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1 Spatial Disorder of Cellular Neural Networks - with Biased Term Jung-Chao Ban Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan Cheng-Hsiung Hsu Λ Department of Mathematics, National Central University, Chung-Li 320, Taiwan Song-Sun Lin y Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan Abstract 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 onedimensional and the spatial entropy of stable stationary solutions can be obtained explicitly as a stair-case function. Key words. spatial disorder, topological entropy, Bernoulli shift, transition matrix. AMS subject classification. 37B0, 37B5, 37B40, 68T05, 68T0. Λ Work partially supported by the NSC under Grant No M and the National Center for Theoretical Sciences Mathematics Division, R. O. C.. y Work partially supported by the NSC under Grant No M , the Lee and MTI Center for Networking Research and the National Center for Theoretical Sciences Mathematics Division, R. O. C..

2 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 [988a, 988b]. The circuit equation of a cell is dx i dt = x i + z + fff(x i )+af(x i )+fif(x i+ ); i 2 Z ; (.) where f(x) is a piecewise-linear output function defined by f(x) = 8 >< >: rx + m r if x ; mx if j x j» ; `x + ` m if x» : (.2) Here r, m and ` are nonnegative 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 function ff, a and fi are real constants and called the space-invariant A-template denoted by A [ff; a; fi]: (.3) For simplicity, f will be denoted by f r,with` = r and m =,i.e., f r (x) = 8 >< >: rx + r if x ; x if j x j» ; rx + r if x» ;: (.4) CNN is applied mainly in image processing and pattern recognition [Chua & Roska, 993], [Chua & Yang, 988a] and [Thiran et al, 995]. A basic and important class of solutions of (.) are the stable stationary solutions of (.). In particular, the complexity of stable stationary solutions of (.) must be investigated. When the output function is f 0, i.e. r = 0 in (.4), much work has subsequently been done in the electrical engineering community, see [Chua & Roska, 993], [Chua & Yang, 988a] and references therein. In addition, [Juang & Lin, 2000], [Hsu & Lin, 999, 2000] and [Hsu et al, 999] recently considered mathematical results involving the complexity of stable stationary solutions and the multiplicity of traveling wave solutions. 2

3 [Juang & Lin, 2000] partitioned the parameters space (a; z) into a finite number of regions in R 2 such that in each region (.) with f = f 0 has the same spatial entropy. However, for z = 0 and r 2 (0; ), [Hsu & Lin, 999] proved that (.) and (.4) can release infinite different spatial entropies and the entropy function is a devil-staircase like function in r. The method used in [Hsu & Lin, 999] considers that the stationary solutions of (.) as an iteration map. In fact, if output v = f(x) is taken as the unknown variable, i.e., let v i = f(x i ) and u i+ = v i : (.5) and if f is invertible with inverse function F, then the stationary solutions of (.) can be written as one- or two-dimensional iteration maps as follows, when ff =0and fi 6= 0 and T (v) = (F (v) z av); (.6) fi T 2 (u; v) =(v ; (F (v) z ffu av)); (.7) fi when ff 6= 0and fi 6= 0. For these maps, each bounded trajectory corresponds to the outputs of bounded stationary solutions. In practice, if the maps are chaotic, then the stationary solutions of (.) are spatially chaos. However, only stable stationary solutions of (.) 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 j S must be computed. If the entropy is positive, then the stable stationary solutions of (.) are spatial chaos. For convenience, T j S is denoted herein as T. [Hsu & Lin, 999] considered (.6) with z = 0, the odd symmetry of the map T makes it is much easier to investigate the complexity of T than the case of z 6= 0. Therefore, this work focuses on the complexity of the one dimensional map T with z 2 R by some complicated computation. According to our results, the entropy function is a stair-case function. As for the two-dimensional map T 2, when r is positive and sufficiently small, the Smale Horseshoe structures of stable stationary solutions of (.) and (.4) are constructed, for details, see [Hsu, 2000]. Carefully examining the orbits of T reveals that the entropy function h is a stair-case function of r for fixed a, z and fi. The main results are Main Theorem. 3

