pone.0340370.t003 -

Nonlinear vehicle dynamics are omnipresent and significantly affect driving performance and safety. Jumping vehicles, in particular, exhibit strong nonlinearities that can lead to severe vibrations and steering instabilities. This study investigates the dynamics of nonlinear jumping vehicle models with and without time delay to clarify their fundamental characteristics. Quarter- and half-car models with jumping nonlinearity are considered, and delayed feedback is introduced as an active suspension control. Numerical simulations under periodic and random excitations reveal several key findings as follows. Both models exhibit sudden, discontinuous transitions from periodic to chaotic dynamics in their bifurcation diagrams. A general relationship between the time period of periodic motions and the forcing period is also identified and further validated using a forced Duffing oscillator with time delay, confirming its generic nature. With the introduction of delay, both in-phase and out-of-phase motions emerge between the front and rear, even under simultaneous excitation. From a practical standpoint, in-phase motion is undesirable, making the realization of out-of-phase behavior within a specific delay range important for improving ride performance. This study identifies the stability regions of nonlinear quarter- and half-car models using bifurcation diagrams, Lyapunov exponents, and frequency response curves. In addition, the inclusion of a time delay is shown to effectively stabilize chaotic motions into periodic responses and to induce both in-phase and out-of-phase oscillations. These findings demonstrate that time delay plays a significant role in enhancing the stability of vehicle models with jumping nonlinearities.