Piecewise two-dimensional maps and applications to cellular neural networks

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International Journal of Bifurcation and Chaos, Vol. 14, No. 7 (2004) 2223–2228 c

World Scientific Publishing Company

PIECEWISE TWO-DIMENSIONAL MAPS AND

APPLICATIONS TO CELLULAR NEURAL NETWORKS

HSIN-MEI CHANG and JONG JUANG

Department of Applied Mathematics, National Chiao Tung University, Hsin-Chu 30050, Taiwan

Received March 13, 2003; Revised September 9, 2003

Of concern is a two-dimensional map T of the form T (x, y) = (y, F (y) − bx). Here F is a three-piece linear map. In this paper, we first prove a theorem which states that a semiconjugate condition for T implies the existence of Smale horseshoe. Second, the theorem is applied to show the spatial chaos of one-dimensional Cellular Neural Networks. We improve a result of Hsu [2000].

Keywords: Cellular Neural Networks; Smale horseshoe; piecewise two-dimensional map.

1. Introduction

We consider a piecewise two-dimensional map of the form T (x, y) = (y, F (y) − bx) , (1) where F (y) =    a1y + a0− a1+ c1 y ≥ 1, a0y + c1 |y| ≤ 1, a−1y + a−1− a0+ c1 y ≤ −1. (2) Here a0 < 0, a1, a−1 > 1, b > 0, and c1 ∈ R is a biased term. The graph of F is given in Fig. 1.

The motivation for studying such a map is, in part, due to the form of the map is a gen-eralized version of Lozi map [Lozi, 1978]. More importantly, the map arises in the study of com-plexity of a set of bounded stable stationary solu-tions of one-dimensional Cellular Neural Networks (CNNs) (see e.g. [Chua, 1998; Chua & Yang, 1998a, 1998b]). In this paper, we first prove a theorem which states that a semiconjugate condition for T implies the existence of Smale horseshoe. Second, we apply the theorem to show the spatial chaos of one-dimensional Cellular Neural Networks. Such CNNs are of the form (e.g. [Ban et al., 2002, 2001;

Hsu, 2000]). dxi

dt = −xi+ z + αf (xi−1) + af (xi)

+ βf (xi+1) , i ∈ Z (3a) where f (x) is a piecewise-linear output function defined by f (x) =    rx + 1 − r x ≥ 1 x |x| ≤ 1 lx + l − 1 x ≤ −1, (3b)

where r and l are positive constants. The quantity z is called threshold or bias term, related to indepen-dent voltage sources in electric circuits. The con-stants α, a and β are the interaction weights be-tween neighboring cells. The study of problems for the case of r = l = 0 and α = β has been estab-lished in [Chua, 1998; Chua & Yang, 1998a; Juang & Lin, 2000]. Here we consider r > 0 and l > 0. Then the main results are the following. Given α and β, if (z, a) is in a certain parameter region Σα,β (see Theorem 3.1), then there exist r and l suffi-ciently small for which Λl,r (see Theorem 3.1) is a hyperbolic invariant set. Consequently, the spatial entropy of the corresponding set of bounded, stable stationary solutions is ln 2.

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y F_{( )}_y

Fig. 1. a1= 1.2, a0= −0.5, a₋1= 1.5, c1= 0.2.

2. Main Results

We first introduce some notations. Let

S = {(x, y) ∈ R2: |x| ≤ p, |y| ≤ p} . (4) Here p > 1. Let the four corners of S be labeled as

K = (p, p) , L = (p, −p) ,

M = (−p, −p) , N = (−p, p) . (5a) Set

K = (p, 1) , L = (p, −1) ,

M = (−p, −1) , N = (−p, 1) . (5b) The x and y coordinates of K are denoted, respec-tively, by Kx _{and K}y_.

We next number the following conditions. K₁y ≥ p > 1 , (6a)

Ny1 ≤ −p , (6b)

Ly1 ≥ p , (6c)

and

M1y≤ −p . (6d)

Here the subscript denotes the iteration index un-der the map T . For instance, K1y denotes the y co-ordinate of T (K) = K1. Suppose (6) holds. Then T (S) ∩ S has three vertical strips. See Fig. 2. Sim-ilarly, T−1

(S)T S has three horizontal strips, and T−1(S)T S T T (S) has 9 components. By induction Tn

j=−n Tj(S) has 9n components. With this infor-mation we can define a semiconjugate

h : Λ → {0, 1, 2}2 (7) which is onto. Here Λ =T∞

j=−∞(Tj(S)T S). If the components of Λ are points, then Λ is a Cantor set.

