Car-following models with delayed feedback: Local stability and Hopf bifurcation

Abstract

Reaction delays are important in determining the qualitative dynamical properties of a platoon of vehicles traveling on a straight road. In this paper, we investigate the impact of delayed feedback on the dynamics of the Modified Optimal Velocity Model (MOVM). Specifically, we analyze the MOVM in three regimes -no delay, small delay and arbitrary delay. In the absence of reaction delays, we show that the MOVM is locally stable. For small delays, we then derive a sufficient condition for the MOVM to be locally stable. Next, for an arbitrary delay, we derive the necessary and sufficient condition for the local stability of the MOVM. We show that the traffic flow transits from the locally stable to the locally unstable regime via a Hopf bifurcation. We also derive the necessary and sufficient condition for non-oscillatory convergence and characterize the rate of convergence of the MOVM. These conditions help ensure smooth traffic flow, good ride quality and quick equilibration to the uniform flow. Further, since a Hopf bifurcation results in the emergence of limit cycles, we provide an analytical framework to characterize the type of the Hopf bifurcation and the asymptotic orbital stability of the resulting non-linear oscillations. Finally, we corroborate our analyses using stability charts, bifurcation diagrams, numerical computations and simulations conducted using MATLAB.

The modified optimal velocity model: stability analyses and design guidelines

“…For example, the Hopf bifurcation analysis provides a mathematical framework to offer a possible explanation for the observed "phantom jams" [10]. In the case of self-driven vehicles, our work offers suggestions for their design guidelines.…”

“…Such an oscillatory behavior in the traffic flow has typically been attributed to a change in the driver's sensitivity, such as a sudden deceleration; for details, see [7], [8]. In general, feedback delays are known to induce oscillations in state variables of dynamical systems [10], [12]. Since the MOVM explicitly incorporates feedback delays, and relative velocities and headways constitute state variables of the MOVM, our work provides a theoretical basis for understanding the emergence and evolution of oscillatory phenomena such as "phantom jams."…”

“…Feedback delays play an important role in determining the qualitative behavior of dynamical systems [11]. In particular, these delays are known to destabilize the system and induce oscillatory behavior [10], [12]. In the context of human-driven vehicles, predominant components of the reaction delay are psychological and mechanical in nature [12].…”

“…To that end, it is imperative to have an in-depth understanding of human behavior and vehicular dynamics. This has led to the development and study of a class of dynamical models known as the car-following models [5]- [10]. Feedback delays play an important role in determining the qualitative behavior of dynamical systems [11].…”

“…Specifically, we consider each vehicle's dynamics to be modeled by the Modified Optimal Velocity Model (MOVM) [10]. Motivated by the wide range of values assumed by reaction delays in various scenarios, we analyze the MOVM in three regimes; namely, (i) no delay, (ii) small delay and (iii) arbitrary delay.…”