Research Article Complex Dynamic Behaviors in Cellular Automata Rule 14

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1 Discrete Dynamics in Nature and Society Volume, Article ID 5839, pages doi:.55//5839 Research Article Complex Dynamic Behaviors in Cellular Automata Rule 4 Qi Han, Xiaofeng Liao, and Chuandong Li State Key Laboratory of Power Transmission Equipment and System Security, College of Computer Science, Chongqing University, Chongqing 43, China Correspondence should be addressed to Qi Han, yiding98@yahoo.com.cn Received November ; Revised February ; Accepted February Academic Editor: Vimal Singh Copyright q Qi Han et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Wolfram divided the 56 elementary cellular automata rules informally into four classes using dynamical concepts like periodicity, stability, and chaos. Rule 4, which is Bernoulli σ τ -shift rule and is a member of Wolfram s class II, is said to be simple as periodic before. Therefore, it is worthwhile studying dynamical behaviors of rule 4, whether it possesses chaotic attractors or not. In this paper, the complex dynamical behaviors of rule 4 of one-dimensional cellular automata are investigated from the viewpoint of symbolic dynamics. We find that rule 4 is chaotic in the sense of both Li-Yorke and Devaney on its attractor. Then, we prove that there exist fixed points in rule 4. Finally, we use diagrams to explain the attractor of rule 4, where characteristic function is used to describe that all points fall into Bernoulli-shift map after two iterations under rule 4.. Introduction Cellular automata CA was first introduced by von Neumann in 95. CA is a mathematical model consisting of large numbers of simple identical components with local interactions. The simple components act together to produce complex global behavior. CA performs complex computation with high degree of efficiency and robustness. Therefore, many applications of CA have been reported, especially in cryptography 3, 4 and image processing 5, 6. Here, we will only consider Boolean automata for which the cellular state x {, }.A cellular automata consists of a number of cells which evolve by a simple local rule identical rule. The value of each cell in the next stage is determined by the values of the cell and its neighbor cells in the current stage under the local rule 4. The identical rule contained in each cell is essentially a finite-state machine, usually specified in the form of a rule table, with an entry for every possible neighborhood of a cell consists of itself and the adjacent cells 7. The cellular array is d-dimensional, where d,, 3 is used in practice. In this paper,

2 Discrete Dynamics in Nature and Society Table : The truth table of Boolean function of rule 4. x i x i x i f 4 x i x i x i x i f 4 x i we will concentrate on d. For a one-dimensional CA, a cell is connected to r local neighbors cells on either side where r is referred to as the radius. A one-dimensional CA has n cells linked in a line or in a circle. Denote the value in the ith cell at the tth stage by x i t. For -state 3-neighborhood CA r, the evolution of ith cell can be represented as a function of the present states of i th, i th, and i th. The local function f i is a deterministic function to determine the next-stage value of the ith cell, x i t f i x i t,x i t,x i t. For example, the rule 4 is a one-dimensional CA, and its rule table is shown in Table.Thus, we have f, f, f, f, f, f, f, and f. In 98s, Wolfram proposed CA as models for physical systems which exhibit complex or even chaotic behaviors based on empirical observations, and he divided the 56 ECA binary one-dimensional CA with radius rules informally into four classes using dynamical concepts like periodicity, stability, and chaos 8. However, some authors 4 found that some rules of Bernoulli σ τ -shift rules are chaotic in the sense of both Li-York and Devaney, where these rules were said to be simple as periodic by Wolfram. Rule 4 is belonging to Bernoulli σ τ -shift rules. Therefore, we need to research the rule 4 and to find its some new dynamical properties. The rest of the paper is organized as follows. In Section,the Boolean function of rule 4 is also presented, and the expressions of its attractors are given. In Section 3, the dynamical behaviors of rule 4 are studied. In Section 4, characteristic function is used to describe that all points fall into Bernoulli-shift map after several iterations under rule 4, and Lameray diagram is used to show clearly the iterative process of an attractor. Section 5 presents some conclusions.. Preliminaries For simplicity, some notations about symbolic dynamics can refer to, 4. In this paper, we will use some notations about CA as follows. Chua et al. 5 mentioned that each rule has three globally equivalent local rules determined by three corresponding global transformations, namely, left-right transformation T, global complementation T, and left-right complementation T. Each equivalence class is identified by ε κ m, where κ is complexity index, and m is index of κth class. In 6, the authors presented that rules of 56 local rules were Bernoulli σ τ -shift rules. Each of the Bernoulli σ τ -shift rules has an ID code B N α, β, τ, where α denotes the number of attractors of rule N, β denotes the slope of the Bernoulli σ τ -shift map, and τ denotes the relevant forward time-τ. Hence, the space-time evolution of any one of the rules on their attractors can be uniquely predicted by two parameters: β ± σ and τ. For example, the attractors of rule 4 are β /, σ, τ and β, σ, τ. It follows from 7 that the Boolean function of rule 4 is [ f4 x ] i x i x i x i x i x i,.

