Exact Stability Boundaries and Chenciner Bifurcation in a Car-Following Map with State Dependent Delayed Response

At uniform following, reaction delay is characterized by its value and headway sensitivity. We derive a three-dimensional car-following map that separates these quantities. The Jury conditions yield the Schur stability region and a closed-form Neimark–Sacker boundary. Headway sensitivity is absent from the Jacobian but enters the nonlinear terms, making the first Lyapunov coefficient quadratic in this sensitivity within the local function class. For a hyperbolic-tangent benchmark, criticality changes while the linear boundary remains fixed, and a fifth-order normal form identifies a Chenciner point numerically. Continuation locates the associated invariant-curve fold and assesses the local predictor. Basin and time-response computations demonstrate coexistence of stable uniform following and an oscillatory attractor. These results concern one leader–follower pair; disturbance propagation along a platoon requires a multi-vehicle model.