細胞類神經網路:馬賽克花樣，分歧點與複雜性

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(1)國立交通大學應用數學系碩士論文. 細胞類神經網路:馬賽克花樣，分歧點與複雜性 Cellular Neural Networks : Mosaic Patterns, Bifurcation and complexity. 研究生：劉明湟指導教授：莊. 重. 教授. 中華民國九十四年六月.

(2) 細胞類神經網路:馬賽克花樣，分歧點與複雜性 Cellular Neural Networks : Mosaic Patterns, Bifurcation and complexity. 研究生：劉明湟. Student：Ming-Huang Liu. 指導教授：莊. Advisor：Jonq Juang. 重. 國立交通大學應用數學系碩士論文. A Thesis Submitted to Department of Applied Mathematics National Chiao Tung University in partial Fulfillment of the Requirements for the Degree of Master in. Applied Mathematics June 2005 Hsinchu, Taiwan, Republic of China. 中華民國九十四年六月.

(3) 細胞類神經網路:馬賽克花樣，分歧點與複雜性. 學生：劉明湟. 指導教授：莊重. 國立交通大學應用數學學系﹙研究所﹚碩士班. 摘. 要. 我們主要探討一個細胞類神經網路模型的馬賽克花樣，在這裡考慮的輸出函數在無窮遠的地方並不是平坦的。許多複雜的參數區域是可以被完整地描繪出來，每一個參數區域的熵是可以藉由轉換矩陣的方法算出來﹔我們也利用參數 z 和 β 來討論一些馬賽克花樣的分歧現象，在這裡 z 是一個偏壓項、 β 是和鄰近細胞的互動比重。特別地，對於一個小的互動比重 β ，我們發現當加入偏壓項之後，許多新的複雜參數區域都會產生。然而當 β 增加到某一個範圍之後，許多上述的複雜參數區域會消失，但是又有一些新的複雜參數範圍會產生。. i.

(4) Cellular Neural Networks : Mosaic Patterns, Bifurcation and Complexity Student : Ming-Huang Liu. Advisor : Jonq Juang. Department of Applied Mathematics National Chiao Tung University Hsinchu, Taiwan,R.O.C. June 2005 Abstract. We study mosaic patterns of a one-dimensional Cellular Neural Network with an output function which is non-flat at infinity. Spatial chaotic regions are completely characterized. Moreover, each of their exact corresponding entropy is obtained via the method of transition matrices. We also study the bifurcation phenomenon of mosaic patterns with bifurcation parameters z and β. Here z is a source (or bias) term and β is the interaction weight between the neighboring cells. In particular, we find that by injecting the source term, i.e. z 6= 0, a lot of new chaotic patterns emerge with a smaller interaction weight β. However, as β increases to a certain range, most of previously observed chaotic patterns disappear, while other new chaotic patterns emerge.. ii.

(5) 誌. 謝. 首先非常感謝我的指導老師莊重教授在各方面的教導與勉勵，讓我在學業上以及待人處事上都受益良多；更感謝老師在生活中的包容與體諒，著實給我很大的幫助。此外，也要感謝謝世峰以及李金龍兩位學長，在我遭遇挫折時適時地給我ㄧ些建議與勉勵，讓我很快地能重拾信心、再接再勵；尤其是金龍學長對這篇論文的貢獻也相當大，很懷念研二升研三那段一起討論、一起奮戰的日子，感覺真的很棒。接著要感謝東海幫的學長姊們在生活上給我的照顧與關懷，讓我在交大這些日子一直都覺得很溫馨，尤其是舉卿和雅文兩位學姊，真的非常感謝妳們。此外，熱心的奕達學長、David Guo 學長以及心眉學姊等等，也在各方面給予我許多協助，謝謝你們對我的照顧。同窗好友郁泉、昱豪、小雅文以及所有 225 的同學們，互相幫忙與砥礪的情誼，陪我一同克服情緒上的低潮以及課業的壓力，真的很感謝你們。另外，我也要謝謝我的同學們，不管是分析組還是組合組的同學，因為有你們，讓我的研究所生活更加充實；還有系排的夥伴們，尤其是同期的戰友：許老、Robbin 、 Cool、明淇、明欣、貓頭以及阿立，和大家一起練球、一起拼戰的感覺是很值得回味的，在此也謝謝傅恆霖老師與白啟光老師對系排的支持與指導。珍惜在交大認識的所有人，尤其是研三這一年，讓我真的覺得過得很充實；校男排的夥伴們、雅欣等等，認識你們值得了，謝謝你們讓我的研三生活過得更加有意義；這一年並沒有白留。最後要感謝我的家人，尤其是辛勞的母親，給予我的支持與栽培，讓我自己選擇我要走的路，也讓我能無後顧之憂地完成學業；也非常謝謝女友小言在這五年多來的陪伴與支持，在此將我的成果與你們一起分享!. iii.

(6) 目中文提要英文提要誌謝目錄 1. 2. 3. 4. Reference. 錄. ……………………………………………………………… ……………………………………………………………… ……………………………………………………………… ……………………………………………………………… Introduction……………………………………………… Basic Mosaic Solutions and Patterns………………… Global Patterns and Their Entropy…………………… The Effect of the Source Term on Patterns………… ………………………………………………………………. iv. i ii iii iv 1 3 7 11 19.

