a2− 2 − az)2
and r∞(z) and r∞(z) are increasing and decreasing functions of z respectively. The proofs from (1) to (5) are also simple and omitted.
The proof of the main theorem in Sec. 3 in-dicates that the case of (1) in Lemma 2.2 is more interesting and complicated.
3. Proof of Main Theorem
In this section, we prove the main theorem by in-troducing some lemmas. If z > 0, the following lemmas will show that unique rm,m−1 lies between rm,m and rm−1,m−1 such that (12) holds.
Lemma 3.1. Assume m≥ 3 and define Γ(a) by Γ(a)≡ min a− 2, −a 3+ 6a2− 4a 3a2− 6a + 4 . If 0 < z < Γ(a), p > q and rp,q satisfies (12) with rm,m < rp,q < rm−1,m−1, then p = m and q = m−1.
Proof. First, we claim that U (ξp,q; p, q) = 0 and E(ξm,m; m) = 0. By simple computation, it is ob-vious that
T−1(1) =rz +1
1−ra and T
−1(−1)=rz−1
1−ra. (15)
Define R and L by
R = T−1(1)− 1 and L = 1 − T−1(−1) . (16) If p > q and r = rp,q satisfies (12), then it is not difficult to compute that ξp,q satisfies
L(1− ξqp,q) 1− ξp,q +R(1− ξ p p,q) 1− ξp,q = 2a− 4 . (17) By (15) and (16), we know that
R + L = 2ξp,q
rp,q − 2 , R =
rp,q(z + a) 1− rp,qa
, and (17) can be rewritten as
2ξ p,q rp,q −2 qX−1 j=0 ξjp,q+R p−1 X j=0 ξp,qj = 2a−4 , (18) ξp,q(a+z) pX−1 j=q ξjp,q+ 2ξp,q rp,q −2 qX−1 j=0 ξp,qj = 2a−4 . (19) According to the definition of ξp,q, we have U (ξp,q; p, q) = 0. Similarly, we have E(ξm,m; m) = 0. Next, we show that rm,m−1 satisfies (12) and rm,m < rm,m−1 < rm−1,m−1. Since z > 0 and ξm,m−1> 0, by (13), (14) and (19), we have a mX−1 i=1 ξm,mi −1 < a− 2 , a mX−1 i=1 ξmi −1,m−1 = a− 2 , (20) and a m X i=1 ξm,mi −1> a− 2 , a m X i=1 ξm,mi = a− 2 . (21) From (20) and (21), rm,m−1 satisfies (12) and rm,m< rm,m−1< rm−1,m−1, for m > 2.
Now, we claim that no rp,q satisfies (12) and rm,m < rp,q < rm−1,m−1 except for p = m and q = m−1. For convenience, let h = rm−1,m−1, k = rm,m and ξ = rp,q, where p = m + n, q = m− n − 1 and 1≤ n < m − 2. By (14) and elementary computa-tion, we have U (h; p, q) < 0 if and only if 2a− (a + z)hn+ (z− a)h−(n+1)< 0 (22) and U (k; p, q) < 0 if and only if 2a− (a + z)kn+1+ (z− a)k−n< 0 . (23) Obviously U0(x; p, q) > 0 and if U (h; p, q)U (k; p, q) > 0; by intermediate value theorem, no ξ lies between h and k and satisfies (12). Therefore, we claim that U (h; p, q) < 0 and U (k; p, q) < 0, if a, z satisfy 0 < z < Γ(a). Denote P (x) and Q(x) by
P (x) = 2a− (a + z)xn+1+ (z− a)x−n and
Q(x) = 2a− (a + z)xn+ (z− a)x−(n+1), then P (x) and Q(x) are concave functions in (0, 1] and P (1) = Q(1) = 0. By elementary computation or [Hsu & Lin, 1999], we know that r2,2 = r2 = (a− 2)/(a2 − a) and 0 < k < h < 1 1 r2,2 − a = a− 2 a . (24)
If a, z satisfy 0 < z < Γ(a), we have P ((a− 2)/ a) < 0. Since P (x) is concave, by (23) we ob-tain that U (k; p, q) < 0. Furthermore, the zero of Q(x) is obviously larger than the zero of P (x) in (0, 1). By the concavity of Q(x), we also obtain P ((a− 2)/a) < 0 and this implies U(h; p, q) < 0. Hence, the proof is complete.
Corollary 3.1. Under the same assumptions of Lemma 3.1, we have rm+1,m < rm,m−1 for all in-teger m > 1.
