In this paper, two classes of techniques, based on multiple scales perturbation analysis and the averaged energy (or Lyapunov function) method, are employed to investigate interesting nonlinear dynamical regimes in a system of coupled Rayleigh-Van der Pol oscillators with time-delayed displacement and velocity feedback. Such systems are currently of topical interest in many applications. The multiple scales analysis is employed to derive a normal form or reduced system for our delayed-coupled oscillators in the vicinity of points of Hopf bifurcation. Detailed analysis of this normal form for various parameter sets reveals periodic dynamics in various post-supercritical Hopf bifurcation regimes, and aperiodic behavior in other post-subcritical Hopf domains. The averaged energy or ‘Lyapunov’ function is constructed next for our oscillators. Although this approach sometimes allows an alternative treatment of the Hopf bifurcation that proves to be difficult in our case. However, parameter regimes where our coupled oscillators may exhibit amplitude death or quenching are comprehensively treated within this second approach.

coupled nonlinear oscillators, delayed displacement and velocity feedback, supercritical and subcritical Hopf bifurcations, amplitude death.