4 Assume fi =, 0 < z < (a)(see Lemma 3.). Denote r (z) = a + z 2 a 2 2+az (.8) and h(r) is the entropy function of T in (.6). Then there exists p(z) 2 Z + and a strictly decreasing sequence fr p;p (z)g; p =3; 4; ;p(z) with r (z) <r p;p and r p <r p;p <r p such that (i) If 3» p» p(z) and r 2 [r p;p (z);r p ;p 2 (z)) then h(r; z) =ln p ;p 2. Where p;p is the largest root of 2 [ 2p 3 p 3 P i p 2 P j=0 j ]=0. (ii) if r 2 (r (z);r p(z);p(z) ) then h(r; z) =ln p(z);p(z). (iii) if r 2 (μr (z);r (z)] then h(r; z) =0. (iv) if r 2 [0; μr (z)] then h(r; z) =ln2. Moreover, p(z) is a decreasing function of z and lim p(z) =. z!0 + h(r;z) ln 2 0 r (z) r (z) r 4,3 r 3,2 r 2, r p(z),p(z)- Fig.. Entropy of T with z 6= 0. r The above results or the proof of the main theorem in section 3 indicate that the nonzero bias z causes a situation in which map T does not have enough 4

5 periodic orbits when r 2 (μr (z);r (z)] and it makes the entropy equal to zero. Therefore, the entropy function of T has a stair-case structure as shown in Fig.. This differs from those results of a devil-staircase like function in [Hsu & Lin, 999] with z = 0 as shown in Fig. 2. Additionally, the results of [Hsu & Lin, 999] recalled in the following corollary can be considered as the limiting case of the main theorem when z tends to 0. Corollary. Assume fi > 0, z = 0 and a>fi+. Denote a fi r = r (a; fi) = a(a ) + fi(a 2) ; a fi r 2 = r 2 (a; fi) = a(a ) + fi(fi ) ; and h(r) is the entropy function of T in (.6) with F = F r = f r, r > 0. Then there exists a strictly decreasing sequence fr p g; p =2; 3; ; with lim r p = r p! ; such that (i) If r 2» r<, then h(r) =0. a + fi (ii) If r 2 [r p ;r p ), p =3; 4;,thenh(r)isln p where p is the largest root of 2p 2 ( p 2 P i ) 2 =0. Moreover, p is strictly increasing in p with + p 5 2 (iii) If r 2 [0;r ], then h(r) =ln2. h(r) 6 = 3 < p < 2; for p =4; 5; ln2 ffl ffl ffi 0 r ffl ffi ffl r 3 ffi ffl r 2 ffl a+fi - r Fig. 2. Entropy of T with z =0. 5

6 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 dynamics, 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 z 6= 0. If a>, fi > 0, and m =, then the inverse function F of f r is F (v; r) = 8 >< r v + if v ; r v if j v j» ; (2.) >: r v + r and the map T can be rewritten as T (v; a; fi; r) = 8 >< if v» ; fi ( r v + av z) if v ; r (v av z) if j v j» ; fi (2.2) >: fi ( r v + av z) if v» : r Instead of F (v; r) and T (v; a; fi; r), F (v) andt (v) willbeusedifitdoesnot cause any confusion. For simplicity, assume that fi = and z 0 hereinafter. The graph of T can be found in the following figure. 6

7 C u u=t(v) A u=v - 0 v D B Fig. 3. Graph of T. An elementary computation produces that rz r + rz r + A =(A ;A 2 )=( ; ra r ra r ); B =(B ;B 2 )=(; a z); C =(C ;C 2 )=( ;a z); D =(D ;D 2 )=( rz + r ra r ; rz + r ra r ): According to [Hsu, 2000] and [Juang & Lin, 2000], any orbit ft k (v)g of T with jt k (v)j» for some k 0 is unstable. Hence, only trajectories of T lying outside the unit rectangle in (u; v) plane should be considered. Therefore, assume that B 2 < and C 2 > while these conditions are equivalent to 2 a < z < a 2: For further computation, we give the following notations. Definition 2. Assume a>2. (i) Define functions r (z) and μr (z) by r (z) = a + z 2 a 2 2+az and μr (z) = a z 2 a 2 2 az : (2.3) (ii) Let m; n 2 Z +, if the slope of f, r = r m;n satisfies T m (B 2 )= and T n (C 2 )=; (2.4) then we call map T is of (m; n) type and denote r m;m, k m;n and ο m;n by r m;m = r m, k m;n = r m;n a and ο m;n = k m;n: 7

