We propose a discrete traffic flow model with discrete time. Continuum limit
of this model is equivalent to the optimal velocity model. It has also an
ultradiscrete limit and a piecewise-linear type of traffic flow model is
obtained. Both models show phase transition from free flow to jam in a
fundamental diagram. Moreover, the ultradiscrete model includes the
Fukui-Ishibashi model in a special...
We propose discrete mappings of second order that have a discrete analogue of
Lyapunov function. The mappings are extensions of the integrable
Quispel-Roberts-Thompson (QRT) mapping, and a discrete Lyapunov function of the
mappings is identical to an explicit conserved quantity of the QRT mapping.
Moreover we can obtain a differential and an ultradiscrete limit of the
mappings preserving the ex...
We show that the third-order difference equations proposed by Hirota,
Kimura and Yahagi are generated by a pair of second-order difference
equations. In some cases, the pair of the second-order equations are equivalent
to the Quispel-Robert-Thomson(QRT) system, but in the other cases, they are
irrelevant to the QRT system. We also discuss an ultradiscretization of the
equations.
Recently, we have proposed a {\em Euler-Lagrange transformation} for cellular
automata(CA) by developing new transformation formulas. Applying this method to
the Burgers CA(BCA), we have succeeded in obtaining the Lagrange representation
of the BCA. In this paper, we apply this method to multi-value generalized
Burgers CA(GBCA) which include the Fukui-Ishibashi model and the quick-start
model a...