N1 K1 K( p, p) K( P, 1) L( p, -1) L( p, -p) N1 K1 L1 M1 N(-p, p) N(-p, 1) M(-p, -1) M(-p, -p) M1 L1 V1 U1 S1 Fig. 2.

This, in turn, implies that the semiconjugacy h is one to one and so is a conjugacy. This motivates the following definition.

Definition 1.1. _{Conditions on b, a}−1, a0, and a1 so that there exists a p > 1 for which (6) holds are called a semiconjugate condition for T .

To prove the main theorem, we need to intro-duce more notations. Now, T (S)T S, has three ver-tical strips, say S1, U1and V1. The one on the right, see Fig. 2, is labeled as S1. Clearly, T (S1)T S also has three vertical strips. The strip of T (S1)T S1 is to be denoted by S2. We then define Sninductively. Note that Sn, n ∈ N, are all parallelograms. Us and Vn are defined similarly.

The parallelogram N1K1K1N1, see Fig. 2, is to be denoted by S1. Likewise, Sndenotes the parallel-ogram NnKnKnNn. The length of the shorter side of the parallelogram Sn (resp. Sn) is to be denoted by

dn(resp. cn) . (8a) The slope of the longer side of the parallelogram Sn is to be denoted by

mn. (8b)

Lemma 2.1. _{The following recursive relations}

hold.

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c

i L

K

( , )

p p

l

i L

K

( , )

p 1

K

i

K

i

d

i L

d

i l Fig. 3. (i) di = _mci_i, ci+1= bdi, (ii) mi+1= a1−_mb i, m1 = a1.

Proof. The first recursive relation is obvious. To see (ii), let li be given as in Fig. 3. We then see that Ki = (p − (li/mi), p) and Ki = (p − (li + p − 1)/mi, 1). Now, the slope mi+1 = the slope of T (Ki)T (Ki) = Ki+1Ki+1 = F (p) − F (1) + b((1 − p)/mi)/(p − 1) = a1− (b/mi).

Lemma 2.2. _If _{b > 0 and a}₁ _{≥ 2(1 + b), then}

limn→∞ cn= 0.

Proof. We first prove that limn→∞ mn = (a1 + pa2

1− 4b)/2. To this end, we see that an induc-tion would yield that mi ≥ 1 for all i ∈ N and that mi is decreasing in i. Suppose x is the limit of {mn}. Then x must satisfy equation x = a1− (b/x). Upon using the fact that m1 = a1, we conclude that x = a1+pa21− 4b/2 as asserted. Now, using Lemma 2.1(i), we get dn= bn−1d1/Qn_i=2mi. Thus,

dn≤ 2b a1+pa21− 4b !n−1 d1 ≤ 2b a1 n−1 d1 ≤ b 1 + b n−1 d1.

We have just completed the proof of the lemma. Similarly, we have the following lemma.

Lemma 2.3. _If _{b > 0 and a}₋₁ _{> 2(1 + b), then}

the length of the shorter side of the parallelogram Vn shrinks to zero as n → ∞.

Using Lemmas 2.2 and 2.3, we have the follow-ing lemma.

Lemma 2.4. If b > 0, min{a1, a−1} > 2(1 + b), then the length of the shorter side of the parallelo-gram Un shrinks to zero as n → ∞.

Remark. The assumptions on Lemmas 2.2–2.4 would also yield thatT−∞

j=0(Tj(S)T S) are pairwise disjoint horizontal line segments.

We are now ready to state our main results.

Theorem 2.1. _Let _{F be a piecewise linear map}

defined as in (2) and the bias term c1 satisfy the inequality

max{−1 − b, a0+ 1 + b}

< c1< min{1 + b, −a0− 1 − b} , (9) then a semiconjugate condition for T implies the conjugate of h.

Proof. Note that K1y ≥ p, (6b) and (6d) are equiv-alent to the following inequalities.

p(a1− 1 − b) ≥ a1− a0− c1, (10a) −a0+ c1 ≥ p(1 + b) , (10b) −a0− c1≥ p(1 + b) , (10c) and

p(a−1− 1 − b) ≥ a−1− a0+ c1, (10d) respectively. We remark (10b) and (10c) to ensure that −a0 − 1 − b > 0, as a result, inequality (9) makes sense. Using (10a) and (10b), we see imme-diately that −a0+ c1 b + 1 ≥ p ≥ a1− a0− c1 a1− b − 1 . (11)

Note that a1 − b − 1 being positive is guaranteed by the fact that p > 1 and the assumptions on c1. Using (10), we get that

a1 ≥ −2a0(b + 1) c1− a0− 1 − b = 2(b + 1) 1 +1 + b − c1 a0 ≥ 2(b + 1) . (12a)

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The last inequality is justified by the assumptions on c1. Similarly, we see that

a−1≥ 2a0(b + 1) c1+ a0+ 1 + b = 2(b + 1) 1+1 + b + c1 a0 ≥ 2(b + 1) . (12b) It then follows from Lemmas 2.2–2.4 that T∞

j=−∞(Tj(S)T S) is a Cantor set. We thus com-plete the proof of the main theorem.