8 (iii) Define polynomials E(x; m) and U(x; m) by E(x; m) = a m i= U(x; m; n) = (a + z) x i a +2; (2.5) m i=n+ x i +2a n i= x i 2a +4: (2.6) From Fig. 3, the relative positions of A; B; C and D are easily obtained in the following. Lemma 2. Assume a > 2, then r (z) and μr (z) are increasing and decreasing functions of z, respectively. Moreover, we have () If r 2 (r (z); ), then A 2 >C 2 and B 2 >D 2. (2) If r = r (z) then A 2 >C 2 and B 2 = D 2. (3) If r 2 (μr (z);r (z)), then A 2 >C 2 and D 2 >B 2. (4) If r =μr (z), then A 2 = C 2 and D 2 >B 2. (5) If r 2 (0; μr (z)), then A 2 <C 2 and D 2 >B 2. Proof. By elementary computation, we have r 0 (z) = 2a 2 (a 2 2+az) 2 and μr 0 (z) = 2 2a (a 2 2 az) 2 and r (z) and μr (z) are increasing and decreasing functions of z respectively. The proofs from () to (5) are also simple and omitted. The proof of the main theorem in section 3 indicates that the case of () in Lemma 2.2 are more interesting and complicated. 3 Proof of Main Theorem In this section, we prove the main theorem by introducing some lemmas. If z > 0, the following lemmas will show that unique r m;m lies between r m;m and r m ;m such that (2.4) holds. Lemma 3. Assume m 3 and define (a) by (a) minfa 2; a3 +6a 2 4a 3a 2 6a +4 g: If 0 < z < (a), p > q and r p;q satisfies (2.4) with r m;m < r p;q < r m ;m, then p = m and q = m. 8

9 Proof. First, we claim that U(ο p;q ; p; q) = 0 and E(ο m;m ; m) = 0. computation, it is obvious that By simple Define R and L by T () = rz + ra and T ( ) = rz ra : (3.) R = T () and L = T ( ): (3.2) If p > q and r = r p;q satisfies (2.4), then it is not difficult to compute that ο p;q satisfies By (3.) and (3.2), we know that and (3.3) can be rewritten as L( ο q p;q) ο p;q + R( οp p;q) ο p;q =2a 4: (3.3) R + L = 2ο p;q r p;q 2, R = r p;q(z + a) r p;q a, j=0 ( 2ο q p p;q 2) ο j p;q + R ο j p;q =2a 4; (3.4) r p;q j=0 p ο p;q (a + z) ο j p;q +( 2ο q p;q 2) ο j p;q =2a 4: (3.5) r p;q j=q According to the definition of ο p;q, we have U(ο p;q ; p; q) = 0. Similarly, we have E(ο m;m ; m) =0. Next, we show that r m;m satisfies (2.4) and r m;m < r m;m < r m ;m. Since z > 0 and ο m;m > 0, by (2.5), (2.6) and (3.5), we have m a i= ο i m;m <a 2; m a i= j=0 ο i m ;m = a 2; (3.6) and a m i= ο i m;m >a 2; a m i= ο i m;m = a 2: (3.7) From (3.6) and (3.7), r m;m satisfies (2.4) and r m;m < r m;m < r m ;m, for m>2. 9

10 Now, we claim that no r p;q satisfies (2.4) and r m;m < r p;q < r m ;m except for p = m and q = m. For convenience, let h = r m ;m, k = r m;m and ο = r p;q, where p = m + n, q = m n and» n<m 2. By (2.6) and elementary computation, we have and U(h; p; q) < 0ifand only if 2a (a + z)h n +(z a)h (n+) < 0 (3.8) U(k; p; q) < 0ifand only if 2a (a + z)k n+ +(z a)k n < 0: (3.9) Obviously that U 0 (x; p; q) > 0andifU(h; p; q)u(k; p; q) > 0; by intermediate value theorem, no ο lies between h and k and satisfies (2.4). Therefore, we claim that U(h; p; q) < 0 and U(k; p; q) < 0, if a; z satisfy 0 < z < (a). Denote P (x) and Q(x) by and P (x) =2a (a + z)x n+ +(z a)x n Q(x) =2a (a + z)x n +(z a)x (n+) ; then P (x) and Q(x) are concave functions in (0,] and P () = Q() = 0. By elementary computation or [3], we know that r 2;2 = r 2 = a 2 a 2 a and 0 <k<h< r 2;2 a = a 2 a : (3.0) If a; z satisfy 0 < z < (a), we have P ( a 2 ) < 0. Since P (x) is concave, a by (3.9) we obtain that U(k; p; q) < 0. Furthermore, the zero of Q(x) is obviously larger than the zero of P (x) in(0; ). By the concavity ofq(x), we also obtain P ( a 2 ) < 0 and which implies U(h; p; q) < 0. Hence, the proof is a complete. Corollary 3. Under the same assumptions of lemma 3., we have r m+;m < r m;m for all integer m>. Now, if z is fixed, since p! lim r p = r and r (z) is an increasing function of z, by lemma 3., we obtain that there exists a maximal positive integer p(z) such that (2.4) holds for sequence fr p;p (z)g with p =3; 4; ;p(z) and no r m;m (z) satisfies (2.4) with m>p(z). As demonstrated later that this observation reveals the stair-case structure of entropy function h of T. For completeness, this study recalls the definitions and some results of entropy for a dynamical system. Details can be found in [Bowen, 973] or [x.6, Afraimovich &Hsu,998]. 0

