关于微观交通建模中出现的动力系统
About Dynamical Systems Appearing in the Microscopic Traffic Modeling
摘要：
受微观交通建模的启发，我们分析了具有分段线性凹动力学不一定单调的动力系统。我们介绍了一个确定性的Petri网扩展的边缘可能有负的权重。这些Petri网的动力学是定义良好的，可以用一个广义矩阵来描述，其中一个子矩阵在标准代数中可能有负项，另一个子矩阵在minplus代数中。当动力学是加性齐次时，可以引入广义加性特征值，并且可以使用遍历论来定义附加技术假设下的增长率。在交通的两条道路与一个路口的例子中，我们明确计算的特征值，我们表明，通过数值模拟，这两个量（附加特征值和增长率）是不相等的，但彼此接近。有了这个结果，我们能够扩展的基本交通图（平均流量作为一个函数的道路上的汽车密度）的情况下，两条道路与一个路口，并给出了一个非常简单的解析近似这个图中出现四个阶段与明确的交通解释。仿真结果表明，所得到的基本图形形状对具有多个结点的系统也是有效的。为了模拟这些系统，我们必须计算它们的动力学，这并不简单。为了建立他们在一个模块化的方式，我们介绍了广义并联，串联和反馈组合分段线性凹动力学。
英文摘要：
Motivated by microscopic traffic modeling, we analyze dynamical systems which have a piecewise linear concave dynamics not necessarily monotonic. We introduce a deterministic Petri net extension where edges may have negative weights. The dynamics of these Petri nets are well-defined and may be described by a generalized matrix with a submatrix in the standard algebra with possibly negative entries, and another submatrix in the minplus algebra. When the dynamics is additively homogeneous, a generalized additive eigenvalue may be introduced, and the ergodic theory may be used to define a growth rate under additional technical assumptions. In the traffic example of two roads with one junction, we compute explicitly the eigenvalue and we show, by numerical simulations, that these two quantities (the additive eigenvalue and the growth rate) are not equal, but are close to each other. With this result, we are able to extend the well-studied notion of fundamental traffic diagram (the average flow as a function of the car density on a road) to the case of two roads with one junction and give a very simple analytic approximation of this diagram where four phases appear with clear traffic interpretations. Simulations show that the fundamental diagram shape obtained is also valid for systems with many junctions. To simulate these systems, we have to compute their dynamics, which are not quite simple. For building them in a modular way, we introduce generalized parallel, series and feedback compositions of piecewise linear concave dynamics.
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