Stability and Hopf bifurcation of a nonlinear model for a four-wheel-steering vehicle system. (Q1428078)

In the past decade, a series of papers has been devoted to modelling the dynamics of a four-wheel steering vehicle-driver system in turning. The authors present a modified model yielding a five-dimensional nonlinear autonomous ordinary differential system for unknowns like the lateral velocity and the yaw angular velocity. The system depends on a number of parameters, such as the forward velocity and several constants characterizing the geometry of the vehicle, the tyre properties, and the steering process. Besides some nontrivial equilibria, this system has the trivial solution corresponding to the uniform motion of the vehicle. The further investigation is restricted to the problem of stability of the trivial solution. The authors prove that only Hopf bifurcation can occur. Choosing the forward velocity as the bifurcation parameter, the conditions for Hopf bifurcation are shown to be satisfied, and finally, some numerically generated stability boundaries for different parameters point out that four-wheel steering has a larger stability region than the two-wheel steering. The numerical results also show that the critical value of the forward velocity may dramatically drop if the mass center of the vehicle moves to the rear.