Global bifurcation investigation of an optimal velocity traffic model with driver reaction time

We investigate an optimal velocity model which includes the reflex time of drivers. After an analytical study of the stability and local bifurcations of the steady-state solution, we apply numerical continuation techniques to investigate the global behavior of the system. Specifically, we find branches of oscillating solutions connecting Hopf bifurcation points, which may be super- or subcritical, depending on parameters. This analysis reveals several regions of multistability.

Figure 1

Sequence of cars on a single-lane road showing their positions, velocities, and headways.

Figure 2

Three different rescaled optimal velocity (OV) functions with an enlargement of the region around $h=1$ (a), and the derivatives of the OV functions with respect to $h$ (b).

Figure 3

Stability charts of the three-car system, where shading denotes the stable region.(a) Slope of the OV function ${\mathrm{V}}^{\prime }\left(h^{*}\right)$ vs the sensitivity $\alpha$; average headway $h^{*}$ vs $\alpha$ for (b) $v^0=0.6$ and for (c) $v^0=0.8$. (This corresponds to ${\mathrm{V}}_{\max }^{\prime }\simeq 0.504$ and ${\mathrm{V}}_{\max }^{\prime }\simeq 0.672$, respectively).

Figure 4

Amplitude of oscillations of the velocity of $i$th car vs average headway $h^{*}$. The horizontal axis represents the equilibrium state. Solid curves denote stable, and dashed curves denote unstable states; the dotted curve represents the collision region. The value of $\alpha$ is depicted in each panel (a)–(d) and $v^0=0.35$ in all panels.

Figure 5

Oscillations of the velocity of the first car over one period, shown as solid curves to the scale on the left; oscillations of the headway of the first car over one period, shown as dashed curves to the scale on the right. Cases A–F correspond to the marks in Fig. 4.

Figure 6

Amplitude of oscillations of the velocity of the $i$th car vs average headway $h^{*}$. The horizontal axis represents the equilibrium state. Solid curves denote stable and dashed curves denote unstable states. The value of $v^0$ is depicted in each panel (a)–(d) and $\alpha =0.1$ in all panels.

Figure 7

Stopping motion for $v^0=1.0$ and $\alpha =1.0$. (a) Amplitude of oscillations of the velocity of the $i$th car vs average headway $h^{*}$. (b) Equilibrium state $\mathrm{V}\left(h^{*}\right)$ (upper curve) and minimum velocity $v_i^{\min }$ (lower curve) of oscillations of the velocity of $i$th car vs average headway $h^{*}$. In panels (a) and (b) solid curves denote stable, and dashed curves denote unstable states. Panels (c)–(e) show oscillations of the velocity of the first car over one period as solid curves to the scale on the left-hand side, and oscillations of the headway of the first car over one period as dashed curves to the scale on the right-hand side. Cases G–I correspond to the marks in panels (a) and (b).

Figure 8

Two-dimensional bifurcation diagrams for different values of $v^0$ as indicated. The horizontal dotted lines in panels (a) and (c) correspond to the values of $\alpha$ used in Figs. 4, 7a, respectively.