11 Definition 3. Let G :! be a dynamical system on the complete metric space and S ρ be an invairant set. (i) The set n (x) = fg k (x)g n k=0 is called an orbit segment of temporal length n. Two segments n (x) and n (y) are said to be (n; ffl)-separated if thers exists k 2 Z ; 0» k» n, such that dist(g k (x);g k (y)) ffl. (ii) Let S n;ffl be a set of segments of temporal length n such that (a) if n (x); n (y) 2 S n;ffl, then they are (n; ffl)-separated; (b) if w 2 S and n (w) =2 S n;ffl, then thereisx 2 S such that n (x) 2 S n;ffl and dist(g k x; G k w) <fflfor each k =0; ; ;n. Define ~C n;ffl = ]S n;ffl, the number of elements of the set S n;ffl and C n;ffl = inf Cn;ffl ~. Then, the entropy function of G, denoted by h(g), is defined as S n;ffl follows: lnc n;ffl h(g) =limlim ffl!0 n! n : (3.) Proposition 3. ([x2.4,afraimovich & Hsu, 998] & [Robinson, 995]) Let ff M : ± M! ± M being a subshift of finite type with the transition matrix M on N symbols. Denoted by K n the number of admissible words of length n +, then the entropy of ff M is equal to lnk n h(ff M )= n! lim n = lnj j; where is the real eigenvalue of M such that j j j j j for all other eigenvalues j 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 In( ; ) with the covering relation as shown in the proof of the main theorem later. Definition 3.2 An interval I T -covers an interval I 2 provided I 2 T (I ). This study writes I! I 2. Proof of Main Theorem. First, we consider the case r > r (z), i.e. A 2 > C 2 and B 2 > D 2. Let R + (r) and R (r) be the first components of the intersection points of AB with u = + and u =, respectively. A simple computation produces R (r) = 2r + rz ra and R + (r) = +rz ra : (3.2) Then, the continuity oft (v; r) with respect to r and lemma 3. make it easy to prove that for any positive integer 2 < p» p(z), there exists a unique

12 r p;p > 0 such that ft i (C 2 ; r p;p )g i= i= is a 2p -periodic orbit, i.e. of (p; p ) type. Where p(z) is the largest integer such that r p(z) less than r (z). Restated, after 2p iteration, (v; T(v; r p;p )) maps C to B and B to C respectively. Denote R + =(R + ;R + 2 )=AB T fu =g; R =(R ;R 2 )=AB T fu = g; L + =(L + ;L + 2 )=CD T fu =g, L =(L ;L 2 )=CD T fu = g, ra 2r + ra 2r + Ω r = f(v; u)j jvj» and juj» g; ra ra here fu = D 2 g T 2r ra CD =( ; a z) and Ω r ρ Ω. Figures 4 and 5 ra give the 5-periodic orbit and 7-periodic orbit of T at r 3;2 and r 4;3, respectively. Now, if 3» p» p(z) and r p;p» r < r p ;p 2 or r (z) < r < r p(z);p(z), define the 2p stable subintervals by and I p+ =(;R ); I 2 p+k =(T k+ (R + 2 );T k (R 2 )) for k =to p 2: I p =(L + 2 ; ); I p k =(T k (L + 2 );T k+ (L 2 )) for k =top : C u u=t(v) u=v L + R v L - R - D B I 3 I I 2 Fig. 4. Graph of T in (3; 2) type and its stable subintervals. I 4 I 5 2

