Orbit quantization in a retarded harmonic oscillator

We study the dynamics of a damped harmonic oscillator in presence retarded potential with state-dependent time-delayed feedback. In limit small time-delays, we show that is equivalent to Li\'enard system. This allows us analytically predict value first Hopf bifurcation, unleashing self-oscillatory motion. compute bifurcation diagrams for several model parameter values and analyse multistable domains detail. Using Lyapunov energy function, two well-resolved levels represented by coexisting stable cycles are discerned. Further exploration space reveals existence superposition cycle, encompassing degenerate at fundamental level. When system driven very far from equilibrium, multiscale strange attractor displaying intrinsic robust intermittency uncovered.