Kohei Higashi, Junkichi Satsuma, Tetsuji Tokihiro
The rule 184 fuzzy cellular automaton is regarded as a mathematical model of traffic flow because it contains the two fundamental traffic flow models, the rule 184 cellular automaton and the Burgers equation, as special cases. We show that the fundamental diagram (flux-density diagram) of this model consists of three parts: a free-flow part, a congestion part and a two-periodic part. The two-periodic part, which may correspond to the synchronized mode region, is a two-dimensional area in the diagram, the boundary of which consists of the free-flow and the congestion parts. We prove that any state in both the congestion and the two-periodic parts is stable, but is not asymptotically stable, while that in the free-flow part is unstable. Transient behaviour of the model and bottle-neck effects are also examined by numerical simulations. Furthermore, to investigate low or high density limit, we consider ultradiscrete limit of the model and show that any ultradiscrete state turns to a travelling wave state of velocity one in finite time steps for generic initial conditions.
Aoi Araoka, Tetsuji Tokihiro
This paper explores cellular automata (CA) constructed from Yang-Baxter maps over finite fields $F_{2^n}$. We define $R$-matrices using a map $f$ on $F_{2^n}$ and establish necessary and sufficient conditions for $f$ to satisfy the Yang-Baxter equation. We show that these conditions become remarkably streamlined in characteristic two. An exhaustive search for bijective solutions in fields of order 4, 8, and 16 yields 16, 736, and 269,056 maps, respectively. Analysis of the resulting CA under helical boundary conditions reveals a consistent alignment between the temporal period and the field order. We propose the conjecture that this periodic identity holds generally for $F_{2^n}$, supported by analytical proofs for $n=2$ and $n=3$. Our results further indicate that bijectivity is a fundamental requirement for this periodic behavior.
Seiryu Shimizu, Tetsuji Tokihiro
We propose a general method for constructing a fuzzy cellular automaton from a given cellular automaton. Unlike previous approaches that use fuzzy distinctive normal form, whose update function is restricted to third-order polynomials, our method accommodates a wide range of fuzzification functions, enabling the generation of diverse and complex time-evolution patterns that are unattainable with simpler heuristic models. We demonstrate that phase transitions in pattern formation can be observed by changing the parameters of the fuzzification function or the mixing ratio between two distinct evolution rules of elementary cellular automata. Remarkably, the resulting generalized fuzzy elementary cellular automata exhibit rich dynamical properties, including stable manifolds and chaos, even in minimal systems composed of just three cells.
Ryo Kamiya, Masataka Kanki, Takafumi Mase, Tetsuji Tokihiro
We introduce an equation defined on a multi-dimensional lattice, which can be considered as an extension to the coprimeness-preserving discrete KdV like equation in our previous paper. The equation is also interpreted as a higher-dimensional analogue of the Hietarinta-Viallet equation, which is famous for its singularity confining property while having an exponential degree growth. As the main theorem we prove the Laurent and the irreducibility properties of the equation in its "tau-function" form. From the theorem the coprimeness of the equation follows. In Appendix we review the coprimeness-preserving discrete KdV like equation whichis a base equation for our main system and prove the properties such as the coprimeness.
Jun Mada, Tetsuji Tokihiro
We investigate correlation functions in a periodic box-ball system. For the second and the third nearest neighbor correlation functions, we give explicit formulae obtained by combinatorial methods. A recursion formula for a specific $N$-point functions is also presented.
Masataka Kanki, Jun Mada, K. M. Tamizhmani, Tetsuji Tokihiro
We investigate the discrete Painleve II equation over finite fields. We treat it over local fields and observe that it has a property that is similar to the good reduction over finite fields. We can use this property, which seems to be an arithmetic analogue of singularity confinement, to avoid the indeterminacy of the equations over finite fields and to obtain special solutions from those defined originally over fields of characteristic zero.
Ryo Kamiya, Masataka Kanki, Takafumi Mase, Tetsuji Tokihiro
We consider a two dimensional extension of the so-called linearizable mappings. In particular, we start from the Heideman-Hogan recurrence, which is known as one of the linearizable Somos-like recurrences, and introduce one of its two dimensional extensions. The two dimensional lattice equation we present is linearizable in both directions, and has the Laurent and the coprimeness properties. Moreover, its reduction produces a generalized family of the Heideman-Hogan recurrence. Higher order examples of two dimensional linearizable lattice equations related to the Dana-Scott recurrence are also discussed.
