Dynamics of a delay limit cycle oscillator with self-feedback

This paper concerns the dynamics of the following nonlinear differential-delay equation: ẋ = -x (t-T) - x^3 + α x in which T is the delay and α is a coefficient of self-feedback. Using numerical integration, continuation programs and bifurcation theory, we show that this system exhibits a wide range of dynamical phenomena, including Hopf and pitchfork bifurcations, limit cycle folds and relaxation oscillations.