# A novel construction of chaotic dynamics base gliders in diffusion rule B2/S7

## Abstract

The dynamical behaviors of gliders (mobile localizations) in diffusion rule B2/S7 are quantitatively analyzed from the theory of symbolic dynamics in two-dimensional symbolic sequence space. Their intrinsic complexity is demonstrated by exploiting the relationship between one-dimensional and two-dimensional subshifts. Based on this rigorous approach and technique, the underlying chaos of the extant gliders and their combinations is characterized in a subtle way. It is demonstrated that they can be identified to distinct subsystems with very rich and complicated dynamics; that is, diffusion rule is topologically mixing and possesses positive topological entropy on each subsystem. This analytical assertion provides the fact that diffusion rule is covered with complex subsystems “almost everywhere”. Finally, it is worth mentioning that the procedure proposed in this paper is also applicable to all other gliders arising from the two-dimensional cellular automata therein. It is an extended discovery in both cellular automata and chaos theory.

### Keywords

Cellular automata Diffusion rule Symbolic dynamic Topologically mixing Topological entropy Chaos## 1 Introduction

Cellular Automata (CAs) are non-linear dynamical systems consisting of a regular lattice of variables which evolve according to the uniform local interactions. They are able to produce rich and diversified phenomena by means of simple local rules, and have a long tradition as modeling widely divergent application areas, including physical system simulation, cryptography, circuit testing, biological process, traffic flow, and many other problems of self-organization [1, 2, 3, 4]. The study of dynamical behaviors of CAs can be developed in different frameworks; however, one of the main challenges in this filed is to uncover their quantitative properties.

CAs were formally introduced by Jone von Neumann in \(1951\) for constructing a self-reproducing system [5]. His original discrete model is a two-dimensional CAs (2D CAs) with 29 states per cell, which is the first discrete parallel universal computational model capable of self-reproduction. Twenty years later, John H. Conway designed his now-famous game of life with emergence and self-organization [6, 7]. During the same period, the study of mathematical theory of CAs was firstly treated by Hedlund, who viewed one-dimensional CAs (1D CAs) in the context of symbolic dynamics as homomorphisms of the full shift [8]. In the early 1980s, Stephen Wolfram carried out exhaustive research on dynamical and computational aspects of CAs based almost entirely on empirical observations from computer simulations [9, 10]. He proposed a qualitative classification of 256 elementary CAs (only \(88\) equivalence classes [11]) that distinguished four different complexity classes of their dynamics using dynamical concepts like periodicity, stability and chaos. In 2002, he introduced his achievement *A New Kind of Science* [12]. To provide a rigorous foundation for Wolfram’s empirical observations via mathematical analysis, L. O. Chua et al., subsequently, combined their research on cellular neural network and nonlinear dynamics to derive the mathematically based nonlinear dynamics of 1D CAs via the basic notations and concepts like characteristic function, forward time map, basin tress diagram, and Isle-of-Eden diagraph under the circumstances of finite length and periodic boundary conditions [13, 14, 15, 16]. Some parts of these results are overlapping with those produced in [17]. For the infinite case, chaotic properties of surjective and permutative 1D CAs were studied in [18]. However, many rules in Chua’s classification are not surjective, then they are not chaotic according to Devaney’s definition in phase space [28]. Recently, chaotic phenomena of Chua’s Bernoulli-shift rules with distinct parameters have been rigorously and thoroughly interpreted under the framework of symbolic dynamics [19, 20, 21].