Now, if z is fixed, since limp→∞ rp = r∞ and r∞(z) is an increasing function of z, by Lemma 3.1, we obtain that there exists a maxi-mal positive integer p(z) such that (12) holds for sequence{rp,p−1(z)} with p = 3, 4, . . . , p(z) and no rm,m−1(z) satisfies (12) with m > p(z). As demon-strated later this observation reveals the staircase structure of entropy function h of T . For complete-ness, this study recalls the definitions and some re-sults of entropy for a dynamical system. Details can be found in [Bowen, 1973] or [Afraimovich & Hsu, 1998, Sec. 6].
Definition 3.1. Let G : X → X be a dynamical system on the complete metric space X and S⊂ X be an invairant set.
(i) The set Γn(x) ={Gk(x)}nk=0−1 is called an orbit segment of temporal length n. Two segments Γn(x) and Γn(y) are said to be (n, ε)-separated if there exists k ∈ Z1, 0≤ k ≤ n − 1, such that dist(Gk(x), Gk(y))≥ ε.
(ii) Let Sn,εbe a set of segments of temporal length n such that
(a) if Γn(x), Γn(y)∈ Sn,ε, then they are (n, ε)-separated;
(b) if w ∈ S and Γn(w) /∈ Sn,ε, then there is x ∈ S such that Γn(x) ∈ Sn,ε and dist(Gkx, Gkw) < ε for each k = 0, 1, . . . , n− 1.
Define ˜Cn,ε = ]Sn,ε, the number of elements of the set Sn,ε and Cn,ε = infSn,εC˜n,ε. Then,
the entropy function of G, denoted by h(G), is defined as follows:
h(G) = lim ε→0nlim→∞
ln Cn,ε
n . (25)
Proposition 3.1. ([Afraimovich & Hsu, 1998, Sec. 2.4; Robinson, 1995]). Let σM : ΣM → ΣM be a subshift of finite type with the transition ma-trix M on N symbols. Denoted by Kn the number of admissible words of length n + 1, the entropy of σM is equal to
h(σM) = lim n→∞
ln Kn
n = ln |λ1| ,
where λ1 is the real eigenvalue of M such that |λ1| ≥ |λj| for all other eigenvalues λj of M .
By Proposition 3.4, we must find a subshift of finite type such that T is topologically conjugate to the subshift. The subshift can be constructed by finding some subintervals of I\(−1, 1) with the covering relation as shown in the proof of the main theorem later.
Definition 3.2. An interval I1T -covers an interval I2 provided I2⊆ T (I1). This study writes I1 → I2.
Proof of Main Theorem. First, we consider the case r > r∞(z), i.e. A2 > C2 and B2 > D2. Let R+1(r) and R−1(r) be the first components of the inter-section points of AB with u = +1 and u = −1,
respectively. A simple computation produces R1−(r) = 1− 2r + rz 1− ra and R + 1(r) = 1 + rz 1− ra. (26) Then, the continuity of T (v; r) with respect to r and Lemma 3.1 make it easy to prove that for any positive integer 2 < p ≤ p(z), there exists a unique rp,p−1 > 0 such that {Ti(C2; rp,p−1)}i=i=∞−∞ is a 2p− 1-periodic orbit, i.e. of (p, p − 1) type, where p(z) is the largest integer such that rp(z) less than r∞(z). Restated, after 2p − 1 itera-tion, (v, T (v; rp,p−1)) maps C to B and B to C, respectively. Denote R+= (R+1, R+2) = AB∩ {u = 1} , R−= (R−1, R−2) = AB∩ {u = −1} , L+= (L+1, L+2) = CD∩ {u = 1} , L−= (L−1, L−2) = CD∩ {u = −1} , Ωr= (v, u)| |v| ≤ ra− 2r + 1 1− ra and |u| ≤ ra− 2r + 1 1− ra , here{u = D2}∩CD = (((2r−ra−1)/(1−ra)), 1− a− z) and Ωr ⊂ Ω. Figures 4 and 5 give the five-periodic orbit and seven-five-periodic orbit of T at r3,2 and r4,3, respectively. Now, if 3 ≤ p ≤ p(z) and rp,p−1 ≤ r < rp−1,p−2 or r∞(z) < r < rp(z),p(z)−1, define the 2p− 1 stable subintervals by
Ip+1= (1, R−2) ,
Ip+k= (T−k+1(R+2), T−k(R−2)) for k = 1 to p−2 . and
Ip= (L+2,−1) ,
Ip−k= (T−k(L+2), T−k+1(L−2)) for k = 1 to p−1 . The 2p− 1 subintervals have the following covering relation:
Ii→ Ii+1 for i = 1 to p− 1 , Ip→ Ij for j = p + 1 to 2p− 2 , Ip+1→ Ik for k = 2 to p ,
Il→ Il−1 for l = p + 2 to 2p− 1 .
u=T(v) u=v v u -1 0 1 B C D R+ L+ L- R I5 I4 I3 I2 I1
Fig. 4. Graph of T in (3, 2) type and its stable subintervals. A B D C u v R+ L+ R L -0 1 -1 u=v u=T(v) I6I 7 I5 I4 I1I2 I3
Fig. 5. Graph of T in (4, 3) type and its stable subintervals.
