Orbit quantization in a retarded harmonic oscillator
We study the dynamics of a damped harmonic oscillator in the presence of a retarded potential with statedependent 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.
The author would like to thank Mattia Coccolo for valuable comments on the elaboration of the present manuscript, the discussion of some of its ideas and the computation of the basins of attraction.
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