Among infinitely many local rules, the ones exhibiting plentiful gliders (mobile localizations) and glider guns have received special attention. They display complex behaviors via the interactions of gliders from random initial conditions, and have the potential of power of universal Turing machines. The famous examples include elementary CAs 110, the game of life, diffusion rule, high life, life without death, and so on. The majority of the published issues for these 2D CAs are focused on their computational ability [22, 23, 24]. While the analytical assertions of Chua’s Bernoulli-shift rules provide the fact that glider implies chaos for 1D CAs, as proved in [25]. This sheds light on the importance of gliders in complex 2D CAs from the theory of symbolic dynamics. Indeed, concerning Devaney chaos of elementary CAs, an easy-to-check property (left or rightmost permutivity) of local rule was enucleated in [26]. However, it is well known that dynamics of multidimensional shifts is vastly more complex, and much more difficult to unveil than that of 1D case. For instance, it is undecidable whether a given set of rules defines a nonempty subshift of finite type [27].

This paper is not to report expressions or discoveries of gliders, but devoted to developing analytical methods that are applicable to all the gilders arising from 2D CAs therein. To enhance clarity and convenience, this work is concerned exclusively with four gliders with the simplest configurations and constructions of diffusion rule B2/S7, denoted by g1, g2, g3, and g4, and their combinations [22, 24]. This rule exhibits diffusion-like dynamics of propagating patterns and possesses potential applications in unconventional computing. Findings discussed in the preceding references are based on computational experiments with exhaustive search of mobile and stationary localizations emerged in rule B2/S7. However, their dynamical behaviors nevertheless are missed in the framework of 2D symbolic dynamics.

From the viewpoint of 2D symbolic dynamics, their chaotic dynamical properties are characterized in subtle detail, such as topological entropy and topologically mixing, and some quantitative explanations of their intrinsic complexity are obtained. The rest of the paper is structured as follows. Section 2 reviews the basic definitions and notations of 2D CAs and symbolic dynamics. Based on sound mathematical theory and empirical observations, Sect. 3 proposes an analytical technique that allows one to identify chaotic subsystems of g1 glider. These main results henceforth present its quantitative and productive dynamical behaviors. Through the similar procedure, Sect. 4 shows the comprehensive and elaborate symbolic dynamics of g1, g2, g3, and g4, and their combinations. Finally, Sect. 5 summarizes the main consequences of this article and prospects for future studies.

## 2 2D CAs and symbolic dynamics

1. A cell in state 0 will take state 1 if it has exactly two neighbors in state 1, otherwise cell remains in state 0 |

2. A cell in state 1 remains in state 1 if it has exactly seven neighbors in state 1, otherwise cell takes state 0 |

For CAs, each rule can be expressed by its unique truth table. Naturally, each local rule can induce a global mapping \(f\) defined on the configuration space via its truth table and three logical operations. It is easy to verify that every global mapping is (uniformly) continuous. Therefore, each cellular automata along with its configuration space forms a compact system. As an illustration, the classical right-shift 2D CAs \(\sigma _{_{10}}\) and right-shift 1D CAs \(\sigma \) are defined, respectively, by \([\sigma _{_{10}}(x)]_{ij}=x_{(i+1)j}\), for any \( x \in S^{Z^2}, i, j \in Z\) and \([\sigma (\bar{x})]_i=\bar{x}_{i+1}\), for any \( \bar{x} \in S^{Z}, i \in Z\). Throughout this paper, the diffusion cellular automata rule B2/S7 is written as \(f\).

For a closed invariant subset \(\Lambda \subseteq S^Z\), the subsystem \((\Lambda ,\sigma )\), or simply \(\Lambda \), is called a subshift of \(\sigma \). Let \(\fancyscript{A}\) denote a set of some finite words over \(S\), and \(\Lambda =\Lambda _{\fancyscript{A}}\) be the set consisting of the bi-infinite configurations of all the words in \(\fancyscript{A}\). Then, \({\Lambda _{\fancyscript{A}}}\) is a subshift of finite type of \((S^Z,\sigma )\), where \(\fancyscript{A}\) is said to be the determinative block system of \(\Lambda \). It follows from [28] that the symbolic dynamics of a subshift of finite type is largely determined by the properties of its transition matrix \(A\), which is defined by having its elements \(A_{ij} =1\), if \((i,j)\prec \Lambda \); otherwise, \(A_{ij} =0\). Therefore, it is helpful to briefly review some definitions. A matrix \(A\) is positive if all of its entries are non-negative; irreducible if \(\forall i,j\), there exists \(n\) such that \(A^n_{ij}> 0\); aperiodic if there exists \(N\), such that \(A^n_{ij}> 0\), for \(n>N\) and \(\forall i,j\). If \(\Lambda _{A}\) is a \(2\)-order subshift of finite type, then it is topologically mixing if and only if \(A\) is irreducible and aperiodic, where \(A\) is its associated transition matrix.