3 Discrete Dynamics in Nature and Society 3 for x S Z, i Z, where,, and stand for AND, NOT, and XOR logical operation, respectively. Sometimes, is omitted for simplicity. The truth table of Boolean function of rule 4 is shown in Table. The subsets denoted by Λ 4, Λ4 are derived from the parameters of rule 4: β /, σ, and τ andβ, σ, and τ, respectively, that is, Λ 4 {x S Z [ f 4 x ] } i x i, i Z, Λ 4 {x S Z [ f 4 x ] } i x i, i Z.. 3. Dynamical Behaviors of f 4 on Two Subsystems In this section, we investigate the complexity and chaotic dynamic behaviors of f 4.Inorder to give our results, some definitions need be introduced. Definition 3. see 8. A square {, } matrix A is irreducible if for every pair of indices i and j, there is an n such that A n ij >. Definition 3. see 8. A square {, } matrix A is aperiodic if there exists N, such that A n ij >, n>n, i, j. Definition 3.3 see 8. Suppose that g : X Y is a continuous mapping, where X is a compact topological space. g is said to be topologically mixing if, for any two open sets U, V X, N >, such that g n U V /, n N. Definition 3.4 see 3. Let X, f and Y, g be compact spaces, we say f and g are topologically conjugate if there is a homeomorphism h : X Y, such that h f g h. Based on the above definitions, we investigate the complexity behaviors of f 4 on two subsystems as follows. Proposition 3.5. For rule 4, there exists a subset Λ 4 S Z which satisfies f 4 Λ 4 ς 4 T Λ 4 Γ 4 Λ 4 if and only if x...,x,x,x,... Λ 4, and x i, x i, and x i cannot be simultaneously, i Z. Proof. Necessity Suppose that there exists a subset Λ 4 S Z such that f 4 Λ 4 Γ 4 Λ 4, then x...,x,x,x,... Λ 4, and we have f 4 x i x i, x Z. If x i, then x i x i x i x i x i x i x i,sowegetx i, x i ; x i, x i ; x i, x i ; x i, x i ; if x i, then x i x i x i x i x i x i x i,so we get x i, x i ; if x i, then x i orx i. Sufficiency Suppose that there exists a subset Λ 4 S Z, x Λ 4, x i, x i,andx i cannot be simultaneously, i Z. This yields that x i x i x i. Thus, x i x i x i x i x i, namely, x i x i x i x i x i x i. Therefore, we have proven that f 4 x Γ 4 x, x Λ 4. Remark 3.6. A condition of bit strings according to Bernoulli σ τ -shift evolution under f 4 is obtained in Proposition 3.5. From Proposition 3.5, we can know that a bit string belonging

4 4 Discrete Dynamics in Nature and Society to Λ 4 is evolved on the basis of shifting the bit string to the right by bit and then complementing changing to and changing to to obtain the next bit string after one iteration under f 4. Remark 3.7. From the definition of subsystem, we know that Λ 4,f 4 are subsystems of S Z,f 4. Proposition 3.8. Γ 4 ς 4 T T ς 4. Proposition 3.9. ς 4 : S Z S Z and Γ 4 : S Z S Z are not topologically conjugate. Λ 4,ς 4 is topologically mixing if and only if Λ 4, Γ 4 is topologically mixing. The topological entropy ent & 4 Λ 4 ent Γ 4 Λ 4. Proof. a Suppose that ς 4 and Γ 4 are topologically conjugate, we give a proof by contradiction. Since ς 4 and Γ 4 are topologically conjugate, then there exists a homeomorphism h such that h ς 4 Γ 4 h, so we have h ς 4 T ς 4 h. Because of h ς 4 ς 4 h, then we obtain that T is identical map, which leads to a contradiction. b Let any nonempty open sets U, V Λ 4. Sufficiency Assume that Γ 4 Λ 4 is topologically mixing, then there exists N >, such that Γ 4 Λ 4 n U V /, n N. Since T is a homeomorphism, T V is also a nonempty open set in Λ 4, hence there exists N >, such that Γ 4 Λ 4 n U T V /, n N. Therefore, T Γ 4 Λ 4 n U T V /, n N. Then, we show that there exists a N> such that ς 4 Λ 4 n U V /, n N.LetN max N,N. i If n k, k Z, then ( ) n U n ( ) n U ς 4 Λ 4 V T ς 4 Λ 4 V T ς Λ 4 T ς Λ 4 U V }{{} n k ( ) n U Γ 4 Λ 4 V /. 3. ii If n k, k Z, then ( ) n U (( ς 4 Λ 4 V T T T ( ( T T (( ς 4 Λ 4 Γ 4 Λ 4 ς 4 Λ 4 ) n U V ) ) n U T V ) ) n U T V ) /. 3. Necessity Suppose that ς 4 Λ 4 is topologically mixing, then there exists an N >, such that ς 4 Λ 4 n U V /, n N. Since T is a homeomorphism, T U is also a nonempty open set in Λ 4, hence for open sets T U and V, there exists an N >, such that ς 4 Λ 4 n T U V /. LetN max N,N.

