Parametric resonance, stability and heteroclinic bifurcation in a nonlinear oscillator with time-delay: Application to a quarter-car model

• Parametric forcing in the chaotic regime creates a considerable degree of order compared to when it is absent.
• Increase in time delay lowers the chaotic threshold and enlarges chaotic domain.
• Amplitude response is lower when velocity gain is lower than the position gain.

This work examines dynamical behavior of a nonlinear oscillator with symmetric potential that models a quarter-car forced by the road profile under parametric excitation. The parametric resonance of a harmonically excited nonlinear quarter-car model with position and velocity time-delayed active control are investigated. We focus on the influence of delay and parametric excitation in the system. The influence of parametric excitation, time-delay and feedback gain parameters on the stability of the steady state response are investigated. By means of Melnikov's method, conditions for onset of chaos resulting from heteroclinic bifurcation is derived analytically and numerically.