(7) 1. Introduction. Of concern is one-dimensional Cellular Neural Networks (CNNs) of the form dxi = −xi + z + αf (xi−1 ) + af (xi ) + βf (xi+1 ), i ∈ Z. dt. (1.1a). Here xi denote the state of a cell Ci and f (x) is a piecewise-linear output function defined by.   rx + 1 − r, x, f (x) =  rx − 1 + r,. if x ≥ 1 , if |x| ≤ 1, if x ≤ −1 ,. (1.1b). where r is a positive constant. The quantity z is called a source term or a bias term. The numbers α, a and β are arranged in a vector form [α, a, β], which is called a space-invariant A-template A = [α, a, β].. (1.2). A is called symmetric (resp., antisymmetric) if α = β (resp., α = −β). CNNs were first proposed by Chua and Yang [1988a, 1988b]. Their main applications are in image processing and pattern recognition [Chua, 1998]. For additional background information, applications, and theory, see [Special Issue, 1995; Thiran, 1997; Chua, 1998] among others. A basic and important class of solutions of (1.1) is the stable stationary solutions. Specifically, a stationary solution x= (xi )i∈Z of (1.1) satisfies the following equation 1 f (xi ) = {xi − z − αf (xi−1 ) − βf (xi+1 )}, i ∈ Z. a. (1.3). Let x= (xi )i∈Z be a solution of (1.3). The associated output y= (yi )i∈Z = (f (xi ))i∈Z is called a pattern. The following two types of stationary solutions are of particular interest. Definition 1.1. A solution x= (xi )i∈Z is called a mosaic solution if |xi | > 1 for all i ∈ Z. Its associated pattern y= (yi )i∈Z = (f (xi ))i∈Z is called a mosaic pattern. If |xi | 6= 1 for all i ∈ Z and there are i, j ∈ Z such that |xi | < 1 and |xj | > 1, then x= (xi )i∈Z and y= (f (xi ))i∈Z are called, respectively, a defect solution and a defect pattern. To define the stability of the stationary solution, we consider the following linearized stability. Let ξ= (ξi )i∈Z ∈ `2 , the linearized operator L(x) of (1.1) at a stationary solution x= (xi )i∈Z is given by (L(x)ξ)i = −ξi + αf 0 (xi−1 )ξi−1 + af 0 (xi )ξi + βf 0 (xi+1 )ξi+1 . 1. (1.4).

(8) Definition 1.2. Let x= (xi )i∈Z be a solution of (1.3) with |xi | 6= 1 for all i ∈ Z. The stationary solution x is called (linearized) stable if all eigenvalues of L(x) have negative real parts. The solution is called unstable if there is an eigenvalue λ of L(x) such that λ has a positive real part. It is well-known, see e.g., [Juang and Lin, 2000; Hsu, 2000], that for 1 > r ≥ 0, |a| + |α| + |β|. (1.5). where r, a, α and β are defined as in (1.1), −L(x) is a self-adjoint and positive operator. Therefore, if r is sufficiently small, all mosaic solutions of (1.1) are stable. For r = 0, the complexity of stable stationary solutions of (1.1) with respect to all the parameters has been completely characterized when the template A is symmetric or antisymmetric (see [Thiran et. al., 1995; Juang and Lin, 2000]). For r > 0, sufficiently small, a map approach was introduced to study the complexity of stable stationary solutions of (1.1) with limited success (see e.g., [Hsu, 2000; Chang and Juang, 2004]). Specifically, only the parameters region that would yield Smale horseshoe, hence, the spatial entropy of ln 2, is located in those papers. That is to say, only regions that yield the full shift with 2 symbols are found. For r = 0 [Juang and Lin, 2000], the parameters regions corresponding to the positive entropy less than ln 2 can also be found. Those are the regions that yield the subshift of finite types (see e.g., [Robinson, 1995]). It would be reasonable to expect that for r 6= 0, one can find such regions as well. The purpose of this thesis is to find parameters regions yielding the subshift of finite types when the template A is symmetric. Our approach here makes use of the techniques originated in [Juang and Lin, 2000] and, later, generalized by [Cheng and Shih, 2005]. The thesis is organized as follows. In section 2, we introduce the notion of (local) basic mosaic patterns. We then identify all these basic mosaic patterns. Moreover, the solvability conditions for the existence of such patterns are also given. Section 3 is devoted to the global mosaic patterns for the symmetric template A and z = 0. Specifically, we find parameters regions whose corresponding positive spatial entropy is less than ln 2. The exact entropy of those regions are obtained via the method of the transition matrix. The effect on the pattern formation with the presence of the bias term z and with the intensity of the interaction weight β is recorded in section 4. In particular, with the injection of a source term z (6= 0), a lot of new patterns, which correspond to a certain subshifts of finite types, emerge with a smaller interaction weight β. However, as β increases to a certain range, most of previously observed chaotic patterns disappear, while other new patterns with positive entropy emerge. 2.

(9) 2. Basic Mosaic Solutions and Patterns. As in the map approach case, we seek to find the set of solutions of (1.3) that is uniformly bounded. This is also the essence of the thesis in [Cheng and Shih, 2005]. Specifically, we consider the set of solutions (xi )i∈Z for which |xi | < 1 + δ f or all i ∈ Z,. (2.1a). |f (xi )| < 1 + rδ f or all i ∈ Z,. (2.1b). or equivalently, where δ > 0 is a constant. To study (1.3), we first define the following concepts. Definition 2.1. Given any i ∈ Z, let xi−1 and xi+1 be any real numbers for which |xj | < 1 + δ, j = i − 1, i + 1. If there is a unique xi satisfying (1.3), then [xi−1 , xi , xi+1 ] is called a basic solution of (1.3). Its corresponding output [f (xi−1 ), f (xi ), f (xi+1 )] is called a basic pattern of (1.3). If, in addition, |xj | > 1, j = i−1, i, i+1, then [xi−1 , xi , xi+1 ] (resp., [f (xi−1 ), f (xi ), f (xi+1 )] ) is called a basic mosaic solution (resp., pattern) of (1.3). Note that the template A is space-invariant. Therefore, a basic solution pattern is independent of the spatial variable i. Notation 2.1. For any mosaic pattern {yi }i∈Z , we shall denote by + (resp., −) if yi = f (xi ) > 1 (resp., yi = f (xi ) < −1). There are only 8 types of basic mosaic patterns. We list as below. [+ + +]δ , [− − −]δ , [+ + −]δ , [− + +]δ , [+ − −]δ , [− − +]δ , [− + −]δ and [+ − +]δ . (2.2) Notation 2.2. The parameters regions that would yield the 8 basic mosaic patterns are, − + + − − + − respectively, denoted by Γ+ 2 , Γ−2 , Γ0 , Γ0 , Γ0 , Γ0 , Γ−2 and Γ2 . Remark 2.1. Since the template A under consideration is symmetric, the parameter regions generating [+ + −]δ and [− + +]δ are exactly the same (see Propositions 2.3 and 4.1). Thus, we make no distinction for the region that would yield those two types mosaic patterns. Likewise, the same is true for [+ − −]δ and [− − +]δ . We next study the range of parameters a, α, β, z and r for which the existence of each of 8 basic mosaic patterns is guaranteed. For simplification, we first consider 0 < r < 21 , z = 0 and α = β. We need the following useful proposition. 3.