Therefore, we obtain the following transition matrix M ≡ M[p, p − 1] of the 2p − 1 subshifts of finite type. M = 0 1 0 • • • • • 0 0 0 1 0 • • • • • 0 • • • • • • • • • • • • • • • • 0 • • 0 1 0 • • 0 • • • • 0 1 • • 1 0 0 1 • • • 1 0 • • 0 0 • • • 0 1 0 • • 0 • • • • • • • • • • • • • • • • 0 • • • • • 0 1 0 0 0 • • • • • 0 1 0
pth row with p− 2 terms of 1 J |
J |
(p + 1)th row with p− 1 terms of 1
(2p− 1) × (2p − 1) This study defines spaces Σ2p−1 and ΣM by
Σ2p−1={1, 2, . . . , 2p − 1, 2p − 1}N, ΣM={s∈Σ2p−1: Msksk+1= 1 for k = 0, 1, 2, . . .} , with a metric on ΣM by d(s, t) = ∞ X k=0 δ(sk, tk) 3k , for s = (s0, s1, . . .) and t = (t0, t1, . . .) in ΣM, where δ(i, j) = ( 0 if i = j, 1 if i6= j.
Let σM : ΣM → ΣM be the subshift of finite type for the matrix M , i.e. σ(s) = t where tk = sk+1. Therefore, if rp,p−1≤ r < rp−1,p−2then there exists
an invariant subset Λp in Ω such that T|Λp is
topo-logical conjugate to the 2p− 1 subshift (ΣM, σM) with entropy h equal to ln λp,p−1, where λp,p−1 is the positive maximal root of characteristic polyno-mial of M . To derive λp,p−1, we need the following lemma.
Lemma 3.2. Given p ∈ Z1 and p > 1, then the characteristic polynomial g(x; p, p−1) of transition matrix M [p, p− 1] is g(x; p, p− 1) = x2 x2p−3− pX−3 i=0 xi pX−2 j=0 xj ! .
Proof. By elementary matrix computation, see Appendix A, we obtain g(x; p, p− 1) = xg(x; p − 1, p − 1) − x2 p−3 X i=0 xi,
where, g(x; p− 1, p − 1) is the characteristic poly-nomial of M with z = 0, for details see [Hsu & Lin, 1999]. In [Hsu & Lin, 1999], we also have g(x; p− 1, p − 1) = x2[x2p−4− (Ppi=0−3 xi)2]. Therefore, the result follows by simple computation.
By Lemmas 3.1 and 3.6, we prove results (i) and (ii) of the main theorem. As for the assump-tion (iii) of the main theorem, it is equivalent to the conditions of (2) and (3) in Lemma 2.2. By the same arguments, we obtain the entropy h of T is zero, see e.g. Fig. 6. In case (iv), which is equiva-lent to the conditions of (4) and (5) in Lemma 2.2, we know that D2 > B2 and C2≥ A2 in Fig. 7 such that the behavior of the map T resembles that of the logistic map as discussed in [Robinson, 1995, Theorem 5.2]. Therefore, there exists an invari-ant Cinvari-antor set such that T is topologically con-jugate to a one-sided Bernoulli shift of two sym-bols. Since the entropy of the one-sided Bernoulli shift of two symbols is ln 2, the result follows by Proposition 3.4.
Finally, since limz→0 r∞(z) = r∞, by Lemma 3.1 we obtain that p(z) is a decreasing function of z with limz→0 p(z) = 0. The proof is complete.
