Nonlinear stability and bifurcation-type transition in mixed human-driven and connected automated vehicle traffic

The stability of mixed traffic comprising human-driven vehicles (HDVs) and connected automated vehicles (CAVs) is conventionally assessed through linear eigenvalue analysis of the uniform-flow equilibrium. Yet linear methods cannot distinguish whether the ensuing instability is a smooth, reversible oscillation or an abrupt, hysteretic collapse — a distinction with profound implications for traffic safety. This study investigates the nonlinear string stability of a mixed platoon on a ring road, modelled by the full velocity difference model for HDVs and the leading cruise control protocol for CAVs, across the two-dimensional parameter space spanned by the CAV penetration rate ϕ and driver sensitivity α. We construct the complete (ϕ,α) phase diagram using both the maximum real part of the Jacobian eigenvalue (Remax) and a nonlinear order parameter (Ψ2), revealing that the two stability boundaries do not coincide and their gap widens monotonically with ϕ. Amplitude-scaling analysis and a quasi-static continuation scan for hysteresis identify a critical penetration rate ϕc≈0.15−0.20 at which the Hopf bifurcation transitions from subcritical to supercritical. Below ϕc, the instability onset is catastrophic and hysteretic, with coexisting attractors and a measurable bistability window; above ϕc, the onset is smooth and reversible. At high penetration, an effective-stability regime is observed in which the system remains linearly unstable yet the nonlinear string-stability order parameter remains below unity within the tested observation window and perturbation protocol. Spatiotemporal kymograph simulations confirm these classifications across four distinct dynamical zones. The results demonstrate that the dominant safety benefit of CAVs lies not in reducing oscillation amplitude but in eliminating the catastrophic collapse regime, and that a purely linear stability framework is insufficient for the design of mixed-traffic systems.