13 u u=t(v) u=v L + C A R v L - R - D B I I 2 I 3 I 4 I 5 I 6 I 7 Fig. 5. Graph of T in (4; 3) type and its stable subintervals. The 2p subintervals have the following covering relation: I i! I i+ for i =top ; I p! I j for j = p +to 2p 2; I p+! I k for k =2top; I l! I l for l = p + 2 to 2p : Therefore, we obtain the following transition matrix M M[p; p ] of the 2p subshifts of finite type. M= p-th row with p-2 terms of p+-th row with p- terms of (2p-) (2p-) 3

14 This study defines spaces ± 2p and ± M by ± 2p = f; 2; ; 2p ; 2p g N, ± M = fs 2 ± 2p : M sk s k+ =for k =0; ; 2; g, with a metric on ± M by d(s; t) = P k=0 ffi(s k ;t k ) 3 k, for s =(s 0 ;s ; ) and t =(t 0 ;t ; )in± M, where ( 0 if i = j; ffi(i; j) = if i 6= j: Let ff M : ± M! ± M be the subshift of finite type for the matrix M, i.e. ff(s) = t where t k = s k+. Therefore, if r p;p» r < r p ;p 2 then there exists an invariant subset Λ p in Ω such that T j Λp is topological conjugate to the 2p subshift (± M ;ff M )withentropy h equals to ln p;p,where p;p is the positive maixmal root of characteristic polynomial of M. To derive p;p,we need the following lemma. Lemma 3.2 Given p 2 Z and p >, then the characteristic polynomial g(x; p; p ) of transition matrix M[p; p ] is g(x; p; p ) = x 2 (x 2p 3 p 3 P x i p 2 P j=0 x j ): Proof. By elementary matrix computation, see appendix A, we obtain g(x; p; p ) = xg(x; p ;p ) x 2 p 3 P Where, g(x; p ;p ) is the characteristic polynomial of M with z = 0, for details see [Hsu & Lin, 999]. In [Hsu & Lin, 999], we also have g(x; p ;p ) = x 2 [x 2p 4 ( p 3 P x i ) 2 ]. Therefore, the result follows by simple computation. By lemmas 3. and 3.6, we proof results (i) and (ii) of the main theorem. As for the assumption (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 isequivalent to the conditions of (4) and (5) in lemma 2.2., we knowthatd 2 >B 2 and C 2 A 2 4 x i :

15 in Fig. 7 such that the behavior of the map T resembles that of the logistic map as discussed in (Theorem 5.2. [Robinson, 995]). Therefore, there exists an invariant Cantor set such that T is topologically conjugate to a one-sided Bernoulli shift of two symbols. Since the entropy of the one-sided Bernoulli shift of two symbols is ln2, the result follows by Proposition 3.4. Finally, since lim r (z) =r,by lemma 3. we obtain that p(z) is a decreasing function of z with lim p(z) =0. The proof is complete. z!0 z!0 B u u=t(v)u=v A - 0 v D C I I 2 I 3 I 4 Fig. 6. Graph of T with r = r (z) and its stable subintervals. u D u=t(v) u=v A - 0 v C B I I 2 I 3 I 4 5

16 Fig. 7. Graph of T with r =μr (z) and its stable subintervals. Remark. (i) If we consider the output function are not symmetric, i.e. r 6= ` in (.2), then lemma 3. is no longer valid. In fact, there exists many different m; n such that r = r m;n lies between r p and r p for any p 3 and T is of (m,n) type. Hence, by the similar arguments in the proof of the main theorem, we also obtain transition matrix M[m; n] such that the corresponding characteristic ploynomial g(x; m; n) is g(x; m; n) =x 2 (x m+n 2 m 2 n 2 x i j=0 x j ): (3.3) (ii) By some further computation, the ordering relation of the maximal root m;n of g(x; m; n) can also be obtained as following lemma. Lemma 3.3 Given (m ;n), (m 2 ;n+) and m >m 2, then g( m ;n; m 2 ;n+ ) < 0. Moreover, we have () If n >n 2 then m ;n > m2 ;n 2. (2) If n = n 2 and m >m 2 then m ;n > m2 ;n 2. Proof. Since = x m m 2 +n g( m ;n; m 2 ;n+) n 2 < 0; the results follows. m m 2 +n x i [ i=n+ x i m 2 x i ] m +n 3 m m 2 +2n x i 6