5 Discrete Dynamics in Nature and Society 5 i If n k, k Z, then ( ) n U n ( ) n U Γ 4 Λ 4 V T ς 4 Λ 4 V ( ) n U ς 4 Λ 4 V /. 3.3 ii If n k, k Z, then ( ) n U ( ) n U Γ 4 Λ 4 V T ς 4 Λ 4 V ( ) n ς 4 Λ 4 T U V /. 3.4 c Γ 4 Λ 4 Hence, ent ς 4 Λ 4 Because of f 4 Λ 4 T ς 4 Λ 4 ent Γ 4 Λ 4 T ς 4 Λ 4 ς 4 Λ 4.. The proof is complete. Γ 4 Λ 4, the following corollaries are immediate. Corollary 3.. ς 4 Λ 4 and f 4 Λ 4 are not topologically conjugate. Corollary 3.. Λ 4,ς 4 is topologically mixing if and only if Λ 4,f 4 is topologically mixing. Corollary 3.. The topological entropy ent & 4 Λ 4 ent f 4 Λ 4. Remark 3.3. By Proposition 3.5, we know that the determinative block system of Λ 4 is P {r,r,r,r 3,r 4,r 5,r 6 }, where r, r, r, r 3, r 4, r 5, and r 6. We have Λ 4 Λ P. Let ϖ { rr r b b b,r b b b P, j such that b j / b j }. ϖ is considered as the excluded block system 9 Λ ϖ, where Λ ϖ {r...r,r,r... P r i P,r i r i / ϖ, i Z}. Obviously, ς 4 : Λ ϖ Λ ϖ,ς 4...,r r r,......, r r r,.... Therefore, it is easy to calculate the transition matrix B of the ς Λϖ as B, 3.5

6 6 Discrete Dynamics in Nature and Society r 4 r 5 r r r r 6 r 3 Figure : The corresponding graph G 4 of the matrix B. where B B ij P P B ij, r i r j ϖ,, else. 3.6 Obviously, B is a square {, } matrix. A square {, } matrix corresponds to a directed graph. The vertices of the graph are the indices for the rows and columns of B. There is an edge from vertex i to vertex j if B ij. A square {, } matrix A is irreducible if and only if the corresponding graph is strongly connected. If Λ A is a two-order subshift of finite type, then it is topologically mixing if and only if A is irreducible and aperiodic 8. We give the corresponding graph G 4 of matrix B in Figure, where vertices are the elements of set P. It is obvious that G 4 is a strongly connected graph. Carefully observing Figure, we find that there are several strongly connected subgraphs: r r r 3 r, r r r 4 r r 3 r, r r r 5 r 3 r, r r r 6 r 5 r 4 r r 3 r, r r 4 r, r r 5 r 4 r, r r 6 r 5 r 4 r, r 6 r 6, and so forth. The elements of Λ 4 P will be composed by all vertices of those strongly connected subgraphs, respectively. For example, if x Λ 4 P, x is composed by vertices of subgraph r r 4 r, then we have r,r,r 3,r 5, and r 6 / x, and all vertices of the subgraph will occur in x if x k, k,,...ifx Λ 4 P and x is composed by vertices of subgraph r r 5 r 4 r, then we have r,r,r 3, and r 6 / x, and all vertices of the subgraph will occur in x if x 3k. Therefore, we can deduce the elements of set Λ 4 via the corresponding strongly connected graph of matrix B. Based on the above analysis, we have the following proposition. Proposition 3.4. a ς 4 : Λ 4 Λ 4 and ς 4 : Λ ϖ Λ ϖ are topologically conjugate. b ς 4 Λ 4 is topologically mixing. c The topological entropy ent & 4 Λ