(10) Proposition 2.1. Let A = (a1 , 0), B = (b1 , 0), C = (1, 1), D = (1 + δ, 1 + rδ), C 0 = (−1, −1) and D0 = (−1 − δ, −1 − rδ). Suppose −1 < a1 < b1 < 1 and 0 < r < 12 . Let E ∈ AB be arbitrarily given. The necessary and sufficient condition for any straight line l passing through E with the slope mE and intercepting the open line segment CD (resp., C 0 D0 ) is that the slope mE satisfies the following inequalities. mBD < mE < mAC. (2.3a). (resp., mAD0 < mE < mBC 0 ).. (2.3b). Here mEF means the slope of the line through E and F . Proof. Form Figure 2.1., we see clearly that l ∩ open segment CD 6= ∅ if and only if mED < mE < mEC . Note that we need 0 < r <. 1 2. (2.4). to ensure (2.4) holds. The slopes mED and mEC are. increasing in E as long as E is in between A and B. Thus if mE satisfies (2.3a), then the intersection of l and open segment CD is nonempty. On the other hand, if mE ≥ mEC , we see immediately that the line passing through E with such slope mE either intersects CD at C or does not intersect CD at all, a contradiction. Similarly, if mE ≤ mED , we draw the same conclusion. The proof for second assertion of the proposition is similar. Y D C. O. | | AE B. `. C `. D. Figure 2.1: .. 4. X.

(11) − In the following, we describe the parameters regions, Γ+ 2 and Γ−2 , which are the. same if 0 < r < 12 , α = β and z = 0. Proposition 2.2. Let 1 1 1 0 < r < , z = 0 and − <α=β< . 2 2(1 + rδ) 2(1 + rδ). (2.5). Then the basic mosaic patterns [+ + +]δ and [− − −]δ exist provided that (2.6) or (2.7) holds. Here (2.6) and (2.7) are given in the following. a + 2β > 1,. (2.6a). (1 + rδ)a + 2(1 + rδ)β < 1 + δ,. (2.6b). β > 0,. (2.6c). a + 2β(1 + rδ) > 1,. (2.7a). (1 + rδ)a + 2β < 1 + δ,. (2.7b). β < 0.. (2.7c). and. Proof. We illustrate only the case that β > 0, let xi+1 and xi−1 be any numbers in between 1 and 1 + δ, then 2β < β(f (xi−1 ) + f (xi+1 )) =: p < 2β(1 + rδ) and equation (1.3) reduces to. 1 f (xi ) = [xi − p]. (2.8) a Set A = (2β, 0), B = (2β(1 + rδ), 0) and E = (p, 0). It then follows from Proposition 2.1 that if (2.6) holds, then (2.8) has a unique solution xi with 1 < xi < 1 + δ.. (2.9). Similarly, if xi−1 and xi+1 are any numbers in between −1 − δ and −1. Then 2β(1 + rδ) < p < 2β. Set A = (2β(1 + rδ), 0) and B = (2β, 0), we also conclude that if (2.6) holds, than (2.8) has a unique solution xi with −1 − δ < xi < −1. Since (2.9) holds for any 1 < xi−1 , xi+1 < 1 + δ or −1 − δ < xi−1 , xi+1 < −1, we conclude that [xi−1 , xi , xi+1 ] is indeed a local solution. From Proposition 2.2, we see that for fixed r, 0 < r < 12 , and δ > 0, − Γ+ 2 = Γ−2 = {(a, β) : (2.6) holds or (2.7) holds and |β| < − We next study the parameters regions Γ+ 0 and Γ0 .. 5. 1 } =: Γ2 . 2(1 + rδ).

(12) Proposition 2.3. Suppose (2.5) holds, then the basic mosaic patterns [+ + −]δ , [− + +]δ , [+ − −]δ and [− − +]δ exist provided that (2.10) or (2.11) holds. Here (2.10) and (2.11) are given in the following. (1 + rδ)a + rδβ < 1 + δ,. (2.10a). a − rδβ > 1,. (2.10b). β > 0,. (2.10c). (1 + rδ)a − rδβ < 1 + δ,. (2.11a). a + rδβ > 1,. (2.11b). β < 0.. (2.11c). and. + The parameters regions Γ− 2 and Γ−2 are given in the following.. Proposition 2.4. Suppose (2.5) holds, then the basic mosaic patterns [+−+]δ and [−+−]δ exist provided that (2.12) or (2.13) holds. Here (2.12) and (2.13) are given in the following. a − 2(1 + rδ)β > 1,. (2.12a). (1 + rδ)a − 2β < 1 + δ,. (2.12b). β > 0,. (2.12c). a − 2β > 1,. (2.13a). (1 + rδ)a − 2(1 + rδ)β < 1 + δ,. (2.13b). β < 0.. (2.13c). and. The proof of Propositions 2.3 and 2.4 are similar to that of Proposition 2.2, and is thus omitted. Clearly, we have that for fixed r, 0 < r < 12 , and δ > 0, − Γ+ 0 = Γ0 = {(a, β) : (2.10) holds or (2.11) holds and |β| <. 1 } =: Γ0 , 2(1 + rδ). and + Γ− 2 = Γ−2 = {(a, β) : (2.12) holds or (2.13) holds and |β| <. 6. 1 } =: Γ−2 . 2(1 + rδ).

