Hopf bifurcation analysis for a dissipative system with asymmetric interaction: Analytical explanation of a specific property of highway traffic

A dissipative system with asymmetric interaction, the optimal velocity model, shows a Hopf bifurcation concerned with the transition from a homogeneous motion to the formation of a moving cluster, such as the emergence of a traffic jam. We investigate the properties of Hopf bifurcation depending on the particle density, using the dynamical system for the traveling cluster solution of the continuum system derived from the original discrete system of particles. The Hopf bifurcation is revealed as a subcritical one, and the property explains well the specific phenomena in highway traffic: the metastability of jamming transition and the hysteresis effect in the relation of car density and flow rate.

Figure 1

Space-time plot of cluster formation of the OV model with OV function as $V\left(\mathrm{\Delta }x\right)=tanh(\mathrm{\Delta }x-2)+tanh2$ in $N=100$ on the periodic boundary condition on a circuit. The vertical axis is location on the circuit. The horizontal axis is time evolution.

Figure 2

The profile of a cluster flow solution. “Fast” and “Jam” denote smoothly moving regions and jam clusters, respectively. All vehicles move along the closed loop in the direction of counter clockwise. A dotted curve represents OV function.

Figure 3

The phase diagram of OV model. $(b_c^{*},a_c^{*})$ is the inflection point of the OV function. In phase (i), a homogeneous flow is stable, while a jam flow is unstable. In phase (ii), both a homogeneous flow and a jam flow are stable. In phase (iii), a jam flow is stable, while a homogeneous flow is unstable. The figure is drawn for an OV function as $V\left(\mathrm{\Delta }x\right)=tanh(\mathrm{\Delta }x-2)+tanh2$ and the inflection point is $(b_c^{*},a_c^{*})=(2,2)$, for example.

Figure 4

Profile of the jam flow solution for various values of the sensitivity $a=1.0,1.5,1.9$ with $b=b_c^{*}=2$ in the headway-velocity space in the OV model. The OV function is chosen as $V\left(\mathrm{\Delta }x\right)=tanh(\mathrm{\Delta }x-2)+tanh2$. In this case, the Hopf bifurcation point is $a_c^{*}=2$.

Figure 5

Hopf bifurcation of the OV model in the case of $b_c=b_c^{*}$. The horizontal axis is a sensitivity $a$, the vertical axis is the projection of two-dimensional phase space $(u,v)$. The center line shows the trivial solution (a homogeneous flow). The solid curve shows the limit cycle (a jam flow solution). The solid line and curve denote stable solutions, and the dotted line means an unstable homogeneous flow solution. Arrows are examples of flows from an unstable solution to a stable one in the phase space, which shows a supercritical Hopf bifurcation.

Figure 6

Hopf bifurcation of the OV model in the case of $b\ne b_c^{*}$. The horizontal and vertical axes are the same in Fig. 5. The center line and the curve show the trivial and limit cycle solution, respectively. The solid line and curve mean stable, and dotted ones signify unstable. Arrows are presented as flows from an unstable solution to a stable one in the phase space, which shows subcritical Hopf bifurcation in the bistable (coexistence) phase.

Figure 7

The typical fundamental diagram (the relation between vehicle density and flow rate) by 1-month data measured at a point on freeway. The critical density is nearly 25 (vehicle/$\text{km}$). The data are measured by Japan Highway Public Cooperation.

Figure 8

The time sequence in the $q-\rho$ space from 07:00 to 10:25 on 2 Aug 1996 at the upper stream of Nihonzaka tunnel in the Tomei Expressway [15].