Junkichi Satsuma

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.

We examine the growth properties of second-order mappings which are integrable by linearisation and which generically exhibit a linear growth of the homogeneous degree of initial conditions. We show that for Gambier-type mappings for which the growth proceeds generically with a step of 1 there exist cases where the degree increase by unity every two steps. We examine also mappings belonging to the family known as “of third kind” in relation to the approach of Diller and Favre concerning the regularisable or not character of mappings and show that the anticonfined singularities of these mappings exhibit a linear growth with step 1. (The term anticonfined is used for singularities where the singular values extend all the way to infinity on both sides with just a few regular values in the middle). Moreover we construct specific examples of Gambier-type mappings which have anticonfined singularities and where the degree of the singularity increases linearly but where the average slope can be adjusted so as to be arbitrarily small.