Remarks

(1) If F (y), as defined in 2, is such that a0 > 0, and a1, a−1< −1, then a similar result can also be obtained.

(2) The theorem holds true in general for F being a finitely many piecewise linear map. Specifically, if the bias term c1 is not “too biased”, then a semiconjugate condition for T implies the exis-tence of Smale horseshoe.

In the following, we give conditions on a0, a1, a−1, b and c for which T has a semiconjugate condition.

Theorem 2.2. Let a0 < 0, a1, a−1 > 1 and b > 0. Suppose a0+ 1 + b < 0, min{a1, a−1} > 2(1 + b). Let the bias term c1 satisfy (9), and that

a1 ≥ −2a0(b + 1) c1− a0− 1 − b (13a) and a−1 ≥ 2a0(b + 1) c1+ a0+ 1 + b . (13b)

then there exists a p > 1 such that T has a semi-conjugate condition.

3. Applications to CNNs

A basic and important class of solutions of (1) is the bounded, stable stationary solutions. In the case that r = l = 0 and α = β, the corresponding stable stationary solutions have been studied in [Chua & Yang, 1998a; Juang & Lin, 2000]. The case that r and l are positive is considered in [Ban et al., 2002, 2001; Hsu, 2000]. The techniques in these two cases are quite different. Specifically, in the latter case, the question of complexity of a set of stable station-ary solutions is converted to asking how chaotic is a map. If α or β = 0, then the resulting map is one-dimensional [Ban et al., 2002, 2001]. If α, β 6= 0,

then the resulting map is a two-dimensional of the following form [Hsu, 2000]

T (x, y) = y, 1 β (F (y) − ay − z) − α βx

=: (y, F (y) − bx) . (14a) Here, F (y) =            1 r y − 1 r + 1 y ≥ 1 y |y| ≤ 1 1 l y − 1 + 1 l y ≤ −1. (14b)

Hsu [2000] used a theorem of Afraimovich (see e.g. [Afraimovich, 1993]) as well as a semicon-jugate condition to show that in certain param-eters’ region, the map T has Smale horseshoe structure. However, Afraimovich’s Theorem is not needed in this case. Only a semiconjugate condition is required.

To apply Theorem 2.2, we first note that a−1= 1 β( 1 l − a), a0 = 1 β(1 − a), a1 = 1 β( 1 r − a), c1 = −z β , b = α_β. With the above identifications, we immedi-ately have the following results concerning the com-plexity of a set of bounded, stable stationary mosaic solutions of (3). Here the stationary mosaic solu-tions (xi)∞i=−∞ means that (xi)∞i=−∞ is a stationary solution of (3) and that |xi| > 1 for all i ∈ Z. More-over, the mosaic solutions obtained in the following theorem are bounded and stable (see e.g. [Chua & Yang, 1998a; Hsu, 2000]).

Define s = α + a + β. Assume the bias term z satisfies the following inequality.

max{−s + a, s − 2a + 1}

< z < min{s − a, 2a − 1 − s} . (15) Define, respectively, the regions Σα,β and Σα,β,l,r as follows. Σα,β = {(z, a) ∈ R2| (15) holds} , (16) and Σα,β,l,r = {(z, a) ∈ R2|r < r+, and l < l+} . (17) Here, r+_z,α,a,β = 2a − s − 1 − z a(1 + s − z) − 2s, (18a) and l+_z,α,a,β = 2a − s − 1 + z a(1 + s + z) − 2s. (18b) We are now in a position to state the following results.

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Theorem 3.1. _Let _{α and β be positive numbers}

and let a > 1 + α + β. Suppose (z, a) ∈ P α,β. Then there exist r and l sufficiently small, more precisely 0 < r < r+ _{= r}+

z,α,a,β and 0 < l < l+= l+_z,α,a,β for which T has a hyperbolic invariant

set Λl,r(z, α, a, β) = Λl,r in the (x, y) plane such that T |Λ_l,r is topologically conjugate to a two-side Bernoulli shift of two symbols. Hence, the spatial entropy of the corresponding set of stationary solu-tions equals ln 2.

l

1

p

0

p

0

r

_-1 z a

Fig. 4. = 13, l1 : −z + a(1 − 2) = 1, p0 : z = 2a,

r₋1: z + a(1 − 2) = 1, p0: z = −2a.

l

₁

r

_-1

p

0

p

0 z a

Fig. 5. = 16, l1 : −z + a(1 − 2) = 1, p0 : z = 2a,

r₋1: z + a(1 − 2) = 1, p0: z = −2a. (I) (II) (III) (IV) (V) (VI) (VII) (VIII) (IX) Fig. 6.

