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Multifractal chaotic attractors in a system of delay-differential equations modeling road traffic. (English) Zbl 1080.34564
Summary: 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.
MSC:
34K23Complex (chaotic) behavior of solutions of functional-differential equations
34K18Bifurcation theory of functional differential equations
37N99Applications of dynamical systems