(13) 3. Global Patterns and Their Entropy. To construct the global solutions/patterns from the local solutions/patterns, we need the following notation and proposition. 0. Notation 3.1. Set Γi = R2 − ½ Γi , i = 2, 0, −2. Let Γ(i1 ,i2 ,i3 ) = R1 ∩ R2 ∩ R3 , where ij = Γ−2j+4 , if ij = 1 , 0 For instance, Γ(1,0,1) = Γ2 ∩ Γ0 ∩ Γ−2 . 0 or 1, j = 1, 2, 3, and Rj = 0 Γ−2j+4 , if ij = 0 . The set of basic mosaic patterns whose corresponding parameters are in Γ(i1 ,i2 ,i3 ) is denoted by B(i1 ,i2 ,i3 ) . ȕ 1+į rį. 1 2r. 1+į 2(1+rį). U S. 1 2 O. 1 r. Q R. a. `. T. `. S. U. `. -(1+į) 2(1+rį). T. P `. -1 2. R. -1 2r. -(1+į). rį. 1+δ 2+δ 1−r Figure 3.1: P = (1, 0), Q = ( 1+rδ , 0), U = ( 2+rδ , r(2+rδ) ).. 7.

(14) For fixed r and δ, we put Γi , i = 2, 0, −2, on the a − β plane as in Figure 3.1.. Let ^ 1 = {(a, β) : |β| < }. 2(1 + rδ) Note that Γ(1,1,1) = Γ2 ∩ Γ0 ∩ Γ−2 = Quadrilateral P RQR0 ∩. ^. 6= φ,. 0. Γ(1,1,0) = Γ2 ∩ Γ0 ∩ Γ−2 = T riangular P SR ∪ T riangular QT 0 R0 ∩ 0. Γ(0,1,1) = Γ2 ∩ Γ0 ∩ Γ−2 = T riangular QT R ∪ T riangular P S 0 R0 ∩. ^ ^. 6= φ, 6= φ,. and 0. Γ(1,0,1) = Γ2 ∩ Γ0 ∩ Γ−2 = φ. Proposition 3.1. Suppose (2.5) holds, and that (a, β) ∈ Γ(i1 ,i2 ,i3 ) , ij = 0 or 1, j = 1, 2, 3. If [xi−1 , xi , xi+1 ] := xL is a local mosaic solution of (1.3) for some i, then xL can be extended to be a global solution x¯G = (¯ xj )j∈Z , where xk = x¯k , k = i − 1, i, i + 1, and for all i 6= j, [¯ xj−1 , x¯j , x¯j+1 ] are any local solutions of (1.3) in B(i1 ,i2 ,i3 ) . Proof. We only illustrate the case that (a, β) ∈ Γ(1,0,0) , since the others are similar. In this case either 1 < xk < 1 + δ or −1 − δ < xk < −1 for all k = i − 1, i, i + 1. Now suppose the former holds, then we assign xi+3 to be any number in between 1 and 1 + δ. Since (a, β) ∈ Γ(1,0,0) , xi+2 can be uniquely determined and its value lies between 1 and 1 + δ. By proceeding similarly, we get to a global solution x¯G as claimed. From here on, by a mosaic pattern, we mean that the pattern consists of only + or − sign. That is to say we make distinction on only the signs of f (xi ). Using Proposition 3.1, we see immediately that if (a, β) ∈ Γ(1,0,0) , then the only mosaic patterns are of the following two types ........ + + + + + + + + + + + +.......... ........ − − − − − − − − − − − −.......... Similarly, if (a, β) ∈ Γ(0,1,0) (resp., Γ(0,0,1) ), then the mosaic pattern produced is unique up to the translation. ........ + + − − + + − − + + − −......... (resp., ........ + − + − + − + − + − + −.........) 8.