Remark
(i) If we consider the output function is not sym-metric, i.e. r 6= ` in (2), then Lemma 3.1 is
no longer valid. In fact, there exists many dif-ferent m, n such that r = rm,n lies between rp and rp−1 for any p ≥ 3 and T is of (m, n) type. Hence, by similar arguments in the proof of the main theorem, we also obtain transition matrix M [m, n] such that the corresponding
Fig. 6. Graph of T with r = r∞(z) and its stable subintervals. A B C D I1 I2 I3 I4 1 0 -1 v u u=T(v) u=v
Fig. 7. Graph of T with r = r∞(z) and its stable subintervals.
characteristic ploynomial g(x; m, n) is g(x; m, n) = x2 xm+n−2− mX−2 i=0 xi nX−2 j=0 xj ! . (27) (ii) By some further computation, the ordering re-lation of the maximal root λm,n of g(x; m, n) can also be obtained as following lemma. Lemma 3.3. Given (m1, n), (m2, n + 1) and m1 > m2, then g(λm1,n; m2, n + 1) < 0. Moreover, we have (1) If n1> n2 then λm1,n1 > λm2,n2. (2) If n1= n2 and m1> m2 then λm1,n1 > λm2,n2. Proof. Since xm1−m2+ng(λ m1,n; m2, n + 1) = nX−2 i=0 xi "m 1−mX2+n−1 i=n+1 xi− mX1−2 i=0 xi # − m1X+n−3 m1−m2+2n−1 xi< 0 , the results follows.
References
Afraimovich, V. S. & Hsu, S.-B. [1998] Lectures on Chaotic Dynamical Systems, National Tsing-Hua University, Hsinchu, Taiwan.
Bowen, R. [1973] “Topological entropy for noncompact sets,” Trans. Amer. Math. Soc. 184, 125–136. Cahn, J. W., Chow, S.-N. & Van Vleck, E. S. [1995]
“Spa-tially discrete nonlinear diffusion equations,” Rocky Mount. J. Math. 25, 87–118.
Chow, S.-N. & Mallet-Paret, J. [1995] “Pattern forma-tion and spatial chaos in lattice dynamical systems,” IEEE Trans. Circuits Syst. 42, 746–751.
Chow, S.-N. & Shen, W. [1995] “Dynamics in a discrete Nagumo equation: Spatial topological chaos,” SIAM J. Appl. Math. 55, 1764–1781.
Chow, S.-N., Mallet-Paret, J. & Van Vleck, E. S. [1996] “Pattern formation and spatial chaos in spatially dis-crete evolution equations,” Rand. Comput. Dyn. 4, 109–178.
Chua, L. O. & Yang, L. [1988a] “Cellular neural networks: Theory,” IEEE Trans. Circuits Syst. 35, 1257–1272.
Chua, L. O. & Yang, L. [1988b] “Cellular neural net-works: Applications,” IEEE Trans. Circuits Syst. 35, 1273–1290.
Chua, L. O. & Roska, T. [1993] “The CNN paradigm,” IEEE Trans. Circuits Syst. 40, 147–156.
Chua, L. O. [1998] CNN: A Paradigm for Complexity, World Scientific Series on Nonlinear Science, Series A, Vol. 31.
de Melo, W. & Van strain, S. [1993] One-Dimensional Dynamics (Springer-Verlag).
Hsu, C.-H. & Lin, S.-S. [1999] “Spatial disorder of cel-lular neural networks,” Japan J. Indust. Appl. Math., to appear.
Hsu, C.-H., Lin, S.-S. & Shen, W. [1999] “Traveling waves in cellular neural networks,” Int. J. Bifurcation and Chaos 9, 1307–1319.
Hsu, C.-H. [2000] “Smale horseshoe of cellular neural net-works,” Int. J. Bifurcation and Chaos 10, 2119–2129. Hsu, C.-H. & Lin, S.-S. [2000] “Existence and multiplic-ity of traveling waves in lattice dynamical system,” J. Diff. Eqns. 164, 431–450.
Juang, J. & Lin, S.-S. [2000] “Cellular neural networks: Mosaic pattern and spatial chaos,” SIAM J. Appl. Math. 60, 891–915.
Kevorkian, P. [1993] “Snapshots of dynamical evolution of attractors from Chua’s oscillator,” IEEE Trans. Circuits Syst. 40, 762–780.
Malkin, M. I. [1989] “On continuity of entropy of discon-tinuous mappings of the interval,” Selecta Math. Sov. 8, 131–139.
Mallet-Paret, J. & Chow, S.-N. [1995] “Pattern forma-tion and spatial chaos in lattice dynamical systems,” IEEE Trans. Circuits Syst. 42, 852–756.
Nekorkin, V. I. & Chua, L. O. [1993] “Spatial disorder and wave fronts in a chain of coupled Chua’s circuits,” Chua’s Circuit: A Paradigm for Chaos, ed. Madan, R. N. (World Scientific, Singapore), pp. 351–367. Parry, W. [1964] “Intrinsic Markov chains,” Trans.
Amer. Math. Soc. 112, 55–66.
Robinson, C. [1995] Dynamical Systems (CRC Press, Boca Raton, FL).