Masataka Kanki, Takafumi Mase, Tetsuji Tokihiro
In this article we investigate the coprimeness properties of one and two-dimensional discrete equations, in a situation where the equations are decomposable into several factors of polynomials. After experimenting on a simple equation, we shall focus on some higher power extensions of the Somos-4 equation and the (1-dimensional) discrete Toda equation. Our previous results are that all of the equations satisfy the irreducibility and the coprimeness properties if the r.h.s. is not factorizable. In this paper we shall prove that the coprimeness property still holds for all of these equations even if the r.h.s. is factorizable, although the irreducibility property is no longer satisfied.
Masahiro Kanai, Katsuhiro Nishinari, Tetsuji Tokihiro
In this paper, we study an exact solution of the asymmetric simple exclusion process on a periodic lattice of finite sites with two typical updates, i.e., random and parallel. Then, we find that the explicit formulas for the partition function and the average velocity are expressed by the Gauss hypergeometric function. In order to obtain these results, we effectively exploit the recursion formula for the partition function for the zero-range process. The zero-range process corresponds to the asymmetric simple exclusion process if one chooses the relevant hop rates of particles, and the recursion gives the partition function, in principle, for any finite system size. Moreover, we reveal the asymptotic behaviour of the average velocity in the thermodynamic limit, expanding the formula as a series in system size.
Tatsuya Hayashi, Fumitaka Yura, Jun Mada, Hiroki Kurihara, Tetsuji Tokihiro
A two-dimensional mathematical model for dynamics of endothelial cells in angiogenesis is investigated. Angiogenesis is a morphogenic process in which new blood vessels emerge from an existing vascular network. Recently a one-dimensional discrete dynamical model has been proposed to reproduce elongation, bifurcation, and cell motility such as cell-mixing during angiogenesis on the assumption of a simple two-body interaction between endothelial cells. The present model is its two-dimensional extension, where endothelial cells are represented as the ellipses with the two-body interactions: repulsive interaction due to excluded volume effect, attractive interaction through pseudopodia and rotation by contact. We show that the oblateness of ellipses and the magnitude of contact rotation significantly affect the shape of created vascular patterns and elongation of branches.
Jun Mada, Tetsuji Tokihiro
We discuss the characteristics of the patterns of the vascular networks in a mathematical model for angiogenesis. Based on recent in vitro experiments, this mathematical model assumes that the elongation and bifurcation of blood vessels during angiogenesis are determined by the density of endothelial cells at the tip of the vascular network, and describes the dynamical changes in vascular network formation using a system of simultaneous ordinary differential equations. The pattern of formation strongly depends on the supply rate of endothelial cells by cell division, the branching angle, and also on the connectivity of vessels. By introducing reconnection of blood vessels, the statistical distribution of the size of islands in the network is discussed with respect to bifurcation angles and elongation factor distributions. The characteristics of the obtained patterns are analysed using multifractal dimension and other techniques.
Masataka Kanki, Jun Mada, Tetsuji Tokihiro
We investigate some of the discrete Painleve equations (dPII, qPI and qPII) and the discrete KdV equation over finite fields. The first part concerns the discrete Painleve equations. We review some of the ideas introduced in our previous papers and give some detailed discussions. We first show that they are well defined by extending the domain according to the theory of the space of initial conditions. We then extend them to the field of p-adic numbers and observe that they have a property that is called an `almost good reduction' of dynamical systems over finite fields. We can use this property, which can be interpreted as an arithmetic analogue of singularity confinement, to avoid the indeterminacy of the equations over finite fields and to obtain special solutions from those defined originally over fields of characteristic zero. In the second part we study the discrete KdV equation. We review the previous discussions and present a way to resolve the indeterminacy of the equation by treating it over a field of rational functions instead of the finite field itself. Explicit forms of soliton solutions and their periods over finite fields are obtained. Note: This is a review article on the recent developments in the theory of discrete integrable equations over finite fields based on arXiv:1201.5429, arXiv:1206.4456, arXiv:1209.0223. This article is published as the proceedings of the domestic conference "The breadth and depth of nonlinear discrete integrable systems" in RIMS, Kyoto University, Japan, on August 2012.
