Instabilities in generic second-order traffic models with relaxation

We prove the existence of weak solutions for a class of second-order traffic models with relaxation, without requiring the sub-characteristic stability condition to hold. With the help of numerical simulations, we show how, in this unstable setting, large but bounded oscillations may arise from small perturbations of equilibria, thus reproducing the formation of stop-and-go waves commonly observed in traffic dynamics. An analysis of the corresponding traveling waves completes the study.