## 3 Symbolic dynamics of g1 glider

Due to the simplicity of g1 glider, this section thoroughly investigates complex dynamics of g1 glider of \(f\) in \(S^{Z^2}\). It is also instructive and convenient to interpret the methodology and procedure to prove chaotic dynamics of gliders in 2D CAs therein. In order to observe the very rich and complicated dynamics of g1 glider, this section will be divided into two parts: one for finite strips of g1 glider and the other for infinite case.

### 3.1 Finite strips of g1 glider

To simplify our following analytical interpretation, this part provides a detail and rigorous proof for one strip. With a similar conception, it is possible to conclude the symbolic dynamics for finite case.

**Proposition 1**

- (1)
\((\Lambda _1,f)\) is a subsystem of \((S^{Z^2},f)\);

- (2)
\(R\) is a homomorphism;

- (3)
\((\Lambda _1,f)\) and \((\Lambda _{\fancyscript{A}},\sigma )\) are topologically conjugate.

Recall that two topologically conjugate systems have qualitatively the same topological dynamics. According to Proposition 1, symbolic dynamics of \(f\) on \(\Lambda _1\) can be disclosed via a subshift of finite type \(\Lambda _{\fancyscript{A}}\). It is remarked that the topological entropy of \(f\) on a subsystem \(X\) is denoted by \(ent(f|_{X})\) in the following.

**Proposition 2**

- (1)
\(f\) is topologically mixing on \(\Lambda _1\);

- (2)
\(ent(f|_{\Lambda _1})\)= \(\log {\rho (A)}=\log {\lambda _0}\approx 0.322\), where \(\rho (A)\) is the spectral radius of the transition matrix \(A\) and \(\lambda _0\) is the positive real root of \(\lambda ^4-\lambda ^3-1=0\).

It follows from [29] that the positive topological entropy implies chaos in the sense of Li–Yorke. And the topologically mixing is also a very complex property of dynamical systems. A system with topologically mixing property has many chaotic properties in different senses, such as in the sense of Devaney [28]. Because topologically mixing infers topologically transitivity and density of periodic points. Thus, the mathematical analysis presented above provides the rigorous foundation for the following.

**Theorem 1**

\(f\) is chaotic in the sense of both Li–Yorke and Devaney on \(\Lambda _1\).

*Remark 1*

Note that if one specifies \(R(\Lambda _{\fancyscript{A}})\) and \(y_{ij}=[R(x)]_{ij}\) with \((y_{4i},y_{5i},y_{6i},y_{7i})^T=A_k,~~if~~x_i=A_k~(k=0,1,2),~~\forall i \in Z\) and otherwise \(y_{ij}=0\), then it can infer one more chaotic subsystem, which disjoints with \(\Lambda _1\) apart from only one point. Analogously, one can identify infinite complex subsystems with only one strip of glider. This particularly uncovers that diffusion rule is saturated with chaotic subsystems almost everywhere and genuinely possesses much richer glider dynamics than those in 1D case.

### 3.2 Infinite strips of g1 glider

Remarkably, it follows from Eqs. (1) and (2) that \(R_0\) is a continuous map. For its elaborate and compact definition, \(R_0\) is surjective and injective. Furthermore, it can be verified that \((\Lambda _2,f)\) is a subsystem of \((S^{Z^2},f)\). Therefore, the above fundamental result is summarized in the following.