(15) Theorem 3.1. Suppose (2.5) holds. Then the following are true. (i) If (a,β) ∈ Γ(1,1,1) , then any mosaic pattern (∗i )i∈Z , ∗i = + or −, is a pattern for (1.3). (ii) If (a,β) ∈ Γ(1,1,0) , then any mosaic pattern (∗i )i∈Z , ∗i = + or −, satisfying the rules that any + is adjacent to at least one +, any − is adjacent to at least one −, is a pattern for (1.3). (iii) If (a,β) ∈ Γ(0,1,1) , then any mosaic pattern (∗)i∈Z , ∗ = + or −, satisfying the rules that any + is adjacent to at least one −, any − is adjacent to at least one +, is a pattern for (1.3). Proof. We illustrate only (ii). The other cases are similar. If (a,β) ∈ Γ(1,1,0) , then its corresponding basic mosaic patterns are [+ + +]δ , [− − −]δ , [+ + −]δ , [− + +]δ , [+ − −]δ , [− − +]δ =: B(1,1,0) .. (3.1). In view of (3.1) and Proposition 3.1, we see, immediately, that the assertion in (ii) holds true. We next study the complexity of the patterns for given choices of parameters . Definition 3.1. Let µΓ = {(∗i )i∈Z : ∗i = + or −} be a set of stable mosaic patterns of (1.3) for given choices of parameters in Γ. The spatial entropy h(µΓ ) is defined as the limit. ln ](µnΓ ) . n→∞ n Here ](µnΓ ) = the cardinality of the set µnΓ = {(∗i )ni=1 : ∗i = + or −, (∗i )i∈Z ∈ µΓ } h(µΓ ) = lim. (3.2). Note that µΓ is a translation invariant set and the limit in (3.2) is well-defined (see e.g., [Chow et. al., 1996]). Definition 3.2. We say the system (1.1) or (1.3) exhibits spatial chaos for given choices of parameters in Γ, in case that spatial entropy h(µΓ ) is positive. We say that the system (1.1) or (1.3) exhibits pattern formation for given choices of parameters in Γ in case the spatial entropy h(µΓ ) is zero. We next recall a well-known result (see e.g., Robinson, 1995). Theorem 3.2. Suppose there is a one-to-one and onto correspondence between the set µΓ and the sequence space ΣA . Here A is a matrix of dimension n × n whose elements are 0 and 1, and that ΣA = {(si ) : (A)si ,si+1 = 1 f or all i}. Then h(µΓ ) = ln λ, where λ is the maximal eigenvalue of A. 9.

(16) Theorem 3.3. Suppose (2.5) holds. If (a, β) ∈ Γ(i1 ,i2 ,i3 ) , ij ∈ {0, 1}, j = 1, 2, 3, then system (1.1) exhibits spatial chaos if and only if i2 = 1 and i1 + i3 ≥ 1. Moreover, √. h(µΓ(1,1,1) ) = ln 2 and h(µΓ(1,1,0) ) = h(µΓ(0,1,1) ) = ln 1+2 5 . Consequently, Γ(1,1,1) , Γ(1,1,0) and Γ(0,1,1) are the only chaotic parameters regions. Proof. We first show that the mosaic patterns produced from Γ(1,1,1) , Γ(1,1,0) and Γ(0,1,1) are all stable. Note that the stability condition (1.5) reduces to 1 |a| + 2|β| < . r. (3.3). If 0 < r < 21 , then Q, see Figure 3.1., is to the left of the a-intercept of the line a + 2β = 1r . Moreover, a direct computation could yield that the point U , see Figure 3.1., lies on the line a + 2β =. 1 . r. Similarly, the point U 0 lies on the line a − 2β =. 1 . r. Thus, the. mosaic patterns under consideration are all stable. We illustrate only the cases that (a,β) ∈ Γ(1,1,0) , and (a,β) ∈ Γ(0,1,1) . We assign 4 symbols ++, +−, −+ and −− to be 1, 2, 3 and 4 , respectively. We define il and ir , respectively, to be the left (resp., right ) side of the symbol corresponding to i. For instance, let 2 = + −, then 2l = + and 2r = −. We construct a 4 × 4 transition matrix A = (ai,j ) as follows. ( Set ai,j =. 1, if ir = jl and [il , ir , jr ] is a basic mosaic patterns in B(1,1,0) , 0, otherwise.. (3.4). Thus the transition matrix with the choice of parameters in Γ(1,1,0) is . 1  0   1 0. 1 0 0 0. 0 0 0 1.  0 1   =: A(1,1,0) . 0  1. Now, the set of µΓ(1,1,0) has a one-to-one and onto correspondence with the sequence space ΣA(1,1,0) . Here ΣA(1,1,0) = {(si ) : si ∈ {1, 2, 3, 4}, (A(1,1,0) )si ,si+1 = 1 f or all i}. Clearly, the characteristic polynomial for A(1,1,0) is λ4 − 2λ3 + λ2 − 1 = 0 or equivalently √. (λ2 − λ + 1)(λ2 − λ − 1) = 0. It then follows from Theorem 3.2 that h(µΓ(1,1,0) ) = ln 1+2 5 . If (a,β) ∈ Γ(0,1,1) , we will define the corresponding transition matrix A(1,1,0) as. 10.

(17) Notation Γ+ 2 Γ− −2 Γ+ 0 Γ− 0 Γ+ −2 Γ− 2. Parameters’ regions Corresponding patterns a + z > 1 − 2β, (1 + rδ)a + z < 1 + δ − 2(1 + rδ)β, β > 0. or [+ + +]δ a + z > 1 − 2(1 + rδ)β, (1 + rδ)a + z < 1 + δ − 2β, β < 0 replacing z by −z in the equations right above. [− − −]δ , a + z > 1 + rδβ, (1 + rδ)a + z < 1 + δ − rδβ, β > 0. or [+ + −]δ , [− + +]δ a + z > 1 − rδβ, (1 + rδ)a + z < 1 + δ + rδβ, β < 0 replacing z by −z in the equations right above. [+ − −]δ , [− − +]δ a + z > 1 + 2(1 + rδ)β, (1 + rδ)a + z < 1 + δ + 2β, β > 0. or [− + −]δ a + z > 1 + 2β, (1 + rδ)a + z < 1 + δ + 2(1 + rδ)β, β < 0 replacing z by −z in the equations right above. [+ − +]δ Table 4.1: . . 0  0   1 0. 1 0 1 0. 0 1 0 1.  0 1   =: A(0,1,1) , 0  0. the characteristic polynomial of A(0,1,1) is (λ2 + λ + 1)(λ2 − λ − 1) = 0. Thus h(µΓ(0,1,1) )) = √ ln 1+2 5 .. 4. The Effect of the Source Term on Patterns. In this section, we first consider the effect of the source term z on patterns. With the − presence of the source term z 6= 0, the regions Γ+ 2 and Γ−2 are no longer identical. Same − − + can be said to the two pairs of regions Γ+ 0 and Γ0 , and Γ2 and Γ−2 . Therefore, some new patterns emerge as z moves away from zero.. Proposition 4.1. Suppose 1 −1 + 2|β|(1 + rδ) < z < 1 − 2|β|(1 + rδ) and 0 < r < . 2. (4.1). Then the Table 4.1. holds true. 1 The first two inequalities imply that |β| < 2(1+rδ) =: β4 1 For fixed 0 < r < 2 and δ > 0 to draw parameters regions in z − a space, we need. the following notations. 11.

