The dynamics of passenger aircraft on the ground are influenced by the nonlinear characteristics of several components, including geometric nonlinearities, aerodynamics, and interactions at the tire-ground interface. We present a fully parameterized mathematical model of a typical passenger aircraft that includes all relevant nonlinear effects. The full equations of motion are derived from first principles in terms of forces and moments acting on a rigid airframe, and they include implementations of the local models of individual components. The overall model has been developed from and validated against an existing industry-tested SIMMECHANICS model. The key advantage of the mathematical model is that it allows for comprehensive studies of solutions and their stability with methods from dynamical systems theory, particularly, the powerful tool of numerical continuation. As a concrete example, we present a bifurcation study of how fixed-radius turning solutions depend on the aircraft’s steering angle and center of gravity position. These results are represented in a compact form as surfaces of solutions, on which we identify regions of stable turning and regions of laterally unstable solutions. The boundaries between these regions are computed directly, and they allow us to determine ranges of parameter values for safe operation. The robustness of these results under the variation in additional parameters, specifically, the engine thrust and aircraft mass, are investigated. Qualitative changes in the structure of the solutions are identified and explained in detail. Overall our results give a complete description of the possible turning dynamics of the aircraft in dependence on four parameters of operational relevance.