64 research outputs found
Algebraic entropy of an extended Hietarinta-Viallet equation
We introduce a series of discrete mappings, which is considered to be an
extension of the Hietarinta-Viallet mapping with one parameter. We obtain the
algebraic entropy for this mapping by obtaining the recurrence relation for the
degrees of the iterated mapping. For some parameter values the mapping has a
confined singularity, in which case the mapping is equivalent to a recurrence
relation between irreducible polynomials. For other parameter values, the
mapping does not pass the singularity confinement test. The properties of
irreducibility and co-primeness of the terms play crucial roles in the
discussion.Comment: 22 page
Discrete Painleve equations and discrete KdV equation over finite fields
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.Comment: 20 pages, 4 figures, Review articl
Graphs emerging from the solutions to the periodic discrete Toda equation over finite fields
The periodic discrete Toda equation defined over finite fields has been
studied. We obtained the finite graph structures constructed by the network of
states where edges denote possible time evolutions. We simplify the graphs by
introducing a equivalence class of cyclic permutations to the initial values.
We proved that the graphs are bi-directional and that they are composed of
several arrays of complete graphs connected at one of their vertices.Comment: 20 pages, 5 figures, NOLTA Vol. E7-N, No.3, Jul. 201
Singularity confinement and chaos in two-dimensional discrete systems
We present a quasi-integrable two-dimensional lattice equation: i.e., a
partial difference equation which satisfies a criterion of integrability,
singularity confinement, although it has a chaotic aspect in the sense that the
degrees of its iterates exhibit exponential growth. By systematic reduction to
one-dimensional systems, it gives a hierarchy of ordinary difference equations
with confined singularities, but with positive algebraic entropy including a
generalized form of the Hietarinta-Viallet mapping. We believe that this is the
first example of such quasi-integrable equations defined over a two-dimensional
lattice.Comment: 10 pages, 1 figur
Singularities of the discrete KdV equation and the Laurent property
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)Comment: 17 pages, 2 figure
A stochastic optimal velocity model and its long-lived metastability
In this paper, we propose a stochastic cellular automaton model of traffic
flow extending two exactly solvable stochastic models, i.e., the asymmetric
simple exclusion process and the zero range process. Moreover it is regarded as
a stochastic extension of the optimal velocity model. In the fundamental
diagram (flux-density diagram), our model exhibits several regions of density
where more than one stable state coexists at the same density in spite of the
stochastic nature of its dynamical rule. Moreover, we observe that two
long-lived metastable states appear for a transitional period, and that the
dynamical phase transition from a metastable state to another metastable/stable
state occurs sharply and spontaneously
Algebraic entropy of a multi-term recurrence of the Hietarinta-Viallet type
We introduce a family of extensions of the Hietarinta-Viallet equation to a
multi-term recurrence relation via a reduction from the coprimeness-preserving
extension to the discrete KdV equation. The recurrence satisfies the
irreducibility and the coprimeness property although it is nonintegrable in
terms of an exponential degree growth. We derive the algebraic entropy of the
recurrence by an elementary method of calculating the degree growth. The result
includes the entropy of the original Hietarinta-Viallet equation.Comment: 24 pages, To appear in RIMS Kokyuroku Bessats
Pattern formation of vascular network in a mathematical model of angiogenesis
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
A two dimensional lattice equation as an extension of the Heideman-Hogan recurrence
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.Comment: 13 page
Irreducibility and co-primeness as an integrability criterion for discrete equations
We study the Laurent property, the irreducibility and co-primeness of
discrete integrable and non-integrable equations. First we study a discrete
integrable equation related to the Somos-4 sequence, and also a non-integrable
equation as a comparison. We prove that the conditions of irreducibility and
co-primeness hold only in the integrable case. Next, we generalize our previous
results on the singularities of the discrete Korteweg-de Vries (dKdV) equation.
In our previous paper (arXiv:1311.0060), we described the singularity
confinement test (one of the integrability criteria) using the Laurent
property, and the irreducibility, and co-primeness of the terms in the bilinear
dKdV equation, in which we only considered simplified boundary conditions. This
restriction was needed to obtain simple (monomial) relations between the
bilinear form and the nonlinear form of the dKdV equation. In this paper, we
prove the co-primeness of the terms in the nonlinear dKdV equation for general
initial conditions and boundary conditions, by using the localization of
Laurent rings and the interchange of the axes. We assert that co-primeness of
the terms can be used as a new integrability criterion, which is a mathematical
re-interpretation of the confinement of singularities in the case of discrete
equations. v2, v3: minor revisionsComment: 18 page
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