Orbit quantization in a retarded harmonic oscillator

TL;DR

This paper investigates a damped harmonic oscillator with retarded feedback, revealing complex behaviors like bifurcations, multistability, and strange attractors, and provides analytical and numerical insights into its dynamics.

Contribution

It analytically predicts bifurcation points and uncovers novel phenomena such as superposition limit cycles and multiscale strange attractors in a retarded harmonic oscillator.

Findings

Identification of the first Hopf bifurcation point.

Discovery of coexisting stable limit cycles at different energy levels.

Detection of a multiscale strange attractor with intermittency.

Abstract

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 Li\'enard 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 analyse 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.