7 Discrete Dynamics in Nature and Society 7 Proof. a Define x (...,x, x,x,... h : Λ 4 Λ ϖ, ) (...,r, r,r,... ), 3.7 where r i x i x i x i, i Z. Infact,bythedefinitionofΛ ϖ, we have h x Λ ϖ, x Λ 4 ; thus, h Λ 4 h Λ ϖ. Then, it is easy to check that h is homeomorphism and h ς 4 Λ 4 ς 4 Λϖ h. Therefore, ς 4 Λ 4 and ς 4 Λϖ are topologically conjugate. b Because B n >, n 4, the transition matrix B of subshift of finite type ς is irreducible and aperiodic. By 8, 9, ς 4 Λ 4 is topologically mixing. c Because two topological conjugate systems have the same topological entropy, and the topological entropy ς 4 on Λ ϖ equals log ρ B, where ρ B is the spectral radius of the transition matrix B 4 of the subshift Λ ϖ. So ent ς 4 Λ 4 log ρ B.694. Remark 3.5. By Corollaries 3. and 3., we know that Λ 4,f 4 is topologically mixing and ent f 4 Λ Theorem 3.6. f 4 is chaotic in the sense of both Li-Yorke and Devaney on Λ 4. Proof. It follows from 9 that the positive topological entropy implies chaos in the sense of Li-Yorke, and a system with topologically mixing property has chaotic properties in different senses such as Devaney. Therefore, rule N 4 possesses very rich and complicated dynamical properties on Λ 4. Proposition 3.7. For rule 4, there exists a subset Λ 4 S Z which satisfies f 4 Λ 4 υ 4 Λ 4 if and only if x...,x,x,x,... Λ 4, and x i, x i, and x i have the following relations: i if x i, thenx i, x i, and x i ; x i, x i, and x i, ii if x i, thenx i, x i ; x i, x i ; x i, x i, and x i. Let P {r,r,r,r 3,r 4 } be a new state set, where r, r, r, r 3, and r 4. Remark 3.8. From the definition of subsystem, we know that Λ 4,f 4 is subsystem of S Z,f 4. The transition matrix of subshift Λ 4,υ 4 is B. 3.8 We give the corresponding graph G 4 of matrix B in Figure, where vertices are the elements of set P.

8 8 Discrete Dynamics in Nature and Society r r r r 4 r 3 Figure : The corresponding graph G 4 of the matrix B. Proposition 3.9. a f 4 Λ 4 is topologically mixing. b The topological entropy ent f 4 Λ Theorem 3.. f 4 is chaotic in the sense of both Li-Yorke and Devaney on Λ 4. Theorem 3.. For rule 4, there exist fixed points of f 4. Proof. A example is given as follows. Let y...,,,,,,,,,,,,,,, }{{}}{{} repetition of, repetition of, By Table, we have ( ) f 4 y...,,,,,,,,,,,,,,,... y. }{{}}{{} repetition of, repetition of, 3. Therefore, there exists a set of fixed points of rule 4. Remark 3.. By Theorem 3., we know that rule 4 has garden of Eden An bit string is said to be a garden of Eden of rule N if and only if it does not have a predecessor under the local rule transformation f N in the finite case. By definition of garden of Eden, the incoming and outgoing bit strings are the same in a garden of Eden, while the basin of attraction of an attractor Λ must contain, at least, one point not belonging to Λ. However, in this paper, we consider garden of Eden as a kind of special attractor. Then, we find that,,,,,,, or Λ 4 Λ 4 Λ4. Furthermore, we guess that all initial bit strings after sufficient iterations will belong to Λ 4 under rule f Characteristic Function of Rule 4 First, we give a definition on global characteristic function 6. Given any local rule N, and binary configuration x x x x I x I for CA, where x i {, }, then we can uniquely