Thiran, P., Crounse, K. B., Chua, L. O. & Hasler, M. [1995] “Pattern formation properties of autonomous cellular neural networks,” IEEE Trans. Circuits Syst. 42, 757–774.
Appendix
To compute the g(λ; p, p− 1) of M in the proof of the main theorem, this work only computes the spe-cial case when m = 6. For other m, g(λ; p, p− 1) can be obtained analogously.
If m = 6 then det[M (6, 5)] = det −λ 1 0 0 0 0 0 0 0 0 0 0 −λ 1 0 0 0 0 0 0 0 0 0 0 −λ 1 0 0 0 0 0 0 0 0 0 0 −λ 1 0 0 0 0 0 0 0 0 0 0 −λ 1 0 0 0 0 0 0 0 0 0 0 −λ 1 1 1 1 0 0 1 1 1 1 1 −λ 0 0 0 0 0 0 0 0 0 0 1 −λ 0 0 0 0 0 0 0 0 0 0 1 −λ 0 0 0 0 0 0 0 0 0 0 1 −λ 0 0 0 0 0 0 0 0 0 0 1 −λ = det −λ 1 0 0 0 0 0 0 0 0 0 0 −λ 1 0 0 0 0 0 0 0 0 0 0 −λ 1 0 0 0 0 0 0 0 0 0 0 −λ 1 0 0 0 0 0 0 0 0 0 0 −λ 1 0 0 0 0 0 0 0 0 0 0 −λ 1 1 1 1 0 λ 0 1 1 1 1 −λ 0 0 0 0 0 0 0 0 0 0 1 −λ 0 0 0 0 0 0 0 0 0 0 1 −λ 0 0 0 0 0 0 0 0 0 0 1 −λ 0 0 0 0 0 0 0 0 0 0 1 −λ =−λg(λ; 5, 5) + λ2det 1 1 1 1 1 −λ 0 0 0 1 −λ 0 0 0 1 −λ =−λg(λ; 5, 5) + λ2det " −1 − λ − λ2 −1 1 −λ #
Hence g(λ; 6, 5) = det[M (6, 5)] =−λg(λ; 5, 5) + λ2 P3i=0 λi. Induction produces g(λ; p, p− 1) = −λg(λ; p − 1, p − 1) + λ2det " −λp−4− λp−2· · · − 1 −1 1 −λ # =−λg(λ; p − 1, p − 1) + λ2 p−3 X i=0 λi. By [Hsu & Lin, 1999], we know that
g(λ; p− 1, p − 1) = λ2 " λ2p−4− p−3 X i=0 λi !2# , and the formula of Lemma 3.6 is obtained by simple computation.
This article has been cited by:
1. Liping Li, Lihong Huang. 2010. Equilibrium Analysis for Improved Signal Range Model of Delayed Cellular Neural Networks.
Neural Processing Letters 31:3, 177-194. [CrossRef]
2. WEN-GUEI HU, SONG-SUN LIN. 2009. ZETA FUNCTIONS FOR HIGHER-DIMENSIONAL SHIFTS OF FINITE TYPE. International Journal of Bifurcation and Chaos 19:11, 3671-3689. [Abstract] [References] [PDF] [PDF Plus]
3. JUNG-CHAO BAN, CHIH-HUNG CHANG. 2008. ON THE DENSE ENTROPY OF TWO-DIMENSIONAL INHOMOGENEOUS CELLULAR NEURAL NETWORKS. International Journal of Bifurcation and Chaos 18:11, 3221-3231. [Abstract] [References] [PDF] [PDF Plus]
4. YI WANG, FANG-YUE CHEN. 2004. THE ENTROPY OF STATIONARY SOLUTIONS' MAP OF CELLULAR NEURAL NETWORKS. International Journal of Bifurcation and Chaos 14:12, 4317-4323. [Abstract] [References] [PDF] [PDF Plus]
5. SONG-SUN LIN, WEN-WEI LIN, TING-HUI YANG. 2004. BIFURCATIONS AND CHAOS IN TWO-CELL CELLULAR NEURAL NETWORKS WITH PERIODIC INPUTS. International Journal of Bifurcation and Chaos 14:09, 3179-3204. [Abstract] [References] [PDF] [PDF Plus]
6. HSIN-MEI CHANG, JONG JUANG. 2004. PIECEWISE TWO-DIMENSIONAL MAPS AND APPLICATIONS TO CELLULAR NEURAL NETWORKS. International Journal of Bifurcation and Chaos 14:07, 2223-2228. [Abstract] [References] [PDF] [PDF Plus]
