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Motivated by microscopic traffic modeling, we analyze dynamical systems whichhave a piecewise linear concave dynamics not necessarily monotonic. Weintroduce a deterministic Petri net extension where edges may have negativeweights. The dynamics of these Petri nets are well-defined and may be describedby a generalized matrix with a submatrix in the standard algebra with possiblynegative entries, and another submatrix in the minplus algebra. When thedynamics is additively homogeneous, a generalized additive eigenvalue may beintroduced, and the ergodic theory may be used to define a growth rate underadditional technical assumptions. In the traffic example of two roads with onejunction, we compute explicitly the eigenvalue and we show, by numericalsimulations, that these two quantities (the additive eigenvalue and the growthrate) are not equal, but are close to each other. With this result, we are ableto extend the well-studied notion of fundamental traffic diagram (the averageflow as a function of the car density on a road) to the case of two roads withone junction and give a very simple analytic approximation of this diagramwhere four phases appear with clear traffic interpretations. Simulations showthat the fundamental diagram shape obtained is also valid for systems with manyjunctions. To simulate these systems, we have to compute their dynamics, whichare not quite simple. For building them in a modular way, we introducegeneralized parallel, series and feedback compositions of piecewise linearconcave dynamics.

About Dynamical Systems Appearing in the Microscopic Traffic Modeling