Presentation Information
[MS01-07]Analysis of traffic congestion phases in the OV model via a differential-difference equation
*Kota Ikeda1, Tomoyuki Miyaji2 (1. School of Interdisciplinary Mathematical Sciences, Meiji University (Japan), 2. Department of Mathematics, Kyoto University (Japan))
Keywords:
traffic flow,optimal velocity model,congestion,periodic solution,perturbation theory
Understanding traffic dynamics is essential for effective transportation planning and management, yet the nonlinear and unsteady nature of traffic makes comprehensive mathematical modeling challenging. Mathematical models of traffic flow are broadly categorized into microscopic and macroscopic approaches. Microscopic models, such as car-following models, describe individual vehicle behavior. One of car-following models, called, the Optimal Velocity (OV) model (M. Bando et al., Japan Journal of Industrial and Applied Mathematics, 1994), has significantly contributed to understanding spontaneous traffic congestion. As noted by Hasebe et al. (Physics Letters A, 1999), periodic solutions corresponding to congested phases in the OV model satisfy a differential-difference equation naturally derived from the model. Although many models described by differential-difference equations exhibit solutions with periodic behavior and transition layers corresponding to state changes, analytically constructing such solutions is generally difficult. By the same way as in the previous result (Sugiyama and Yamada, Physical Review E, 1997), a periodic solution can be rigorously constructed when the OV function is assumed to be a step function. Nevertheless, analytical studies for general OV functions remain unexplored.
Our goal in this presentation is to construct a periodic solution for the differential-difference equation when the OV function is given as a generalization of a step function while remaining close to it. We require analyzing a linearized equation around the step function, which naturally introduces Dirac delta functions due to its discontinuity. Furthermore, the position where the delta function appears is affected by the influence of the difference term. By appropriately handling the delta function and applying the contraction mapping principle, we successfully construct a periodic solution. This analytical approach provides new insights into the mathematical structure of traffic flow dynamics and expands the applicability of the OV model to more general cases.
Our goal in this presentation is to construct a periodic solution for the differential-difference equation when the OV function is given as a generalization of a step function while remaining close to it. We require analyzing a linearized equation around the step function, which naturally introduces Dirac delta functions due to its discontinuity. Furthermore, the position where the delta function appears is affected by the influence of the difference term. By appropriately handling the delta function and applying the contraction mapping principle, we successfully construct a periodic solution. This analytical approach provides new insights into the mathematical structure of traffic flow dynamics and expands the applicability of the OV model to more general cases.