Masataka Kanki, Jun Mada, Tetsuji Tokihiro
We study the distribution of singularities for partial difference equations, in particular, the bilinear and nonlinear form of the discrete version of the Korteweg-de Vries (dKdV) equation. By the Laurent property, the irreducibility, and the co-primeness of the terms of the bilinear dKdV equation, we clarify the relationship of these properties with the appearance of zeros in the time evolution. The results are applied to the nonlinear dKdV equation and we formulate the famous integrability criterion (singularity confinement test) for nonlinear partial difference equations with respect to the co-primeness of the terms. (v2,v3,v4: minor revisions have been made)
Jun Mada, Tetsuji Tokihiro
We investigate correlation functions in a periodic box-ball system. For the two point functions of short distance, we give explicit formulae obtained by combinatorial methods. We give expressions for general N-point functions in terms of ultradiscrete theta functions.
Masahiro Kanai, Katsuhiro Nishinari, Tetsuji Tokihiro
In recent works, we have proposed a stochastic cellular automaton model of traffic flow connecting two exactly solvable stochastic processes, i.e., the Asymmetric Simple Exclusion Process and the Zero Range Process, with an additional parameter. It is also regarded as an extended version of the Optimal Velocity model, and moreover it shows particularly notable properties. In this paper, we report that when taking Optimal Velocity function to be a step function, all of the flux-density graph (i.e. the fundamental diagram) can be estimated. We first find that the fundamental diagram consists of two line segments resembling an {\it inversed-$λ$} form, and next identify their end-points from a microscopic behaviour of vehicles. It is otable that by using a microscopic parameter which indicates a driver's sensitivity to the traffic situation, we give an explicit formula for the critical point at which a traffic jam phase arises. We also compare these analytical results with those of the Optimal Velocity model, and point out the crucial differences between them.
Masahiro Kanai, Shin Isojima, Katsuhiro Nishinari, Tetsuji Tokihiro
In this paper, we propose the ultra-discrete optimal velocity model, a cellular-automaton model for traffic flow, by applying the ultra-discrete method for the optimal velocity model. The optimal velocity model, defined by a differential equation, is one of the most important models; in particular, it successfully reproduces the instability of high-flux traffic. It is often pointed out that there is a close relation between the optimal velocity model and the mKdV equation, a soliton equation. Meanwhile, the ultra-discrete method enables one to reduce soliton equations to cellular automata which inherit the solitonic nature, such as an infinite number of conservation laws, and soliton solutions. We find that the theory of soliton equations is available for generic differential equations, and the simulation results reveal that the model obtained reproduces both absolutely unstable and convectively unstable flows as well as the optimal velocity model.
Masataka Kanki, Jun Mada, Tetsuji Tokihiro
We construct generalized solutions to the ultradiscrete KdV equation, including the so-called negative solition solutions. The method is based on the ultradiscretization of soliton solutions to the discrete KdV equation with gauge transformation. The conserved quantities of the ultradiscrete KdV equation are shown to be constructed in a similar way to those for the box-ball system.
Masataka Kanki, Jun Mada, Tetsuji Tokihiro
We investigate the multi-soliton solutions to the generalized discrete KdV equation. In some cases a soliton with smaller amplitude moves faster than that with larger amplitude unlike the soliton solutions of the KdV equation. This phenomenon is intuitively understood from its ultradiscrete limit, where the system turns to the box ball system with a carrier. KEYWORDS: soliton, integrable equation, nonlinear system, discrete KdV equation, cellular automaton (v2: Final form to appear in J. Phys. Soc. Jpn.)
Masataka Kanki, Jun Mada, Tetsuji Tokihiro
We reformulate the singularity confinement of the discrete Toda equation. We prove the co-primeness property, which has been introduced in our previous paper (arXiv:1311.0060) as one of the integrability criteria, for the discrete Toda equation. We study three types of boundary conditions (semi-infinite, molecule, periodic) for the discrete Toda equation, and prove that the same co-primeness property holds for all the types of boundaries. (v2: typos corrected, final version to appear in J. Math. Phys.)
Keisuke Matsuya, Hiroki Kurihara, Tetsuji Tokihiro
Based on recent experiments with time-lapse fluorescent imaging, we propose a cellular automaton model for the dynamics of vascular endothelial cells (ECs) in angiogenic morphogenesis. The model successfully reproduces cell mixing behavior, elongation and bifurcation of blood vessels. The results suggest that the two-body interaction between ECs, which is repulsive in short distance and become attractive in moderately long distance, is essential to the dynamics of ECs, in particular, to the cell mixing behavior. The corresponding analytically solvable differential equation model is also proposed.