(18) Notation 4.1. Denote by z = 1−2β(1+rδ), a+z = 1−2β, (1+rδ)a+z = 1+δ−2(1+rδ)β, a+z = 1+rδβ, (1+rδ)a+z = 1+δ−rδβ, a+z = 1+2(1+rδ)β and (1+rδ)a+z = 1+δ+2β by l0 , l1 , l4 , l2 , l5 , l3 and l6 , respectively. Replacing z and −z in those equations above, we shall denote the corresponding equations by r0 , r1 , r4 , r2 , r5 , r3 and r6 , respectively. Notation 4.2. (i) We shall denote the intersection of the lines li and rj , i, j = 1, 2, ..6 by Ai,j .(ii) We shall denote by the quadrilateral Ai,j Ai,k Al,k Al,j = (li , rk , ll , rj ) = (i, k, l, j). Here the a-coordinate of Ai,j is greater than those of Ai,k , Al,k and Al,j . Note that such tuple is well-defined. (1−r)δ Let 0 < r < 12 and δ > 0 be fixed and 0 < β < 2(1+rδ)(2+rδ) =: β1 . Putting ri and li , i = 0, 1, 2, ...6, on z − a plane, we have the Figure 4.1... a. r6. r5. r4. l4. r3. r2. r1. 0. r0. l5. l6. z l1. 12. l3. l0. Figure 4.1: Orange region: (5, 4, 4, 5), green region: (4, 3, 3, 4) and yellow region: (3, 2, 2, 3).. 12.

(19) − + − + − Notation 4.3. Set Λ1 = Γ+ 2 , Λ2 = Γ−2 , Λ3 = Γ0 , Λ4 = Γ0 , Λ5 = Γ−2 , Λ6 = Γ2 , and for 6 T V ij , j = 1, 2, ...6, ∈ {0, 1}, we define Γ(i1 ,i2 ,i3 ,i4 ,i5 ,i6 ) = Rj ∩ z , where j=1 ½ V Λj , if ij = 1 , Rj = and z = {(a, β) : |z| < 1 − 2β(1 + rδ)}. R2 − Λj , if ij = 0 ,. With z 6= 0 and a small β > 0, we see, in the following, that a lot more chaotic parameters regions emerge. The case for β < 0 is similar and is, thus, omitted. Theorem 4.1. Assume that (4.1) holds and r is sufficiently small. Then the following hold: 2(1−r)δ 2 (i) Suppose 0 < β < min{ (2+rδ)(4+5rδ) , 6+5rδ } =: min{β0 , βˆ0 } and 0 < δ <. 2 . 1−2r. Then all. parameters regions in Table 4.2. are nonempty and all assertions in Table 4.2. hold true. 2 (ii) Suppose min{β0 , βˆ0 } < β < β1 and 0 < δ < 1−2r . Then the last two parameters regions Γ(0,1,0,1,1,1) and Γ(1,0,1,0,1,1) in Table 4.2. are empty, and all other regions are nonempty. (iii) Suppose 0 < β < min{β0 , βˆ0 } and. 2 1−2r. < δ. Then the last two parameters regions. Γ(0,1,1,1,1,0) and Γ(1,0,1,1,0,1) in Table 4.2. are empty, and all other regions are nonempty. 2 (iv) Suppose min{β0 , βˆ0 } < β < β4 and 1−2r < δ. Then the last four parameters regions Γ(0,1,0,1,1,1) , Γ(1,0,1,0,1,1) , Γ(0,1,1,1,1,0) and Γ(1,0,1,1,0,1) in Table 4.2. are empty, and all other regions are nonempty. Proof. We illustrate only (i). To see the non-emptiness of the parameters regions in Table 4.2., we first check that the z-coordinates of both A4,3 and A5,4 are smaller than 2 z = 1 − 2β(1 + rδ). A direction computation would yield so provided that 0 < δ < 1−2r and 0 < β < βˆ0 . We then need to verify that the intersection A of r3 and r4 lies above l5 .. We see, via direct computations, that only if 0 < β < β0 , then A lies above l5 . Note also that if r is sufficiently small, the stability condition (1.5) is satisfied. The verification of the other assertions in the theorem is then similar to the above and those in Theorem 3.1 and is thus omitted. 2 and 0 < r < 12 , 1−2r√ λ2 > 1+2 5 . Thus, Table. Remark 4.1. (i) If 0 < δ <. then β4 > β1 .. (ii) Note that 2 > λ1 >. 4.2. is arranged in the following way :. the higher row the parameters region is placed the more complex its corresponding patterns are. (iii) It is clear that the chaotic patterns produced from the regions Γ(1,1,1,1,1,1) and Γ(1,1,1) are the same. Similarly, the pairs Γ(0,0,1,1,1,1) , Γ(0,1,1) and Γ(1,1,1,1,0,0) , Γ(1,1,0) generate the 13.