**Proposition 3**

- (1)
\((\Lambda _2,f)\) and \((\Lambda ,\sigma _{10})\) are topologically conjugate;

- (2)
\((\Lambda ,\sigma _{10})\) and \((\Lambda _{\fancyscript{A}},\sigma )\) are topologically semi-conjugate.

In other words, \((\Lambda _{\fancyscript{A}},\sigma )\) is the factor of \((\Lambda ,\sigma _{10})\) and \((\Lambda ,\sigma _{10})\) is the expansion of \((\Lambda _{\fancyscript{A}},\sigma )\). Note that topological entropy of factor is equal or greater than that of expansion. Thus, \(ent(f|_{\Lambda _2})=ent(\sigma _{10}|_{\Lambda })\ge ent(\sigma |_{\Lambda _{\fancyscript{A}}})=\log {\lambda _0}\approx 0.322\). Its precise value is evaluated as follows.

**Proposition 4**

The topological entropy of \(f\) on \(\Lambda _2\) is \(+\infty \); that is, \(ent(f|_{\Lambda _2})=+\infty \).

*Proof*

It follows from [27] that the topological entropy of \(\sigma _{10}\) equals \(\mathop {\lim }\limits _{n \rightarrow \infty }\displaystyle \frac{log|B_n(\Lambda )|}{n}\), where \(B_n(\Lambda )\) is the set of square blocks of size \(n\) of \(\Lambda \), and \(|B_n(\Lambda )|\) is its cardinal number. Since each row of \(B_n(\Lambda )\) is the finite word of length \(n\) appearing in \(\Lambda _{\fancyscript{A}}\), and the total number of such allowable words is \(||A^{n-1}||=\sum _{i,j}A_{ij}^{n-1}\). The rows in \(B_n(\Lambda )\) are independent, it is therefore clear that \(|B_n(\Lambda )|=||A^{n-1}||^n\). Finally, \(ent(f|_{\Lambda _2})=ent(\sigma _{10}|_{\Lambda })=\mathop {\lim }\limits _{n \rightarrow \infty }\displaystyle \frac{log|B_n(\Lambda )|}{n}=\mathop {\lim }\limits _{n \rightarrow \infty }log|B_n(\Lambda )|=+\infty \).\(\square \)

*Remark 2*

If the definition of topological entropy of \(\sigma _{10}\) in 2D case is adopted as \(\mathop {\lim }\limits _{n \rightarrow \infty }\displaystyle \frac{log|B_n(\Lambda )|}{n^2}\), then \(ent(f|_{\Lambda _2})= ent(\sigma _{10}|_{\Lambda })=\mathop {\lim }\limits _{n \rightarrow \infty }\displaystyle \frac{log|B_n(\Lambda )|}{n^2}=\mathop {\lim }\limits _{n \rightarrow \infty }\displaystyle \frac{log||A^{n-1}||}{n}=\log {\lambda _0}\approx 0.322\). Similarly, the topological entropy of \(f\) on \({\Lambda _1}\) equals zero, as a comparison of that shown in Proposition 2. This asserts that zero entropy can coexist with rather complex dynamics in 2D CAs. It is necessary to recall that these two definitions are both invariant under conjugacy. This paper has opted for the classical one which is applicable for all compact systems. Additionally, topologically mixing of \(f\) on \(\Lambda _2\) is certified as follows.

**Proposition 5**

\(f\) is topologically mixing on \(\Lambda _2\).

