Orbit quantization in a retarded harmonic oscillator

We study the dynamics of a damped harmonic oscillator in the presence of a retarded potential with state-dependent time-delayed feedback. In the limit of small time-delays, we show that the oscillator is equivalent to a Liénard system. This allows us to analytically predict the value of the first Hopf bifurcation, unleashing a self-oscillatory motion. We compute bifurcation diagrams for several model parameter values and analyze multistable domains in detail. Using the Lyapunov energy function, two well-resolved energy levels represented by two coexisting stable limit cycles are discerned. Further exploration of the parameter space reveals the existence of a superposition limit cycle, encompassing two degenerate coexisting limit cycles at the fundamental energy level. When the system is driven very far from equilibrium, a multiscale strange attractor displaying intrinsic and robust intermittency is uncovered. •A retarded harmonic oscillator with state-dependent delay is studied.•The oscillator is related to a Liénard system proving that the zero point fluctuations arise via Hopf bifurcation.•Multistability domains with two well-resolved quantized orbits are described by means of bifurcation diagrams and the Lyapunov energy function.•When the retarded potential is comparable to the external well, a superposition limit cycle encompassing two degenerate limit cycles appears.•For high time-delays we also find a multiscale strange attractor displaying robust intermittency.