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Remarks

(1) Note that if (z, a) ∈ Σα,β, then −2s + a(1 + s − z) = a(−z − 1 − s + 2a) + 2(a − 1)(s − a) > 0 and −2s + a(1 + s + z) = a(z − 1 − s + 2a) + 2(a − 1)(s − a) > 0. Consequently, those r+_and l+ _{are positive.}

(2) Adapting the notations in [Juang & Lin, 2000] we let α = β = a. Then the set Σα,β = Σ is given in the following figure.

Note that for 0 < ε < 1₄, Σ ( [3, 3] (see Fig. 5.1 of [Juang & Lin, 2000] for the definition of [3, 3]), and for 1₄ ≤ < 1₂, Σ = [3, 3] (see Figs. 4 and 5). Applying Theorem 3.1, we conclude that let α = β = a, 1₄ ≤ ε < 1₂, and if (z, a) ∈ Σ= [3, 3], then there exist r and l sufficiently small for which Λl,r is a hyperbolic invariant set. This result gener-alized those in [Chua, 1998; Chua & Yang, 1998a; Juang & Lin, 2000]. For 0 < < 1

4, if (z, a) ∈ Σ and r, l > 0 is sufficiently small, then the corre-sponding set of stable, bounded stationary solutions also has spatial entropy ln 2.

(3) To get a feel of how small r and l are required to be, set = 1₄ and z = 0. We see easily that r+_{= l}+ _{has a maximum} 1

16 for 2 < a < ∞. (4) Figure 6 is a collection of a computer

sim-ulation with a set of parameters, satisfying a > 1 + α + β, 0 < r < r+ = r+_z,α,a,β and 0 < l < l+ _{= l}+

z,α,a,β. Specifically, we choose α = β = 1, r = l = 0.005, z = 0, a = 4. Each collection in Fig. 6 contains two arrays of col-ors. The first array is the initial outputs. The second array represents the final outputs. If the state xj of a cell cj is such that |xj| < 1, then

we color it green. If the state xj of a cell cj is less than −1 (greater than 1, respectively), then we color it blue (red, respectively).

Acknowledgment

We thank Dr. C. J. Yu for providing the simulation work in the paper.

References

Afraimovich, V. S., Bykov, V. V. & Shil’nikov, L. P. [1993] “On the structurally unstable attracting limit sets of the Lorentz attractor type,” Trans. Mosc. Math. Soc. 2, 153–215.

Ban, J.-C., Chien, K.-P., Lin, S.-S. & Hsu, C.-H. [2001] “Spatial disorder of CNN — with asymmetric out-put function,” Int. J. Bifurcation and Chaos 11, 2085–2095.

Ban, J.-C., Lin, S.-S. & Hsu, C.-H. [2002] “Spatial dis-order of cellular neural networks – with biased term,” Int. J. Bifurcation and Chaos 12, 525–534.

Chua, L. O. [1998] CNN: A Paradigm for Complexity (World Scientific, Singapore).

Chua, L. O. & Yang, L. [1998a] “Cellular neural net-works: Theory,” IEEE Trans. Circuits Syst. 35, 1257–1272.

Chua, L. O. & Yang, L. [1998b] “Cellular neural net-works: Applications,” IEEE Trans. Circuits Syst. 35, 1273–1290.

Hsu, C. H. [2000] “Smale Horseshoe of cellular neu-ral networks,” Int. J. Bifurcation and Chaos 10, 2119–2127.

Juang, J. & Lin, S.-S. [2000] “Cellular neural networks: Mosaic pattern and spatial chaos,” SIAM J. Appl. Math. 60, 891–915.

Lozi, R. [1978] “Un attracteur ´etrange du type attracteur de H´enon,” J. Phys. (Paris) 39, 9–10.

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Nonlinearity 22:5, 1123-1144. [CrossRef]

4. JONQ JUANG, CHIN-LUNG LI, MING-HUANG LIU. 2006. CELLULAR NEURAL NETWORKS: MOSAIC PATTERNS, BIFURCATION AND COMPLEXITY. International Journal of Bifurcation and Chaos 16:01, 47-57. [Abstract] [References] [PDF] [PDF Plus]