*Proof*

To prove \(f\) is topologically mixing on \(\Lambda _2\), one needs to check that for any two open cylinder sets \(U, V\subset \Lambda _2\), there exists \(N>0\) such that \({f}^{n}(U)\bigcap V \ne {\emptyset }\), for \(n\ge N\). Since \(\Lambda _2=R_0(\Lambda )\), it immediately follows that there exist \(2t\) open sets \(U_1,\ldots ,U_t,V_1,\ldots ,V_t\) of \(\Lambda _{\fancyscript{A}}\) such that for any \(x \in U_i\), \(y \in V_i\), \(i=1,2,\ldots ,t\), \(x\) appears in \(U\) and \(y\) appears in \(V\). Observe that \(\sigma \) is topologically mixing on \(\Lambda _{\fancyscript{A}}\). This means that there exist \(N_i\), \(i=1,2,\ldots ,t\), such that \({\sigma }^{n}(U_i)\bigcap V_i \ne {\emptyset }\), for \(n\ge N_i\). Now, let N be \(max\{N_1,\ldots ,N_t\}\), then for any \(n\ge N\) and \(i=1,2,\ldots ,t\), \({\sigma }^{n}(U_i)\bigcap V_i \ne {\emptyset }\). Consequently, \({\sigma _{10}}^{n}(U)\bigcap V \ne {\emptyset }\) which implies \({f}^{n}(U)\bigcap V \ne {\emptyset }\), for \(n\ge N\). \(\square \)

**Theorem 2**

\(f\) is chaotic in the sense of both Li–Yorke and Devaney on \(\Lambda _2\).

## 4 Symbolic dynamics of g1, g2, g3, and g4 gliders

The main results of Sect. 3 have provided a constructive and explicit proof that for each glider in diffusion rule, there exist chaotic subsystems in the sense that they are topologically mixing and have positive entropies. Indeed, all these complicated topological properties depend only on their associated transition matrix as specified in previous section; the same happens with the combinations of g1, g2, g3, and g4 gliders. As an example, the transition matrix of the combination of g1 and g2 gliders is formalized in the following.

**Proposition 6**

For the combination of g1 and g2 gliders, \(\fancyscript{B}=\{(A_0,A_0,A_0), (A_0,A_0,A_1), (A_0,A_1,A_2), (A_1,A_2,A_0), (A_2,A_0,A_0),(A_0,A_3,A_4), (A_3,A_4,A_2), (A_4,A_2,A_0), (A_0,A_0,A_3) \}\), and its transition matrix is \(B=\left[ \begin{array}{lllllllll} 1&{}\ 1&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 1 \\ 0&{}\ 0&{}\ 1&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0 \\ 0&{}\ 0&{}\ 0&{}\ 1&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0 \\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 1&{}\ 0&{}\ 0&{}\ 0&{}\ 0\\ 1&{}\ 1&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 1\\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 1&{}\ 0&{}\ 0\\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 1&{}\ 0\\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 1&{}\ 0&{}\ 0&{}\ 0&{}\ 0\\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0&{}\ 0 \end{array}\right] \), where “\(A_0=(0,0,0,0)^{T}\)”, “\(A_1=(1,0,0,1)^{T}\)”, “\(A_2=(0,1,1,0)^{T}\)”, “\(A_3=(0,0,0,1)^{T}\)” and “\(A_4=(1,0,0,0)^{T}\)”.

For \(n>7\), \(B_{ij}^n>0\), \(\forall i,j\), and its spectral radius \(\log {\rho (A)}=\log {\lambda }\approx 0.404\), where \(\lambda _0\) is the positive real root of \(\lambda ^5-\lambda ^4-\lambda -1=0\); therefore, the following result holds.

**Proposition 7**

- (1)
\(\Lambda _3=R(\Lambda _{\fancyscript{B}})\) and \(\Lambda _4=R_0(\prod \limits _{n=-\infty }^{+\infty }\Lambda _{\fancyscript{B}})\);

- (2)
\(f\) is topologically mixing on \(\Lambda _3\) and \(\Lambda _4\);

- (3)
\(ent(f|_{\Lambda _3})\approx 0.404\) and \(ent(f|_{\Lambda _4})=+\infty \);

- (4)
\(f\) is chaotic in the sense of both Li–Yorke and Devaney on \(\Lambda _3\) and \(\Lambda _4\).