9 Discrete Dynamics in Nature and Society φ φ 7 4 Figure 3: All points fall into Bernoulli-shift map after seven iterations under rule 4, where I 3. associate the Boolean string x with the binary expansion of a real number x x x I x I on the unit interval,, x x x x I x I φ x x x I x I, 4. where φ I i i x i is the decimal form of Boolean string x x x x I x I.TheCA characteristic function χ N of rule N is defined as χ N : Q, Q,, i.e. φ n χ N( φn ), 4. where Q denotes rational numbers. By 9, the express of characteristic function of rule 4 is I [ χ 4 δ x i x i x i ] i, 4.3 i where, w >, δ{w}, w. 4.4 In order to transform every concrete bit string into abstract decimal digit, we use characteristic function χ N. Next, we plot the characteristic function diagram for rule 4. We choose I 4. Figure 3 describes that all points fall into Bernoulli-shift map after eight iterations under rule 4, which shows that Λ 4 is global attractor of f 4. The phenomenon also shows that the prediction in Remark 3. is correct. Two sets Λ 4 and Λ 4 can also be found in Figure 3. The dots in the two straight lines, whose slope is /, are in the set Λ 4 ;thedots in the two straight lines, whose slope is, are in the set Λ 4. Two isles of Eden are shown in Figure 4. We find that the bit string y...,,,... }{{} repetition of, is isle of Eden, when the length y is even. Some attractors are shown in Figure 5. Figure 5 a shows that the period of attractor is ; Figure 5 b shows that the period of attractor is

10 Discrete Dynamics in Nature and Society Figure 4: Two isles of Eden are shown, where x 6. a b c d Figure 5: Some attractors of rule 4, where I 5, and the white pixel stands for and black for. six, where β, σ, and τ ; Figure 5 c shows that the period of attractor is six, where β /, σ, and τ ; Figure 5 d shows that after two iterations, the initial string is attracted. We find that the possible periods of attractors are and 6 if I 5. By Proposition 3.5, the attractors of Figures 4 and 5 c belong to Λ 4.ByProposition 3.7, the attractors of a, b, and d belong to Λ 4. There is a interesting phenomenon, that is, Figure 5 d, where one iteration accords with Proposition 3.5; however, the two iterations accord with Proposition 3.7, and the attractor belongs to Λ 4. The phenomenon shows that these two propositions can switch in some situation. The bit string of attractor in Figure 5 c corresponds with strongly connected subgraph r r r 5 r 4 r r 3 r in Figure. Next, we use Lameray diagram to present our attractors. The diagrams show clearly the iterative process of attractors. The evolution of characteristic function of period- 5 attractor is shown in Figure 6, where I 4, and the values of characteristic function are.938,.875,.375,.75, and.533, respectively. The five points fall into the time- map φ n χ N φ n in Figure 5. Then we can associate this particular period-5 attractor of rule 4 as a period-5 point of a continuous map f :,, which we know that it is chaotic because period-5 implies chaos based on Li-York theorem and Sarkovskii theorem 3. According to Sarkovskii theorem, period-5 implies that the number of periods of f 4 are infinite.

11 Discrete Dynamics in Nature and Society φn φ n Figure 6: The evolution of characteristic function of the period-5 attractor, where the values of characteristic function of the attractor are.938,.875,.375,.75, and.533, respectively. In recent years, associative memory was researched in many papers 4, 5. Itis obvious that rule 4 can be used in associative memory. By strongly connected graph of rule 4, we can get the elements on its attractors. Then, we can choose a bit string which belongs to an attractor as memory pattern. For example, bit string in Figure 5 b can be chosen as a memory pattern. Since there are infinite orbits, the storage capability is very large. The associative memory model provides a solution to problem where time to recognize a pattern is independent of the number of patterns stored. 5. Conclusion In this paper, the dynamical behaviors of rule 4 of global equivalence class ε 3 in ECA, where rule 4 is Bernoulli σ τ -shift rule, are carefully investigated from, the viewpoint of symbolic dynamics. We derive the conditions according to Bernoulli σ τ -shift evolution for rule 4. Then, we prove that rule 4 is chaotic in the sense of both Li-Yorke and Devaney on their attractors, respectively. We use diagrams to explain the attractor of rule 4, where characteristic function and Lameray diagram are used to describe that all points fall into Bernoulli-shift map after several iterations and to show clearly the iterative process of an attractor, respectively. Acknowledgments This work was supported in part by the National Natural Science Foundation of China under Grant 69734, Grant 6749, and Grant 6347, in part by the Natural Science Foundation project of CQCSTC under Grant 9BA4, in part by the State Key Laboratory of Power Transmission Equipment & System Security and New Technology, Chongqing University, under Grant 7DA576, and in part by Teaching & Research Program of Chongqing Education Committee KJ4.

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