(20) Parameters region Γ(1,1,1,1,1,1). Exact location in Figure 4.1. V (4, 3, 3, 4) ∩ z. Basic mosaic patterns contained. Spatial Entropy. [+ + +]δ , [− − −]δ , [+ + −]δ , ln 2 [− + +]δ , [+ − −]δ , [− − +]δ , [− + −]δ , [+ − +]δ . V Γ(0,1,1,1,1,1) (5, 3, 4, 4) ∩ z [− − −]δ , [+ + −]δ , [− + +]δ , ln λ1 [+ − −]δ , [− − +]δ , [− + −]δ , [+ − +]δ . V Γ(1,0,1,1,1,1) (4, 4, 3, 5) ∩ z [+ + +]δ , [+ + −]δ , [− + +]δ , ln λ1 [+ − −]δ , [− − +]δ , [− + −]δ , [+ − +]δ . V Γ(1,1,1,1,0,1) (3, 3, 2, 4) ∩ z [+ + +]δ , [− − −]δ , [+ + −]δ , ln λ2 [− + +]δ , [+ − −]δ , [− − +]δ , [+ − +δ ]. V Γ(1,1,1,1,1,0) (4, 2, 3, 3) ∩ z [+ + +]δ , [− − −]δ , [+ + −]δ , ln λ2 [− + +]δ , [+ − −]δ , [− − +]δ , [− + −]δ . √ V Γ(0,0,1,1,1,1) (5, 4, 4, 5) ∩ z [+ + −]δ , [− + +]δ , [+ − −]δ , ln 1+2 5 [− − +]δ , [− + −]δ , [+ − +]δ . √ V Γ(1,1,1,1,0,0) (3, 2, 2, 3) ∩ z [+ + +]δ , [− − −]δ , [+ + −]δ , ln 1+2 5 [− + +]δ , [+ − −]δ , [− − +]δ . √ V Γ(0,1,1,1,1,0) (5, 2, 4, 3) ∩ z [− − −]δ , [+ + −]δ , [− + +]δ , ln 1+2 5 [+ − −]δ , [− − +]δ , [− + −]δ . √ V Γ(1,0,1,1,0,1) (3, 4, 2, 5) ∩ z [+ + +]δ , [+ + −]δ , [− + +]δ , ln 1+2 5 [+ − −]δ , [− − +]δ , [+ − +]δ . √ V Γ(0,1,0,1,1,1) (6, 3, 5, 4) ∩ z [− − −]δ , [+ − −]δ , [− − +]δ , ln 1+2 5 [− + −]δ , [+ − +]δ . √ V Γ(1,0,1,0,1,1) (4, 5, 3, 6) ∩ z [+ + +]δ , [+ + −]δ , [− + +]δ , ln 1+2 5 [− + −]δ , [+ − +]δ . Here λ1 and λ2 are the maximal roots of (λ3 − λ2 − λ − 1) = 0 and (λ3 − 2λ2 + λ − 1) = 0, respectively. Table 4.2: .. 14.

(21) exact patterns. Thus, with the presence of the bias term z 6= 0, some new chaotic patterns would emerge. Specifically, the patterns whose parameters regions are from Γ(0,1,1,1,1,1) , Γ(1,0,1,1,1,1) , Γ(1,1,1,1,0,1) , Γ(1,1,1,1,1,0) , Γ(0,1,1,1,1,0) , Γ(1,0,1,1,0,1) , Γ(0,1,0,1,1,1) , and Γ(1,0,1,0,1,1) are new and chaotic. (iv) Note that in Figure 4.1., we have 0 < β < β1 . Such condition is to ensure that the β-intercept of l3 is smaller than that of l4 . We also remark that β1 is the β coordinate of R in Figure 3.1.. Therefore when β¯ (< β1 ) is fixed, we see in Figure 3.1. that the line β = β¯ passes through Γ(1,1,0) , Γ(1,1,1) and Γ(0,1,1) , which corresponds to the line z = 0 in Figure 4.1. going through Γ(1,1,1,1,0,0) , Γ(1,1,1,1,1,1) and Γ(0,0,1,1,1,1) . (v) In the case that. (1−r)δ (2+rδ)(2+3rδ). < β < β0 , (6, 3, 5, 4) reduces to a triangular A5,4 A5,3 A.. Here A is the intersection of lines r3 and r4 . Likewise, (4, 5, 3, 6) reduces to a triangular too. (vi) In the case that β0 < β < β1 , (5, 2, 4, 3) and (3, 4, 2, 5) both reduce to a triangular. Moreover, (6, 3, 5, 4) and (4, 5, 3, 6) disappear. (1−r)δ (1−r)δ For β1 < β < min{ (1+rδ)(2+rδ) , 2(1+rδ) 2 +rδ , β4 } =: min{β2 , β3 , β4 }, we have Figure. 4.2. and Table 4.3. a. r6. l5. r5 r4. r3. r2. l6. z. r1. l1. 0. r0. l2. l3. l4. l0. Figure 4.2: Orange region: (5, 3, 3, 5) and yellow region: (4, 2, 2, 4).. 15.