Summary of quantitative properties of transition matrix of g1, g2, g3 and g4 gliders in diffusion rule cellular automata B2/S7 along with their chaotic statements

Type | Transition matrix | Spectral radius | Aperiodicity | Chaotic property |
---|---|---|---|---|

g1 | \(A_1\) | 0.322 | Yes | Li–Yorke and Devaney |

g2 | \(A_2\) | 0.322 | Yes | Li–Yorke and Devaney |

g3 | \(A_3\) | 0.322 | Yes | Li–Yorke and Devaney |

g4 | \(A_4\) | 0.322 | Yes | Li–Yorke and Devaney |

g1 and g2 | \(A_5\) | 0.404 | Yes | Li–Yorke and Devaney |

g1 and g3 | \(A_6\) | 0.404 | Yes | Li–Yorke and Devaney |

g1 and g4 | \(A_7\) | 0.404 | Yes | Li–Yorke and Devaney |

g2 and g3 | \(A_8\) | 0.372 | Yes | Li–Yorke and Devaney |

g2 and g4 | \(A_9\) | 0.372 | Yes | Li–Yorke and Devaney |

g3 and g4 | \(A_{10}\) | 0.372 | Yes | Li–Yorke and Devaney |

g1, g2 and g3 | \(A_{11}\) | 0.456 | Yes | Li–Yorke and Devaney |

g1, g2 and g4 | \(A_{12}\) | 0.456 | Yes | Li–Yorke and Devaney |

g1, g3 and g4 | \(A_{13}\) | 0.456 | Yes | Li–Yorke and Devaney |

g2, g3 and g4 | \(A_{14}\) | 0.430 | Yes | Li–Yorke and Devaney |

g1, g2, g3 and g4 | \(A_{15}\) | 0.413 | Yes | Li–Yorke and Devaney |

## 5 Discussion and conclusions

Cellular automata theory has been around for over fifty years, the qualitative dynamics, however, is surprising hard to pin down. To address this question, scientists from diverse disciplines have employed many techniques and approaches. 2D CAs supporting large variety of gliders have drawn more and more investigators’ attention in the last 20 years. This is because the computation can be implemented via collisions between the mobile gilders. One famous example is the diffusion rule cellular automata B2/S7, which may be the minimal model exhibiting glider in 2D CAs. Various authors have already discovered 26 types of distinct gliders, their dynamical behaviors nevertheless are missed in the framework of 2D symbolic dynamics. This paper takes this rule as an example on account of its ample types of gliders. Moreover, Sects. 3 and 4 just apply our proposed techniques to only simply gliders, and accounts for some consequences presented in the foregoing references.

In this research, chaotic dynamics of g1, g2, g3, and g4 gliders of diffusion rule is accurately determined by applying the releasing and extended releasing transformations. The main result is that each glider can infer subsystems with rich and complex symbolic dynamics as well as their combinations. Especially, the diffusion rule is chaotic in the sense of both Devaney and Li–Yorke on each subsystems. This analytical assertion concludes that the diffusion rule B2/S7 is filled with infinite disjoint chaotic subsystems, big and small. Meanwhile, one can, therefore, precisely predict chaotic properties of a 2D cellular automata rule via probing the gliders at its evolution from a random initial configuration. Noticeable, the method and procedure presented in this paper are also applicable to all other gliders in diffusion rule and 2D CAs. The results obtained here are just a tip of an iceberg. Frankly, more approaches to uncover dynamics of 2D CAs should be designed and the connections between universal properties of a cellular automaton and its topological properties are needed to affirm. These are all important topics for further research in the near future.

## Notes

### Acknowledgments

This research was jointly supported by NSFC (Grants no. 11171084 and 60872093), and Foundation of Zhejiang Chinese Medical University (Grant no. 17211076).

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