Multifractal chaotic attractors in a system of delay-differential equations modeling road traffic

Abstract

We study a system of delay-differential equations modeling single-lane road traffic. The cars move in a closed circuit and the system's variables are each car's velocity and the distance to the car ahead. For low and high values of traffic density the system has a stable equilibrium solution, corresponding to the uniform flow. Gradually decreasing the density from high to intermediate values we observe a sequence of supercritical Hopf bifurcations forming multistable limit cycles, corresponding to flow regimes with periodically moving traffic jams. Using an asymptotic technique we find approximately small limit cycles born at Hopf bifurcations and numerically preform their global continuations with decreasing density. For sufficiently large delay the system passes to chaos following the Ruelle-Takens-Newhouse scenario (limit cycles-two-tori-three-tori-chaotic attractors). We find that chaotic and nonchaotic attractors coexist for the same parameter values and that chaotic attractors have a broad multifractal spectrum. (c) 2002 American Institute of Physics.

Self-control of traffic lights and vehicle flows in urban road networks

“…Switched flow networks are known to exhibit chaotic behavior under certain conditions [65,109]. In principle, this is also expected to apply to traffic light controlled road networks.…”

Traffic jams: dynamics and control

“…For more details about the techniques used and results presented, one may consult Gasser et al (2004), Orosz & Stépán (2006) and Orosz et al (2009). Some claims about multi-stability are also given in Krauss et al (1997), Lee et al (1998), Igarashi et al (2001), Safonov et al (2002) and Helbing & Moussaid (2009) by using different tools.…”

“…For more details about the techniques used and results presented, one may consult Gasser et al (2004), Orosz & Stépán (2006) and Orosz et al (2009). Some claims about multi-stability are also given in Krauss et al (1997), Lee et al (1998), Igarashi et al (2001), Safonov et al (2002) and Helbing & Moussaid (2009) by using different tools.Here we consider the OVM (3.10), (3.12), but one may reproduce the analysisand obtain qualitatively similar behaviour-for any dynamic model that admits a uniform flow equilibrium (4.1), (4.2) with an equilibrium function that looks like figure 3b; for example, see the IDM (3.10), (3.13) except for small equilibrium headways. We consider the human driver set-up t = s > 0, k = 0 for the reaction times and assume the optimal and relative velocity functions The function V is shown in figure 3b.…”

Stochastic switching in delay-coupled oscillators

“…Time delays arise naturally, and might play a role in many areas of physics, biology and technology, such as nonlinear optics [3,4], gene regulatory circuits [5], population dynamics [6,7], traffic flows [8,9], neuroscience [10], and social or communication networks [11,12].…”