(22) Parameters region Γ(0,0,1,1,1,1). Exact location in Figure 4.2. V (5, 3, 3, 5) ∩ z. Basic mosaic patterns contained. Spatial Entropy √. [+ + −]δ , [− + +]δ , [+ − −]δ , ln 1+2 5 [− − +]δ , [− + −]δ , [+ − +]δ . √ V Γ(1,1,1,1,0,0) (4, 2, 2, 4) ∩ z [+ + +]δ , [− − −]δ , [+ + −]δ , ln 1+2 5 [− + +]δ , [+ − −]δ , [− − +]δ . V Γ(0,1,1,1,0,0) (3, 2, 4, 4) ∩ z [− − −]δ , [+ + −]δ , [− + +]δ , ln λ3 [+ − −] , [− − +] . δ δ V Γ(1,0,1,1,0,0) (4, 4, 2, 3) ∩ z [+ + +]δ , [+ + −]δ , [− + +]δ , ln λ3 [+ − −]δ , [− − +δ ]. V Γ(0,0,1,1,0,1) (3, 3, 4, 5) ∩ z [+ + −]δ , [− + +]δ , [+ − −]δ , ln λ4 [− − +]δ , [+ − +]δ . V Γ(0,0,1,1,1,0) (5, 4, 3, 3) ∩ z [+ + −]δ , [− + +]δ , [+ − −]δ , ln λ4 [− − +]δ , [− + −]δ . Here λ3 and λ4 are the maximal roots of λ√4 − λ3 − 1 = 0 and λ4 − λ − 1 = 0, respectively. Clearly, 1+2 5 > λ3 > λ4 > 1. Table 4.3: .. Theorem 4.2. Let (4.1) hold, 0 < δ <. 2 1−2r. and r be sufficiently small. In the case that. β1 < β < min{β2 , β3 , β4 }, the parameters regions in Table 4.3. are nonempty, and all assertions in Table 4.3. hold true. Remark 4.2. (i) If β1 < β < min{β2 , β3 , β4 }, then the a-intercept of l3 is greater than that of l4 . We also note that β2 and β3 are the β-coordinates of S and T , respectively. So when β1 < β < min{β2 , β3 , β4 }, we see from Figure 3.1. that Γ(1,1,1) disappears. Thus, not surprisingly, most of regions in Figure 4.1. are destroyed; however, there are some new chaotic parameters regions as opposed to the case that 0 < β < β1 appear. Specifically, the parameters regions with indexes containing three zeros newly emerge. (ii) For β > min{β2 , β3 , β4 }, most of chaotic regions are destroyed and yield no new chaotic regions. We thus skip the discussion of the case. We conclude the thesis with the following remarks. (i) The antisymmetric template for (1.1) can be similarly done. Moreover, the generalization of the work to two-dimensional CNNs with output function (1.1) and with the symmetric and antisymmetric templates is also straightforward. (ii) It is of considerable interests to study the defect patterns for (1.1).. 16.

(23) )2* !.G! )2-2-2-2-2-2*. )3* !.G! )1-2-2-2-2-2*. !.G! )4* )2-1-2-2-2-2*. )5* !.G! )2-2-2-2-1-2*. )6* !.G! )2-2-2-2-2-1*. )7* !.G! )1-1-2-2-2-2*. )8* !.G! )2-2-2-2-1-1*. )9* !.G! )1-2-2-2-2-1*. ):* !.G! )2-1-2-2-1-2*. !.G! )21* )1-2-1-2-2-2*. )22* !.G! )2-1-2-1-2-2*. !.G! )1-2-2-2-1-1* )23*. )24* !.G! )2-1-2-2-1-1*. !.G! )25* )1-1-2-2-1-2*. )26* !.G! )1-1-2-2-2-1*. Figure 4.3:. (iii) Figure 4.3. is a collection of a computer simulation with sets of parameters chosen from the parameters regions in Tables 4.2. and 4.3.. Specifically, we set r = 0.25 and δ = 2 for all cases. The first eleven cases in Figure 4.3. correspond to the first eleven parameters regions in Table 4.2.. The last four cases in Figure 4.3 correspond to the last four parameters regions in Table 4.3.. Each collection in Figure 4.3. contains two arrays of colors. 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). Moreover, the final outputs in each of the collection consist of all 17.

(24) basic mosaic patterns allowed in their corresponding parameters region. For instance, the final outputs in (1) consist of all 8 basic mosaic patterns. Likewise, in (6) − Γ(0,0,1,1,1,1) and (12) − Γ(0,1,1,1,0,0) , their corresponding outputs contain 6 and 5 basic mosaic patterns listed in Table 4.2. and 4.3., respectively.. 18.

(25) References [1] Afraimovich, V. S. and Hsu, S. B. [2003] Lecture on Chaotic Dynamical Systems, American Mathemtional Society, International Press. [2] Chang, H. M. and Juang, J. [2004] ”Piecewise two-dimensional maps and applications to cellular neural networks,” Int. J. Bifurcation and Chaos, Vol.14, No.7, 2223-2228. [3] Chow, S. N., Mallet-Paret, J. and Van Vleck, E. S. [1996a] ”Pattern formation and spatial chaos in spatially discrete evolution equatrion,” Rand. Comput. Dyn. 4(2 and 3), 109-178 [4] Cheng, C. Y. and Shih, C. W. [2005] ”Pattern formations and spatial entropy for spatially discrete reaction diffusion equations”, to appear in Physica D. [5] Chua, L. O. and Yang, L. [1988a] ”Cellular neural networks : Theory,” IEEE Trans. Circuits Syst. 35, 1257-1272. [6] Chua, L. O. and Yang, L. [1988b] ”Cellular neural networks : Applications,” IEEE Trans. Circuits Syst. 35, 1273-1290. [7] Chua, L. O. [1998] CNN : A Paradigm for Complexity, World Scientific Series on Nonlinear Science, Series A, Vol. 31 (World Scientific, Singapore). [8] Hsu, C. H. [2000] ”Smale horseshoe of cellular neural networks,” Int. J. Bifurcation and Chaos, Vol.10, No.9, 2119-2127. [9] Juang, J. and Lin, S. S. [2000] ”Cellular neural networks : Mosaic pattern and spatial chaos,” SIAM J. Appl. Math. 60, 891-915. [10] Robinson, C. [1995] Dynamical Systems : Stability, Symbolic Dynamics and Chaos, CRC Press, Boca Raton, FL. [11] Special Issue, [1995] ”Nonlinear waves, patterns and spatio-temporal chaos in dynamic arrays,” IEEE Trans. Circuits Syst. I 42(10). [12] Thiran, P. [1997] Dynamics of Self-Organization of Locally Coupled Neural Networks, Presses Polytechniques et Universitaires Romandes, Lausanne.. 19.

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