17 Appendix To compute the g( ; p; p ) of M in the proof of the main theorem, This work only computes the special case when m =6. For other m, g( ; p; p ) can be obtained analogously. If m = 6 then det[m(6; 5)] = det = det = g( ;5; 5) + 2 det " 2 = g( ;5; 5) + 2 det Hence g( ;6; 5) = det[m(6; 5)] = g( ;5; 5) + 2 Induction produces g( ; p; p ) = g( ; p ;p ) + 2 det 7 3 P i : # " p 4 p 2 #

18 p 3 = g( ; p ;p ) + 2 i : By [Hsu & Lin, 999], we know that p 3 g( ; p ;p ) = 2 [ 2p 4 ( i ) 2 ]; and the formula of Lemma 3.6 is obtained by simple computation. 8

19 References Afraimovich, V.S. & Hsu, S.-B. [998] Lectures on chaotic dynamical systems, National Tsing-Hua University, Hsinchu, Taiwan. Bowen, R. [973] Topological entropy for noncompact sets," Trans. Amer. Math. Soc. 84, Cahn, J.W., Chow, S.-N. & Van Vleck, E.S. [995] Spatially discrete nonlinear diffusion equations," Rocky Mount. J. Math. 25 (995), Chow, S.-N. & Mallet-Paret, J. [995] Pattern formation and spatial chaos in lattice dynamical systems," IEEE Trans. Circuits Syst. 42, Chow, S.-N., Mallet-Paret, J. & Van Vleck, E.S. [996] Pattern formation and spatial chaos in spatially discrete evolution equations," Random and Comp. Dyn. 4, Chow, S.-N. & Shen, W. [995], Dynamics in a discrete Nagumo equation: spatial topological chaos," SIAM J. Appl. Math. 55, Chua, L.O. & Roska, T. [993] The CNN paradigm," IEEE Trans. Circuits Syst. 40, Chua, L.O. [998] CNN : A Paradigm for Complexity, World Scientific Series on Nonlinear Science, Series A, Vol. 3. Chua, L.O. & Yang, L. [988a] Cellular neural networks: Theory," IEEE Trans. Circuits Syst. 35, Chua, L.O. & Yang, L. [988b] Cellular neural networks: Applications," IEEE Trans. Circuits Syst. 35, Hsu, C.-H. [2000] Smale horseshoe of cellular neural networks," Int. J. Bifurcation and Chaos 0, Hsu, C.-H. & Lin, S.-S. [999] Spatial disorder of cellular neural networks," preprint. Hsu, C.-H. & Lin, S.-S. [2000] Existence and multiplicity of traveling waves in lattice dynamical system", J. Diff. Eqns. 64, Hsu, C.-H., Lin, S.-S. & Shen, W. [999] Traveling waves in cellular neural networks," Int. J. Bifur. and Chaos 9,

20 Juang, J. & Lin, S.-S. [2000] Cellular neural networks: mosaic pattern and spatial chaos," SIAM J. Appl. Math. 60, Kevorkian, P. [993] Snapshots of dynamical evolution of attractors from Chua's oscillator," IEEE Trans. Circuit Syst. 40, Malkin, M. I. [989] On continuity of entropy of discontinuous mappings of the interval," Selecta Mathenatica Sovietiea 8, Mallet-Paret, J. & Chow, S. -N. [995] Pattern formation and spatial chaos in lattice dynamical systems," IEEE Trans. Circuit Syst. 42, de Melo, W. & Van strain, S. [993] One-dimensional dynamics, Spriger- Verlag. Nekorkin, V. I. & Chua, L.O. [993] Spatial disorder and wave fronts in a chain of coupled Chua's circuits," from R.N. Madan, Chua's Circuit: A Paradigm for Chaos, World Scientific Publishing Co. Pte. Ltd., Parry, W. [964] Intrinsic markov chains," Trans. Amer. Math. Soc. 2, Robinson, C. [995] Dynamical Systems, CRC Press, Boca Raton, FL. Thiran, P., Crounse, K.B., Chua, L.O. & Hasler, M. [995] Pattern formation properties of autonomous cellular neural networks," IEEE Trans. Circuits Syst., 42 (995),

MARKOV PARTITIONS FOR HYPERBOLIC SETS

MARKOV PARTITIONS FOR HYPERBOLIC SETS TODD FISHER, HIMAL RATHNAKUMARA Abstract. We show that if f is a diffeomorphism of a manifold to itself, Λ is a mixing (or transitive) hyperbolic set, and V is a neighborhood